Theory, Numerics and Applications of Hyperbolic Problems I: Aachen, Germany, August 2016 9783319915456, 3319915452
The first of two volumes, this edited proceedings book features research presented at the XVI International Conference o
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English Pages 706 [685] Year 2018
Contents
Proceedings of the 16th International Conference on Hyperbolic Problems: Theory, Numerics and Application
Organizer’s Introduction
Reference
Speakers at the 16th International Conference on Hyperbolic Problems that Contributed to these Proceedings
The Sharp-Interface Limit for the Navier–Stokes–Korteweg Equations
1 Introduction
2 A Priori Estimates and Compactness Properties
3 The Sharp-Interface Limit and Main Theorem
4 Conclusions
References
Asymptotic Behavior of a Solution of Relaxation System for Flow in Porous Media
1 Introduction
2 Physical and Mathematical Models
2.1 Mathematical Analysis
3 Numerical Experiments
4 Concluding Remarks
References
Optimal Control of Level Sets Generated by the Normal Flow Equation
1 Introduction
2 Optimal Control of Level Set Dynamics
3 Approximate Solutions with the Extended Ritz Method
4 Adjoint Equation and Optimization Procedure
5 Simulation Results
6 Conclusions
References
Emergent Dynamics for the Kinetic Kuramoto Equation
1 Introduction
1.1 The Kuramoto Model for N Oscillators
1.2 The Kinetic Kuramoto Equation
2 BV Solutions and the Front-Tracking Algorithm
2.1 Identical Oscillators
2.2 Nonidentical Oscillators
3 Phase Concentration of the Kuramoto–Sakaguchi Equation
3.1 Identical Oscillators
3.2 Nonidentical Oscillators
References
A Hyperbolic Model of Nonequilibrium Phase Change at a Sharp Liquid–Vapor Interface
1 Context
2 Mathematical Model
2.1 Conservation Equations
2.2 Hyperbolicity
2.3 Comparison with Other Models
2.4 Limitation
3 Discretization and Test Cases
3.1 Discretization Strategy
3.2 Numerical Results
4 Conclusion
References
The Cauchy Problem for the Maxwell–Schrödinger System with a Power-Type Nonlinearity
1 Introduction
2 Proof of Theorem1
3 Global Existence
4 Quantum Magnetohydrodynamics
References
Construction and Approximation of the Polyatomic Bitemperature Euler System
1 Introduction
2 The Mathematical Models
2.1 The Bitemperature Euler System
2.2 The Kinetic Model
3 Construction of the Fluid Model
3.1 Scaling on the One Dimensional BGK Model
3.2 Hydrodynamic Limit
4 Numerical Approximation
4.1 Derivation of the Numerical Scheme
4.2 Numerical Results
References
An Implicit–Explicit Scheme Accurate at Low Mach Numbers for the Wave Equation System
1 Introduction
2 Low Mach Number Limit of the Wave Equation
3 An IMEX Scheme Accurate at Low Mach Number
3.1 Time Discretisation
3.2 Space Discretisation
4 L2-Stability Analysis
5 Numerical Test Problems
5.1 Order of Convergence and Accuracy
5.2 Asymptotic Preserving Property
6 Conclusions
References
Bose–Einstein Condensation and Global Dynamics of Solutions to a Hyperbolic Kompaneets Equation
1 Introduction
2 Definitions and Main Results
3 Contraction
4 Regularity and Compactness
5 Proof of the Main Theorem
6 Future Directions
References
Finite Volume Methods for Hyperbolic Partial Differential Equations with Spatial Noise
1 Model Problem
2 Stochastic Entropy Solution
3 Numerical Experiments
3.1 Noise Approximation
3.2 Finite Volume Discretization
3.3 Numerical Experiments
References
A Hyperbolic Approach for Dissipative Magnetohydrodynamics
1 Introduction
2 The Hyperbolic Method
2.1 Steady State: Discretization and Implementation
3 Hyperbolic Method for the 2D MHD Equations
3.1 Steady State: Discretization and Implementation
4 Results on Steady-State Magnetic Reconnection
5 Extension to Time-Dependent Problems
6 Conclusion
References
A General Well-Balanced Finite Volume Scheme for Euler Equations with Gravity
1 Introduction
2 One-Dimensional Euler Equations with Gravity
2.1 Hydrostatic States
3 One-Dimensional Finite Volume Scheme
3.1 Well-Balanced Property
4 Numerical Tests
4.1 One-Dimensional Numerical Results
4.2 Two-Dimensional Numerical Results
References
A Second-Order Well-Balanced Scheme for the Shallow Water Equations with Topography
1 Introduction
2 A Second-Order Accurate Convex Combination
2.1 Time Update of the Well-Balanced Scheme with Respect to the Steady-State Deviation
2.2 Proof of Theorem 1
3 Numerical Experiments
3.1 Order of Accuracy Verification
3.2 Well Balance of the Scheme: Capture of a Steady State
3.3 Dam Break Experiment
4 Conclusion
References
A Lagrangian Approach to Scalar Conservation Laws
1 Introduction
2 Lagrangian Representation and Structure of the Solution
2.1 Lagrangian Representation
2.2 Structure of the Solution
3 Concentration of Entropy Dissipation
References
On Uniqueness of Weak Solutions to Transport Equation with Non-smooth Velocity Field
1 Introduction
2 Preliminaries and Useful Results
2.1 Ambrosio's Superposition Principle
2.2 Level Sets of Lipschitz Functions and Disintegration of Lebesgue Measure
2.3 The Weak Sard Property
3 Proof I: The Local Argument
3.1 Partition and Curves
3.2 Local Disintegration of the Equation div(ub) = µ
3.3 Local Disintegration of a Balance Law
3.4 Matching Lemma and Weak Sard Property for HB
3.5 Level Sets and Trajectories
4 Proof II: The Global Argument
4.1 Covering Property of the Regular Level Sets
4.2 Selection of Appropriate Trajectories
4.3 Conclusion: Solutions Are Constant Along η-a.e. Trajectory
References
Johnson–Segalman–Saint-Venant Equations for a 1D Viscoelastic Shallow Flow in Pure Elastic Limit
1 Setting of the Problem
2 Hyperbolic Structure of the System of Equations
3 Elementary Waves Solutions
3.1 Contact Discontinuities and Shocks
3.2 Rarefaction Waves
4 Solution to the General Riemann Problem
References
On the Exact Dimensional Splitting for a Scalar Quasilinear Hyperbolic Conservation Law
1 Introduction
2 Exact Dimensional Splitting
3 Locally One-Dimensional Compact and Bicompact Schemes
4 Conclusions
References
On the Derivation of Newtonian Gravitation from the Brownian Agitation of a Regular Lattice
1 Introduction
2 The Stochastic Model of a Lattice with Brownian Agitation
3 ``Surfing'' the ``Heat Wave''
4 Zero Noise Limit of the ``Heat Wave'' ODE
5 Large Deviations of the ``Heat Wave'' ODE
6 Γ-Limit of the Freidlin–Vencel Action
7 Application of the Least Action Principle
8 Obtention of the Vlasov–Monge–Ampère System
References
Traffic Flow Models on a Network of Roads
1 Introduction
2 Modeling Traffic Flow at a Road Intersection
3 Intersection Models with Buffers
4 Global Optima and Nash Equilibria
References
Chemotaxis and Haptotaxis on Cellular Level
1 Introduction
2 The Model and the Method
3 Cellular-Level Chemotaxis-Driven Cell Migration
4 Cellular-Level Haptotaxis-Driven Cell Migration
5 Comparison of Chemotaxis and Haptotaxis on Cellular Level
6 Discussion
References
Improved Accuracy of High-Order WENO Finite Volume Methods on Cartesian Grids with Adaptive Mesh Refinement
1 Introduction
1.1 The Dimension-by-Dimension WENO Method
1.2 The Multi-dimensional WENO Reconstruction
2 The Modified WENO-FV Method
3 Rotated Vortex Test Problem
References
Explicit Construction of Effective Flux Functions for Riemann Solutions
1 Introduction
2 The 2times2 System of Conservation Laws
2.1 Terse Review of Fundamental Waves
2.2 The Wave Curve Method
3 The Effective Flux Function Construction
3.1 Construction of the Base Curve Γ(ell)
3.2 EFF Construction: The First Wave is a Shock Wave
3.3 EFF Construction: The First Wave is a Rarefaction Fan
3.4 The Complete EFF Construction
4 Some Examples and Applicability
4.1 Simplified Quadratic Corey Model
4.2 Example 1: Critical Solution Along the Separatrix
4.3 Example 2: Choosing a Parametrization Coordinate
5 Conclusions
References
Fractional Spaces and Conservation Laws
1 Introduction
2 One-sided Condition and Degeneracy of the Flux
3 BVs Smoothing Effect
3.1 BVs Spaces
3.2 The Short Proof of the BVs Smoothing
4 Optimal Smoothing Effect in Sobolev Spaces Ws,1
4.1 Usual Results in W1,1
4.2 The W+s,1(mathbbR) Case
4.3 Optimal Smoothing Effect in Fractional Sobolev Spaces
References
Jacobian-Free Incomplete Riemann Solvers
1 Preliminaries
2 PVM-Type Riemann Solvers: A Review
3 Internal Polynomial Approximations to |x|
4 Jacobian-Free Implementation
5 Numerical Results
5.1 One-Dimensional Test Problems
5.2 Two-Dimensional Test Problems
6 Conclusions
References
A Finite-Volume Tracking Scheme for Two-Phase Compressible Flow
1 Introduction
2 Isothermal Euler Equations
3 An FV Moving Mesh Method with Interface Tracking
4 Numerical Results
4.1 Numerical Results for the One-Dimensional Euler Equations
4.2 Numerical Results for the Two-Dimensional Euler Equations
References
Arbitrary Lagrangian–Eulerian Discontinuous Galerkin Method for 1D Euler Equations
1 Introduction
2 Euler Equations
3 Mesh and Solution Space
4 Discontinuous Galerkin Method
4.1 Mesh Velocity
5 Computing the Predictor
6 Positivity Property
7 Grid Coarsening and Refinement
8 Numerical Results
8.1 Order of Accuracy
8.2 Sod Problem
8.3 Shu–Osher Problem
8.4 Low-Density Problem
8.5 Blast Problem
9 Summary
References
A Runge–Kutta Discontinuous Galerkin Scheme for the Ideal Magnetohydrodynamical Model
1 Introduction
2 Ideal MHD Equations
3 Discontinuous Galerkin Semi-discrete Scheme for the MHD System
4 Numerical Implementation
4.1 Polarized Alfvèn Wave Test Case
4.2 Orszag–Tang Vortex Test Case
5 Conclusions
References
Well-Balanced Central-Upwind Schemes for 2times2 Systems of Balance Laws
1 Introduction
2 Numerical Method
2.1 Second-Order Central-Upwind Scheme
2.2 Well-Balanced Modification
3 Computational Results
4 Conclusion
References
On The Relative Entropy Method For Hyperbolic-Parabolic Systems
1 The Relative Entropy Method
2 The General Framework
3 The Relative Entropy Identity
4 Application in Thermoviscoelasticity
References
A Multispecies Traffic Model Based on the Lighthill-Whitham and Richards Model
1 Introduction
2 Multispecies Extension of the LWR Model
2.1 The Riemann Problem When a Species Is Absent
2.2 Perturbation of the Riemann Problem
2.3 Conclusion
References
Semi-Lagrangian Particle Methods for Hyperbolic Equations
1 Particle Methods for Conservation Laws
1.1 Classical Examples
1.2 Sketch of Numerical Analysis and Overlapping Condition
2 From Grid-Free to Semi-Lagrangian Particles
3 Semi-Lagrangian Particle Methods for Linear Hyperbolic Equations
4 Semi-Lagrangian Particles for Nonlinear Conservation Laws
5 Non-oscillatory Semi-Lagrangian Particles
5.1 The Linear Case
5.2 The Nonlinear Case
6 Conclusion
References
Convergence for PDEs with an Arbitrary Odd Order Spatial Derivative Term
1 Introduction
2 Order of Accuracy for an Initial Datum in mathbbH4p+2(mathbbR)
2.1 Consistency Estimate
2.2 Stability
2.3 Error Estimates
3 Less Smooth Initial Data
3.1 Initial Datum in mathbbHm(mathbbR) with m2p+2
3.2 Initial Datum in mathbbHm(mathbbR) with m0
4 Numerical Results
5 Conclusion
References
A Cell-Centered Lagrangian Method for 2D Ideal MHD Equations
1 Introduction
2 Governing Equations and Their Properties
3 Numerical Schemes
3.1 Semi-discrete Scheme of the 2D Ideal MHD Equations
3.2 Conservation Principles
3.3 Thermodynamic Consistency
3.4 Construction of the Semi-positive Matrix mathbbM"0362Mpc
3.5 The 2D Nodal Solver at p
4 Solution Procedure
5 Numerical Tests
5.1 The 1D Noh's Problem With Magnetic Field
5.2 The Rotor Problem
References
The Riemann Problem for a General Phase Transition Model on Networks
1 Introduction
2 The PT Models
3 The Riemann Problem on a Single Unidirectional Road
4 The Riemann Problem at a Junction
References
Residual Error Indicators for Discontinuous Galerkin Schemes for Discontinuous Solutions to Systems of Conservation Laws
1 Introduction
2 Numerical Method and Reconstruction
2.1 Numerical Scheme
2.2 Limiting
2.3 Reconstruction
3 Numerical Experiments
3.1 Two Rarefaction Wave Problems
3.2 Sod Problem
4 Conclusions
References
Effective Boundary Conditions for Turbulent Compressible Flows over a Riblet Surface
1 Introduction
2 Mathematical Formulation
3 Numerical Results
3.1 Cell Problem
3.2 Effective Problem
4 Conclusions
References
A Deterministic Particle Approximation for Non-linear Conservation Laws
1 Introduction
2 LWR
3 The IBVP for LWR with Dirichlet Boundary Conditions
4 The Hughes Model
5 ARZ
6 Numerical Simulations
References
Splash Singularity for a Free-Boundary Incompressible Viscoelastic Fluid Model
1 Introduction
2 Conformal and Lagrangian Transformations
3 Local Existence of Smooth Solutions
3.1 Iterative Scheme
3.2 Analysis of the System (9)
3.3 The Fixed Point Argument
4 Stability
5 Existence of Splash Singularity (Proof of Theorem 1)
References
An Asymptotic Preserving Mixed Finite Element Method for Wave Propagation in Pipelines
1 Introduction
2 Analysis of a Single Pipe
3 A Mixed Finite Element Runge–Kutta Scheme
4 Extension to Pipe Networks
5 Numerical Validation
References
Non-existence of Irrotational Flow Around Solids with Protruding Corners
1 Equations
2 Protruding Corners
3 Flow Around Bounded Bodies
4 Smooth Infinite Angles
References
A Splitting Approach for Freezing Waves
1 Introduction
2 The Splitting Scheme
2.1 First-Order Scheme
2.2 Second-Order Scheme
3 Numerical Results
References
Metastability for Hyperbolic Variations of Allen–Cahn Equation
1 Introduction
2 Energy Approach
3 Dynamical Approach
4 Concluding Remarks
References
Cell-Centred Lagrangian Lax–Wendroff HLL Hybrid Schemes in Cylindrical Geometry
1 Introduction
2 Governing Equations
3 Numerical Method
3.1 Meshes
3.2 Summary of Method for Cartesian Geometry
3.3 Method for Cylindrical Geometry
3.4 Time Step Control
4 Numerical Results
4.1 Noh Problem in Cartesian Geometry
4.2 Noh Problem in Cylindrical Geometry
4.3 Sedov Problem
4.4 Spherical Sod Problem
5 Conclusions
References
Semilinear Shifted Wave Equation in the de Sitter Spacetime with Hyperbolic Spatial Part
1 Introduction
2 Lp-Lq Estimates
3 Proof of Theorem 1
4 Strichartz Type Estimates
References
Convergence Rates of a Fully Discrete Galerkin Scheme for the Benjamin–Ono Equation
1 Background
2 Analysis of Convergence Rates
2.1 Full Line Problem
2.2 Periodic Problem
3 Numerical Experiments
3.1 Full Line Problem
3.2 Periodic Problem
References
The Simulation of a Tsunami Run-Up Using Multiwavelet-Based Grid Adaptation
1 Introduction
2 Numerical Model
2.1 Discontinuous Galerkin Scheme for Shallow Water Equations
2.2 Review of Multiwavelet-Based Grid Adaptation
3 Initial Adaptive Grid: Projection of Initial Data and Bottom Topography
4 Results—Tsunami Run-Up onto a Complex Three-Dimensional Beach
References
Constrained Reconstruction in MUSCL-Type Finite Volume Schemes
1 Introduction
2 Stabilization of MUSCL-Type Finite Volume Schemes
3 Constrained Linear Reconstruction
4 Numerical Results
4.1 Nonlinear Problem Admitting a Smooth Solution
4.2 Linear Problem
4.3 Euler Equations of Gas Dynamics
5 Conclusion
References
A Posteriori Analysis for the Euler–Korteweg Model
1 Introduction
2 The Euler–Korteweg (EK) System
3 Reduced Relative Entropy
4 Discretization and A Posteriori Setup
4.1 Mixed Formulation
4.2 Discrete Scheme
4.3 A Posteriori Setup
4.4 A Posteriori Error Estimate
References
Conservation Laws Arising in the Study of Forward–Forward Mean-Field Games
1 Introduction
2 First-Order, One-Dimensional, Forward–Forward Mean-Field Games as Nonlinear Wave Equations
3 One-Dimensional Forward–Forward Mean-Field Games as Conservation Laws
3.1 Hyperbolicity and Genuine Nonlinearity
3.2 Riemann Invariants and Invariant Domains
4 Convergence of One-Dimensional, Forward–Forward, Parabolic Conservation Laws
References
On the Relaxation Approximation for 2times2 Hyperbolic Balance Laws
1 Introduction
2 Natural Relaxation Approximation for Hyperbolic Balance Laws
3 Uniform Exponential Decay
4 Application to the Isothermal Euler Equations
4.1 Horizontal Pipe with Nonzero Gas Velocity
4.2 Constant Stationary States
5 Conclusion
References
Numerical Solutions for a Weakly Hyperbolic Dispersed Two-Phase Flow Model
1 Introduction
2 The Two-Phase Flow Model
2.1 Equations of State
2.2 Exact Solutions for the Riemann Problem
3 Numerical Method
4 Numerical Results
5 Summary and Conclusions
References
Optimal Controls in Flux, Source, and Initial Terms for Weakly Coupled Hyperbolic Systems
1 Introduction
2 The Initial Value Problem for Weakly Coupled Hyperbolic Systems
3 Classical Solutions of the Weakly Coupled Parabolic Initial Value Problem
3.1 The Weakly Coupled Viscous Initial Value Problem
3.2 The Basic Existence and Uniqueness Result
4 Weak Solutions and Entropy Solutions
4.1 Globally Limited Growth
5 Total Variation and Stability
6 A Class of Optimal Control Problems
6.1 Assumptions
6.2 Existence of Optimal Controls
References
On Convergence of Numerical Methods for Optimization Problems Governed by Scalar Hyperbolic Conservation Laws
1 Introduction
2 Numerical Method
2.1 Iterative Algorithm
2.2 Numerical Schemes
3 Convergence Analysis
4 Numerical Results
References
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Springer Proceedings in Mathematics & Statistics
Christian Klingenberg Michael Westdickenberg Editors
Theory, Numerics and Applications of Hyperbolic Problems I Aachen, Germany, August 2016
Springer Proceedings in Mathematics & Statistics Volume 236
Springer Proceedings in Mathematics & Statistics This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.
More information about this series at http://www.springer.com/series/10533
Christian Klingenberg Michael Westdickenberg •
Editors
Theory, Numerics and Applications of Hyperbolic Problems I Aachen, Germany, August 2016
123
Editors Christian Klingenberg Department of Mathematics Würzburg University Würzburg Germany
Michael Westdickenberg Department of Mathematics RWTH Aachen University Aachen Germany
ISSN 2194-1009 ISSN 2194-1017 (electronic) Springer Proceedings in Mathematics & Statistics ISBN 978-3-319-91544-9 ISBN 978-3-319-91545-6 (eBook) https://doi.org/10.1007/978-3-319-91545-6 Library of Congress Control Number: 2018941540 Mathematics Subject Classification (2010): 35Lxx, 35M10, 35Q30, 35Q35, 35Q60, 35Q72, 35R35, 65Mxx, 65Nxx, 65Txx, 65Yxx, 65Z05, 74B20, 74Jxx, 76L06, 76Rxx, 76Txx, 80A32, 80Mxx, 83C55, 83F05 © Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer International Publishing AG part of Springer Nature The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Contents
The Sharp-Interface Limit for the Navier–Stokes–Korteweg Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Helmut Abels, Johannes Daube, Christiane Kraus and Dietmar Kröner
1
Asymptotic Behavior of a Solution of Relaxation System for Flow in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Abreu, A. Bustos and W. J. Lambert
15
Optimal Control of Level Sets Generated by the Normal Flow Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Angelo Alessandri, Patrizia Bagnerini, Roberto Cianci and Mauro Gaggero
29
Emergent Dynamics for the Kinetic Kuramoto Equation . . . . . . . . . . . . Debora Amadori and Jinyeong Park
43
A Hyperbolic Model of Nonequilibrium Phase Change at a Sharp Liquid–Vapor Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matthieu Ancellin, Laurent Brosset and Jean-Michel Ghidaglia
59
The Cauchy Problem for the Maxwell–Schrödinger System with a Power-Type Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Paolo Antonelli, Michele D’Amico and Pierangelo Marcati
71
Construction and Approximation of the Polyatomic Bitemperature Euler System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Denise Aregba-Driollet and Stéphane Brull
85
An Implicit–Explicit Scheme Accurate at Low Mach Numbers for the Wave Equation System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. R. Arun, A. J. Das Gupta and S. Samantaray
97
Bose–Einstein Condensation and Global Dynamics of Solutions to a Hyperbolic Kompaneets Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Joshua Ballew
v
vi
Contents
Finite Volume Methods for Hyperbolic Partial Differential Equations with Spatial Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Andrea Barth and Ilja Kröker A Hyperbolic Approach for Dissipative Magnetohydrodynamics . . . . . . 137 Hubert Baty and Hiroaki Nishikawa A General Well-Balanced Finite Volume Scheme for Euler Equations with Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Jonas P. Berberich, Praveen Chandrashekar and Christian Klingenberg A Second-Order Well-Balanced Scheme for the Shallow Water Equations with Topography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Christophe Berthon, Raphaël Loubère and Victor Michel-Dansac A Lagrangian Approach to Scalar Conservation Laws . . . . . . . . . . . . . 179 Stefano Bianchini and Elio Marconi On Uniqueness of Weak Solutions to Transport Equation with Non-smooth Velocity Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Paolo Bonicatto Johnson–Segalman–Saint-Venant Equations for a 1D Viscoelastic Shallow Flow in Pure Elastic Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Sébastien Boyaval On the Exact Dimensional Splitting for a Scalar Quasilinear Hyperbolic Conservation Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Michael D. Bragin and Boris V. Rogov On the Derivation of Newtonian Gravitation from the Brownian Agitation of a Regular Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Yann Brenier Traffic Flow Models on a Network of Roads . . . . . . . . . . . . . . . . . . . . . 237 Alberto Bressan Chemotaxis and Haptotaxis on Cellular Level . . . . . . . . . . . . . . . . . . . . 249 A. Brunk, N. Kolbe and N. Sfakianakis Improved Accuracy of High-Order WENO Finite Volume Methods on Cartesian Grids with Adaptive Mesh Refinement . . . . . . . . . . . . . . . 263 Pawel Buchmüller, Jürgen Dreher and Christiane Helzel Explicit Construction of Effective Flux Functions for Riemann Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Pablo Castañeda Fractional Spaces and Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . 285 Pierre Castelli, Pierre-Emmanuel Jabin and Stéphane Junca
Contents
vii
Jacobian-Free Incomplete Riemann Solvers . . . . . . . . . . . . . . . . . . . . . . 295 Manuel J. Castro, José M. Gallardo and Antonio Marquina A Finite-Volume Tracking Scheme for Two-Phase Compressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 Christophe Chalons, Jim Magiera, Christian Rohde and Maria Wiebe Arbitrary Lagrangian–Eulerian Discontinuous Galerkin Method for 1D Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 Jayesh Badwaik and Praveen Chandrashekar A Runge–Kutta Discontinuous Galerkin Scheme for the Ideal Magnetohydrodynamical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 Praveen Chandrashekar, Juan Pablo Gallego-Valencia and Christian Klingenberg Well-Balanced Central-Upwind Schemes for 2 2 Systems of Balance Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 Alina Chertock, Michael Herty and Şeyma Nur Özcan On The Relative Entropy Method For Hyperbolic-Parabolic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 Cleopatra Christoforou and Athanasios Tzavaras A Multispecies Traffic Model Based on the Lighthill-Whitham and Richards Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 Rinaldo M. Colombo, Christian Klingenberg and Marie-Christine Meltzer Semi-Lagrangian Particle Methods for Hyperbolic Equations . . . . . . . . 395 Georges-Henri Cottet Convergence for PDEs with an Arbitrary Odd Order Spatial Derivative Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 Clémentine Courtès A Cell-Centered Lagrangian Method for 2D Ideal MHD Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 Zihuan Dai The Riemann Problem for a General Phase Transition Model on Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 Edda Dal Santo, Massimiliano D. Rosini and Nikodem Dymski Residual Error Indicators for Discontinuous Galerkin Schemes for Discontinuous Solutions to Systems of Conservation Laws . . . . . . . . 459 Andreas Dedner and Jan Giesselmann
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Contents
Effective Boundary Conditions for Turbulent Compressible Flows over a Riblet Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 G. Deolmi, W. Dahmen, S. Müller, M. Albers, P. S. Meysonnat and W. Schröder A Deterministic Particle Approximation for Non-linear Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 Marco Di Francesco, Simone Fagioli, Massimiliano D. Rosini and Giovanni Russo Splash Singularity for a Free-Boundary Incompressible Viscoelastic Fluid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 Elena Di Iorio, Pierangelo Marcati and Stefano Spirito An Asymptotic Preserving Mixed Finite Element Method for Wave Propagation in Pipelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 Herbert Egger and Thomas Kugler Non-existence of Irrotational Flow Around Solids with Protruding Corners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 Volker Elling A Splitting Approach for Freezing Waves . . . . . . . . . . . . . . . . . . . . . . . 539 Robin Flohr and Jens Rottmann-Matthes Metastability for Hyperbolic Variations of Allen–Cahn Equation . . . . . 551 Raffaele Folino Cell-Centred Lagrangian Lax–Wendroff HLL Hybrid Schemes in Cylindrical Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565 David Fridrich, Richard Liska and Burton Wendroff Semilinear Shifted Wave Equation in the de Sitter Spacetime with Hyperbolic Spatial Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 Anahit Galstian Convergence Rates of a Fully Discrete Galerkin Scheme for the Benjamin–Ono Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 Sondre Tesdal Galtung The Simulation of a Tsunami Run-Up Using Multiwavelet-Based Grid Adaptation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 Nils Gerhard and Siegfried Müller Constrained Reconstruction in MUSCL-Type Finite Volume Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 Christoph Gersbacher and Martin Nolte A Posteriori Analysis for the Euler–Korteweg Model . . . . . . . . . . . . . . . 631 Jan Giesselmann and Dimitrios Zacharenakis
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ix
Conservation Laws Arising in the Study of Forward–Forward Mean-Field Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 Diogo Gomes, Levon Nurbekyan and Marc Sedjro On the Relaxation Approximation for 2 2 Hyperbolic Balance Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 Martin Gugat, Michael Herty and Hui Yu Numerical Solutions for a Weakly Hyperbolic Dispersed Two-Phase Flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 Maren Hantke, Christoph Matern and Gerald Warnecke Optimal Controls in Flux, Source, and Initial Terms for Weakly Coupled Hyperbolic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677 Maryse Hawerkamp, Dietmar Kröner and Hanna Moenius On Convergence of Numerical Methods for Optimization Problems Governed by Scalar Hyperbolic Conservation Laws . . . . . . . . . . . . . . . 691 Michael Herty, Alexander Kurganov and Dmitry Kurochkin
Proceedings of the 16th International Conference on Hyperbolic Problems: Theory, Numerics and Application Organizers: Christian Klingenberg and Michael Westdickenberg
Organizer’s Introduction This series of bi-yearly conferences began in 1986 and celebrated its 30th anniversary with this conference. From its very beginning, the conference set out to bring together researchers studying theoretical issues, numerical methods and applications in hyperbolic partial differential equations. Initially, this pursuit was under the influence of Glimm’s major result [1], where the convergence of a useful numerical method (in one space dimensions) gave rise to an existence proof. Even though 30 years later, the areas of theory, numerics and applications are no longer as closely intertwined as they were in the beginning, the organizers feel that having them together in one conference today is much more than historical nostalgia. One area may give impulses to another area. Given that fundamental issues in the field of hyperbolic problems are open (e.g. is there an admissibility condition that gives rise to well-posedness for the Euler equations in multiple space dimensions?), new impulses from different areas are thoroughly needed. This conference and these proceedings provide a snapshot of the activity in its field at the time of this conference. The field is quite broad, as seen by a partial list of subjects covered: • • • • • •
hyperbolic conservation laws wave equations partial differential equations of mixed type kinetic equations theoretic questions and numerical schemes for all of the above applications in physics and engineering using all of the above
This conference over the last 30 years has developed into one of the main conferences in applied mathematics. This is due to the vigour of the field, the enormously challenging questions that still lie ahead and its extreme usefulness in applications. Many researches around the world contribute to this field, so we expect that this series of conferences will be vital for many years to come.
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Proceedings of the 16th International Conference …
The organizers want to acknowledge the financial support of the German Science Foundation (DFG). The organizers wish to thank all participants, the local staff and organizers, and everybody else who was involved in this event in one way or another, for making the 2016 edition of the “International Conference on Hyperbolic Problems: Theory, Numerics, and Applications” a success.
Reference 1. J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math. 18(4), 697–715 (1965)
Speakers at the 16th International Conference on Hyperbolic Problems that Contributed to these Proceedings
Abreu, Eduardo Amadori, Debora Ancellin, Matthieu Bagnerini, Patrizia Ballew, Joshua Baskar, S. Baty, Hubert Berberich, Jonas Bonicatto, Paolo Boyaval, Sebastien Bragin, Michael Brenier, Yann Bressan, Alberto Brull, Stephane Castaneda, Pablo Chandrashekar, Praveen Chertok, Alina Christoforou, Cleopatra Colombo, Rinaldo Cottet, Georges-Henri Courtes, Clementine Dai, Zihuan Dal Santo, Edda D’Amico, Michele Das Gupta, Arnab Jyoti Daube, Johannes Deolmi, Giulia Di Iorio, Elena Egger, Herbert Elling, Volker Flohr, Robin
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Speakers at the 16th International Conference …
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Folino, Raffaele Fridrich, David Gallardo, Jose M. Gallego-Valencia, Juan Pablo Galstian, Anahit Galtung, Sondre Tesdal Gerhard, Nils Gersbacher, Christoph Giesselmann, Jan Gugat, Martin Hantke, Maren Helzel, Christiane Herty, Michael Hunter, John K. Jaust, Alexander Jiang, Nan Jin, Shi Junca, Stephane Kabil, Bugra Karite, Touria Kausar, Rukhsana Klima, Matej Klingenberg, Christian Klotzky, Jens Koellermeier, Julian Korsch, Andrea Kränkel, Mirko Kröker, Ilja Kröner, Dietmar Lambert, Wanderson J. Lee, Min-Gi LeFloch, Philippe G. Li, Qin Liu, Hailiang Magiera, Jim Marconi, Elio Michel-Dansac, Victor Mifsud, Clement Modena, Stefano Nguyen, Thinh T. Öffner, Philipp Ohnawa, Masashi Panov, Evgeny Yu. Pareschi, Lorenzo Pelanti, Marica
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Speakers at the 16th International Conference …
Peralta, Gilbert Peshkov, Ilya Pichard, Teddy Pirner, Marlies Prebeg, Marin Ranocha, Hendrik Ray, Deep Roe, Philip Röpke, Friedrich Rosini, Massimiliano Rozanova, Olga S. Sahu, Smita Schnücke, Gero Sedjro, Marc Seguin, Nicolas Sfakianakis, Nikolaos Shu, Chi-Wang Sikstel, Aleksey Straub, Veronika Tang, Tao Ueda, Yoshihiro Wang, Tian-Yi Weber, Frankziska Wiebe, Maria Yagdjian, Karen Zacharenakis, Dimitrios Zakerzadeh, Hamed Zakerzadeh, Mohammad Zumbrun, Kevin
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The Sharp-Interface Limit for the Navier–Stokes–Korteweg Equations Helmut Abels, Johannes Daube, Christiane Kraus and Dietmar Kröner
Abstract We investigate the sharp-interface limit for the Navier–Stokes–Korteweg model, which is an extension of the compressible Navier–Stokes equations. By means of compactness arguments, we show that solutions of the Navier–Stokes–Korteweg equations converge to solutions of a physically meaningful free-boundary problem. Assuming that an associated energy functional converges in a suitable sense, we obtain the sharp-interface limit at the level of weak solutions. Keywords Sharp-interface limit · Diffuse-interface model · Liquid–vapour flow Navier–Stokes–Korteweg system · Free-boundary problem MSC (2010) 35B40 · 76T10 · 35Q35 · 35R35
H. Abels (B) Fakultät für Mathematik, Universität Regensburg, Universitätsstr. 31, 93053 Regensburg, Germany e-mail: [email protected] J. Daube (B) · D. Kröner Abteilung für Angewandte Mathematik, Universität Freiburg, Hermann-Herder-Str. 10, 79104 Freiburg, Germany e-mail: [email protected] D. Kröner e-mail: [email protected] C. Kraus Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, Germany e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_1
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H. Abels et al.
1 Introduction Models describing liquid–vapour flow are basically classified into two different types: sharp- and diffuse-interface models. They differ in how the interface dividing liquid from vapour is represented. In diffuse-interface models, an additional order parameter (here, the density) is introduced, such that the interface is described as follows: the “sharp interface” is replaced by an interfacial layer of positive thickness ε, where the order parameter varies rapidly but smoothly between two values distinguishing the liquid and the vapour phase. The sharp-interface limit encodes the behaviour of diffuse-interface models and their corresponding solutions, as ε tends to zero. We shall investigate the sharp-interface limit of the following “phase-field-like scaling” of the Navier–Stokes–Korteweg equations [13, 15]. In a bounded domain Ω ⊂ Rn , n = 2, 3, with C 2 -boundary ∂Ω and outer unit normal ν, we consider, for the unknowns density ρε and velocity vε , the partial differential equations ∂t ρε + div(ρε vε ) = 0, ∂t (ρε vε ) + div(ρε vε ⊗ vε ) +
1 ∇ p(ρε ) ε
(1)
= 2 div(μ(ρε )Dvε ) + ερε ∇ρε ,
(2)
depending on a parameter ε ∈ (0, 1), in the space-time cylinder Ω × (0, T ) with T ∈ (0, ∞). In (2), D stands for the symmetric part of the gradient. We close the system by adding the boundary and initial conditions ∇ρε · ν = 0 vε = 0
on ∂Ω × [0, T ), on ∂Ω × [0, T ),
(3) (4)
ρε (·, 0) = ρε(i)
in Ω,
(5)
vε(i)
in Ω.
(6)
vε (·, 0) =
The non-monotone pressure function p = p(ρ) is given by the relation p (ρ) = ρW (ρ), where W ∈ C 2 ([0, ∞)) is a non-negative double-well potential, such that W (z) = 0 if and only if z ∈ {β1 , β2 }, and W (z) ≥ C1 |z − a| p∗ −2 for all z ∈ [0, ∞) with |z − a| ≥ b − C2 ,
(7)
2 1 and b = β2 −β , for some constants C1 > 0, C2 ∈ (0, b) and p∗ > 2. where a = β1 +β 2 2 As a direct consequence of (7), there exist constants C1 , C2 > 0, such that
W (z) ≥ C1 |z − a| p∗ − C2 and (|z − a| − b)2 ≤ C1 W (z) for all z ∈ [0, ∞). (8) The viscosity function μ : [0, ∞) → [cμ , Cμ ], 0 < cμ ≤ Cμ , is Lipschitz continuous. For well-posedness results for (1)–(6) and related models see [3–5, 7, 11, 12, 16, 17]. Our work summarizes the results of [8, Chaps. 3 and 5], where detailed proofs are given, and extends [13], where the static case of (1)–(6) is treated, to the dynamic
The Sharp-Interface Limit for the Navier–Stokes–Korteweg Equations
3
case. We study (weak) solutions to (1)–(6) as ε tends to zero, and seek to extract subsequences of (ρε , vε )ε∈(0,1) converging to solutions (ρ0 , v0 ) of an appropriate sharpinterface model. We prove that (ρ0 , v0 ) is a (weak) solution of the two-phase Navier– Stokes equations with surface tension [14]: the free-boundary problem describing the motion of the vapour phase, of constant density β1 , and the liquid phase, of constant density β2 , of an isothermal, viscous, incompressible Newtonian fluid. For each time t ∈ [0, T ], a hypersurface Γ (t) separates Ω into two disjoint open subsets Ω − (t) and Ω + (t) of Ω, i.e. we have Ω = Ω − (t) ∪ Γ (t) ∪ Ω + (t) and Γ (t) = ∂Ω − (t) ∩ Ω. The unknowns are the free boundary Γ (t), the velocity field v(·, t) : Ω \ Γ (t) → Rn and the pressure function p(·, t) : Ω \ Γ (t) → R. The sharp-interface model then reads as in Ω − (t), t ∈ [0, T ],
β1 ∂t v + β1 (v · ∇)v − μ(β1 )v + ∇ p = 0 β2 ∂t v + β2 (v · ∇)v − μ(β2 )v + ∇ p = 0
in Ω (t), t ∈ [0, T ],
(10) in Ω \ Γ (t), t ∈ [0, T ], (11) on Γ (t), t ∈ [0, T ], (12)
div(v) = 0 [v] = 0 V = v · ν− −
(9)
+
[T ] ν = −2σst κν
−
on Γ (t), t ∈ [0, T ],
(13)
on Γ (t), t ∈ [0, T ].
(14)
The stress tensor T is given by T (v(t), p(t)) = 2μ(β1 )Dv(t) − p(t)I in Ω − (t), and by 2μ(β2 )Dv(t) − p(t)I in Ω + (t). For a given quantity f , [ f ] denotes the jump across Γ (t) in the direction of the exterior unit-normal field ν − (·, t) of ∂Ω − (t) + (and pointinginto Ω (t)); that is, [ f ](x, t) = limξ 0 f (x + ξ ν − (x, t), t) − f (x − ξ ν − (x, t), t) for x ∈ Γ (t), and V and κ are the normal velocity and the mean curvature of Γ , both taken with respect to ν − . Moreover, σst is the surface-tension constant given by σst =
β2
β1
min 21 W (z), |z − a|2 + b2 dz.
We close the system by the boundary and initial conditions v(·, t) = 0 −
Ω (0) = Ω v(·, 0) = v
on ∂Ω, t ∈ [0, T ], −,(i)
(i)
,
(15) (16)
in Ω,
(17)
where v(i) and Ω −,(i) are prescribed data satisfying Ω −,(i) ∩ ∂Ω = ∅. Notation and Preliminaries Let U ⊂ Rd , d ∈ N, be open or closed. The space of ∞ (U ) is the smooth and compactly supported functions in U is denoted by C0∞ (U ), C0,σ ∞ subspace of C0 (U ) of divergence-free functions and C0 (U ) is the closure of C0∞ (U ) ∞ (Q) = with respect to the supremum norm. Moreover, for Q ⊂ Rd , we define C(0) {u : Q → R : u = U | Q , U ∈ C0∞ (Rd ), supp(u) ⊂ Q}. For a measurable set M ⊂
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H. Abels et al.
Rd and r ∈ [1, ∞], L r (M) and L r (M; X ) denote the standard Lebesgue spaces of scalar and X -valued functions, respectively. W k,r (U ) is the Sobolev space of order k ∈ N and integrability exponent r . By W0k,r (U ), we denote the closure of C0∞ (U ) in W k,r (U ), and we set H k (U ) = W k,2 (U ) and H0k (U ) = W0k,2 (U ). Furthermore, 1 ∞ (U ) denote the closure of C0,σ (U ) in L 2 (U ) and H 1 (U ), respectively. L 2σ (U ) and H0,σ For a Banach space Y and α ∈ (0, 1), the space C 0 ([0, T ]; Y ) contains all continuous functions f : [0, T ] → Y and the Hölder space C 0,α ([0, T ]; Y ) is the subspace of all f ∈ C 0 ([0, T ]; Y ) with finite norm f C 0,α ([0,T ];Y ) = sup f (t)Y + t∈[0,T ]
sup
0≤t1 0. We introduce the transformation s min 21 W (z), |z − a|2 + b2 dz for s ∈ [0, ∞). Φ(s) = a
By standard arguments [6, p. 276], (rε )ε∈(0,1) , defined by rε = Φ ◦ ρε , is bounded in L ∞ (0, T ; BV (Ω)). For a standard mollifying kernel Θ and sufficiently small η > 0, let Θ(y)ρε (x − ηy, t) dy for (x, t) ∈ Ω × [0, T ], ρε,η (x, t) = B1 (0)
The Sharp-Interface Limit for the Navier–Stokes–Korteweg Equations
7
where ρε is extended to a small neighbourhood of Ω as in [6, Proof of Lemma 3.2]. Proceeding analogously to [6, Eqs. (3.2)–(3.4)], there exists a constant C > 0, such that, for t ∈ [0, T ] and sufficiently small η > 0, 2 √ ρε,η (t) − ρε (t) L 2 (Ω) ≤ C η and ∇ρε,η (t) L 3 (Ω)n ≤ Cη−( 3 n+1) . 1
(23)
1
Then, (ρε )ε∈(0,1) ⊂ C 0, 28 ([0, T ]; L 2 (Ω)) and (rε )ε∈(0,1) ⊂ C 0, 28 ([0, T ]; L 1 (Ω)) are bounded, by the arguments of Chen [6, Lemma 3.2]; see [8, Theorem 3.3.11] for details. For the convenience of the reader, we briefly ensure the existence of a constant C > 0, such that, for all t1 , t2 ∈ [0, T ] with t1 < t2 and |t2 − t1 | sufficiently small,
1
Ω
|ρε (t2 ) − ρε (t1 )|2 dx ≤ C |t2 − t1 | 14 .
(24)
Due to (22) and H 1 (Ω) → L 6 (Ω), (ρε )ε∈(0,1) and (vε )ε∈(0,1) are bounded in L ∞ (0, T ; L 2 (Ω)) and L 2 (0, T ; L 6 (Ω)n ), respectively. From (19), we obtain
t2
ρε (t2 ) − ρε (t1 ) =
t2
∂t ρε (t) dt = −
t1
div(ρε vε )(t) dt in W 1,3 (Ω)∗ .
t1
Hence, recalling that vε ∈ L 2 (0, T ; H01 (Ω)n ), and using (23), we infer (ρε (t2 ) − ρε (t1 ))(ρε,η (t2 ) − ρε,η (t1 )) dx t2 div(ρε vε )(t), ρε,η (t2 ) − ρε,η (t1 ) W 1,3 (Ω) dt = − Ω
t1 t2
= t1
Ω
≤ ρε vε
(ρε vε )(t) · (∇ρε,η (t2 ) − ∇ρε,η (t1 )) dx dt 3
L 2 (t1 ,t2 ;L 2 (Ω)n )
∇ρε,η (t2 ) − ∇ρε,η (t1 ) L 2 (t1 ,t2 ;L 3 (Ω)n ) 1
≤ 2ρε L ∞ (0,T ;L 2 (Ω)) vε L 2 (0,T ;L 6 (Ω)n ) sup ∇ρε,η (t) L 3 (Ω)n |t2 − t1 | 2 t∈[0,T ]
≤ Cη
−( 23 n+1)
1 2
|t2 − t1 | .
Using ρε (t2 ) − ρε (t1 ) = ρε (t2 ) ∓ ρε,η (t2 ) ∓ ρε,η (t1 ) − ρε (t1 ) and Hölder’s inequality, in view of (23), there exists a constant C > 0, such that, for sufficiently small η > 0,
√ 2 1 |ρε (t2 ) − ρε (t1 )|2 dx ≤ C η + η−( 3 n+1) |t2 − t1 | 2 . Ω
1
Since n ∈ {2, 3}, the choice η = |t2 − t1 | 7 implies (24) for |t2 − t1 | sufficiently small.
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H. Abels et al.
Compactness Throughout this paper, we will not relabel subsequences. As a direct consequence of (22), there exists v0 ∈ L 2 (0, T ; H01 (Ω)n ), such that, after passing to a subsequence, vε v0 in L 2 (0, T ; H 1 (Ω)n ) → L 2 (0, T ; L 6 (Ω)n ). We will use the following adaptation of Simon [19, Theorem 3]. Lemma 1. Let 0 < α < β and let X, Y be Banach spaces, such that X →→ Y . Let ( f k )k∈N ⊂ C 0,β ([0, T ]; Y ) be bounded and let f ∈ C 0 ([0, T ]; Y ). If f k → f in C 0 ([0, T ]; Y ) for k → ∞, then f ∈ C 0,β ([0, T ]; Y ), and f k → f in C 0,α ([0, T ]; Y ) as k → ∞. Moreover, L ∞ (0, T ; X ) ∩ C 0,β ([0, T ]; Y ) →→ C 0,α ([0, T ]; Y ). By applying Lemma 1 to (rε )ε∈(0,1) , there exist a subsequence (rε j ) j∈N and r0 ∈ 1 L ∞ (0, T ; BV (Ω)) ∩ C 0, 28 ([0, T ]; L 1 (Ω)), such that there holds 1
rε j → r0 in C 0, 29 ([0, T ]; L 1 (Ω))
(25)
and |∇r0 (t)| (Ω) ≤ lim inf j→∞ ∇rε j (t) (Ω) for every t ∈ [0, T ]. Using the properties of the transformation Φ and Lemma 1, (25) implies that ρ0 = 1 1 Φ −1 ◦ r0 ∈ C 0, 28 ([0, T ]; L 2 (Ω)) and ρε → ρ0 in C 0, 29 ([0, T ]; L 2 (Ω)). In particu2 lar, for any t ∈ [0, T ], ρε (t) → ρ0 (t) in L (Ω), hence, (22) implies ρε (t) ρ0 (t) in L p∗ (Ω). Finally, by interpolation, for any q ∈ [1, p∗ ), it follows ρε (t) → ρ0 (t) in L q (Ω). Note that, by the preceding results, for every t ∈ [0, T ], it follows that ρ0 (t) ∈ BV (Ω, {β1 , β2 }) and ρ0 (t) = (β1 − β2 )χ0 + β2 = (β1 − β2 )χΩ − (t) + β2 a.e. in Ω, where χ0 =
ρ0 −β2 β1 −β2
(26)
and Ω − (t) is the set of finite perimeter in Ω given by
Ω − (t) = x ∈ Ω : lim
1 δ→0 |Bδ (x)|
Bδ (x)
χ0 (y, t) dy = 1 .
(27)
We call Ω − (·) the measure-theoretic representative set of ρ0 . In this way, ρ0 (t) induces the disjoint partition Ω = Ω − (t) ∪ Γ (t) ∪ Ω + (t), where the (sharp) interface Γ (t) and Ω + (t) are, respectively, defined by Γ (t) = ∂ ∗ Ω − (t) ∩ Ω and Ω + (t) = Ω \ (Ω − (t) ∪ Γ (t)). Note that, in view of the generalized Gauß–Green theorem, the generalized measuretheoretic outer normal ν − (t) exists on Γ (t).
3 The Sharp-Interface Limit and Main Theorem We investigate the sharp-interface limit for weak solutions of (1)–(6) along suitable subsequences under the additional assumptions on the energy functionals E ε and E εtot
The Sharp-Interface Limit for the Navier–Stokes–Korteweg Equations
9
given below in (28) and (29). To simplify notation, we assume that any convergence property of (ρε , vε )ε∈(0,1) holds true for the entire sequence and not only for an appropriate subsequence. Assumptions To identify the limit of E εtot , suppose that, for any t ∈ [0, T ], Γ (t) is compactly contained in Ω, and that, for every ϕ ∈ L 1 (0, T ; C0 (Ω)),
T 0
1
W (ρε ) + 2ε |∇ρε |2 ϕ dx dt → 2σst ε
Ω
T
0
Γ (t)
ϕ dH n−1 (x) dt.
(28)
Additionally, assume the following asymptotic behaviour of the kinetic part of E εtot : √ √ ρε vε → ρ0 v0 in L 2 (0, T ; L 2 (Ω)n ).
(29)
Using the reasoning of [18, Lemmata 1 and 2], (28) implies the so-called equipartitionof-energy property T
0
Ω
1
W (ρε ) − 1 ε |∇ρε |2 dx dt → 0, ε 2
(30)
∞ and, moreover, for any ψ ∈ C0∞ ([0, T ); C0,σ (Ω)), there holds
ε 0
T
Ω
∇ρε ⊗ ∇ρε : ∇ψ dx dt → 2σst 0
T Γ (t)
ν − ⊗ ν − : ∇ψ dH n−1 (x) dt.
After passing to a suitable subsequence, in view of (21) and (29), we obtain
2 ∗
ρε jm vε jm dx ρ0 |v0 |2 dx in L ∞ (0, T ) ∼ = L 1 (0, T )∗ . Ω
(31) (32)
Ω
Weak Formulation First, we introduce a weak formulation of (9)–(17). For its derivation and justification, we refer to [8, Chap. 4]. For the prescribed data in (16) 1 and (17), we assume that v(i) ∈ H0,σ (Ω) and Ω −,(i) ⊂⊂ Ω with corresponding char(i) acteristic function χ = χΩ − (0) . Moreover, let ρ (i) ∈ BV (Ω, {β1 , β2 }) given by ρ (i) = (β1 − β2 )χ (i) + β2 . Definition 2. A pair (ρ, v) ∈ L ∞ (0, T ; BV (Ω, {β1 , β2 })) × L ∞ (0, T ; L 2σ (Ω)) ∩ L 2 (0, T ; H01 (Ω)n ) is called a weak solution of (9)–(17) if the following conditions are fulfilled. 1. The measure-theoretic representative set Ω − (t) of ρ(t), cf. (27), is compactly contained in Ω; that is, for a.e. t ∈ (0, T ), there holds Ω − (t) ⊂⊂ Ω. ∞ (Ω)), there holds 2. For each ψ ∈ C0∞ ([0, T ); C0,σ
10
H. Abels et al.
T
0
= −
ρv · ∂t ψ + ρv ⊗ v : ∇ψ − 2μ(ρ)Dv : Dψ dx dt T (i) (i) ρ v · ψ(0) dx − 2σst ν − ⊗ ν − : ∇ψ dH n−1 (x) dt,
Ω
Ω
Γ (t)
0
where Γ (t) = ∂ ∗ (Ω − (t)) with generalized outer unit normal ν − (t). 3. For a.e. τ1 ∈ [0, T ), including τ1 = 0, there holds, for all τ2 ∈ [τ1 , T ),
ρ(τ2 ) |v(τ2 )|2 dx + 2 Ω n−1 ≤ 2σst H (Γ (τ1 )) + 21 ρ(τ1 ) |v(τ1 )|2 dx.
2σst H n−1 (Γ (τ2 )) +
1 2
τ2 τ1
Ω
μ(ρ) |Dv|2 dx dt
Ω
4. Let χ =
ρ−β2 . β1 −β2
∞ For every ϕ ∈ C(0) (Ω × [0, T )), there holds
T 0
Ω
χ (∂t ϕ + v · ∇ϕ) dx dt +
Ω
(33)
χ (i) (x)ϕ(0) dx = 0.
(34)
(35)
Energy Inequality We prove that, for all τ1 ≤ τ2 < T and almost all 0 ≤ τ1 < T , including τ1 = 0, there holds
ρ0 (τ2 ) |v0 (τ2 )|2 dx + 2 Ω ρ0 (τ1 ) |v0 (τ1 )|2 dx. ≤ 2σst H n−1 (Γ (τ1 )) + 21
2σst H n−1 (Γ (τ2 )) +
1 2
τ2
τ1
Ω
μ(ρ0 ) |Dv0 |2 dx dt
(36)
Ω
By (21) and [1, Lemma 4.3], for all τ ∈ W 1,1 (0, T ) with τ ≥ 0 and τ (T ) = 0, we obtain T T tot tot E ε (t)τ (t) dt ≥ 2 μ(ρε ) |Dvε |2 dx τ (t) dt. (37) E ε (0)τ (0) + 0
0
Ω
Since μ is non-negative and Lipschitz continuous, ρε → ρ0 in L ∞ (0, T ; L 2 (Ω)), Dvε Dv0 in L 2 (0, T ; L 2 (Ω)n×n ) and (21) holds, we conclude that, after a possible passage to an appropriate subsequence, there holds μ(ρε )Dvε μ(ρ0 )Dv0 in L 2 (0, T ; L 2 (Ω)n×n ). Using W 1,1 (0, T ) → L ∞ (0, T ), τ ≥ 0 and the lower semi-continuity of the L 2 -norm with respect to weak convergence, we obtain T lim inf ε→0
0
Ω
μ(ρε (t)) |Dvε (t)|2 dx τ (t) dt ≥
T 0
Ω
μ(ρ0 (t)) |Dv0 (t)|2 dx τ (t) dt.
The Sharp-Interface Limit for the Navier–Stokes–Korteweg Equations
11
By the convergence of E εtot,(i) , (28) and (32), taking (37) to the limit leads to
2
(i)
2σst H +
v0 dx τ (0) Ω T 2 n−1 1 2σst H (Γ (t)) + 2 ρ0 (t) |v0 (t)| dx τ (t) dt + n−1
≥2
ρ0(i)
μ(ρ0 (t)) |Dv0 (t)|2 dx τ (t) dt.
Ω
0
1 2
Ω
0 T
(Γ0(i) )
Finally, another application of [1, Lemma 4.3] gives (36). Regularity of Limiting Velocity and Transport Equation Taking (19) to the limit, and using the energy estimate (36), we may conclude that the pair (ρ0 , v0 ) has the desired regularity and that χ0 satisfies (35). As ρ0 (t) ≥ β1 > 0 a.e. in Ω, for t ∈ (0, T ), the energy estimate (36) yields 2σst H n−1 (Γ (t)) + 21 β1
|v0 (t)|2 dx ≤ 2σst H n−1 (Γ0(i) ) +
Ω
1 2
Ω
2
ρ0(i) v0(i) dx.
Recalling that |∇χ0 (t)| (Ω) = H n−1 (Γ (t)), implies that χ0 ∈ L ∞ (0, T ; BV (Ω)) and v0 ∈ L ∞ (0, T ; L 2 (Ω)n ). Taking (19) to the limit leads to
T
ρ0 ∂t ϕ + ρ0 v0 · ∇ϕ dx dt +
Ω
0
Ω
ρ0(i) ϕ(0) dx = 0
(38)
∞ for any ϕ ∈ C(0) (Ω × [0, T )). By [10, Theorem 10.29], ρ0 is a renormalized solution in the sense of DiPerna and Lions [9], which means that there holds T
0
Ω
b(ρ0 )∂t ϕ + b(ρ0 )v0 · ∇ϕ − (ρ0 b (ρ0 ) − b(ρ0 )) div(v0 )ϕ dx dt = 0
for any b ∈ C 1 ([0, ∞)) ∩ W 1,∞ (0, ∞) and any ϕ ∈ C0∞ (Ω × (0, T )). Choosing b such that b(β1 ) = b(β2 ) = 0, b (β1 ) = β11 and b (β2 ) = β12 , and recalling that ρ0 (t) ∈ {β1 , β2 } a.e. in Ω, yields
T Ω
0
T
v0 · ∇ϕ dx dt = − 0
Ω
div(v0 )ϕ dx dt = 0.
(39)
Hence v0 ∈ L ∞ (0, T ; L 2σ (Ω)). Plugging (18) and (26) into (38) implies
T
(β1 − β2 )
T
= − β2 0
Ω
0
Ω
χ0 (∂t ϕ + v0 · ∇ϕ) dx dt +
v0 · ∇ϕ dx dt
Ω
χ0(i) ϕ(0) dx
(40)
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H. Abels et al.
∞ for any ϕ ∈ C(0) (Ω × [0, T )). Due to (39), we finally infer that χ0 satisfies (35). ∞ Variational Formulation For any ψ ∈ C0∞ ([0, T ); C0,σ (Ω)), by (20), there holds
T 0
= −
ρε vε ∂t ψ + ρε vε ⊗ vε : ∇ψ − 2μ(ρε )Dvε : Dψ dx dt T (i) (i) ρε vε · ψ(0) dx − ε ∇ρε ⊗ ∇ρε : ∇ψ dx dt.
Ω
Ω
0
(41)
Ω
Finally, we use the convergence properties established in Sect. 2 to take (41) to the limit. As μ is Lipschitz continuous, μ(ρε ) inherits the convergence properties of ρε . Hence, using (29) and (31), we conclude that (ρ0 , v0 ) satisfies (33). Main Theorem We perform the sharp-interface limit and gather together the results of Sects. 2 and 3 in the following theorem. Theorem 1. Let (28) and (29) hold true. Then there exist a subsequence (ρε j , vε j ) j∈N of (ρε , vε )ε∈(0,1) and a pair (ρ0 , v0 ) with the following properties. 1. 2. 3. 4.
ρ0 ∈ C 0, 28 ([0, T ]; L 2 (Ω)) ∩ L ∞ (0, T ; BV (Ω, {β1 , β2 })). v0 ∈ L 2 (0, T ; H01 (Ω)n ) ∩ L ∞ (0, T ; L 2σ (Ω)). (ρ0 , v0 ) is a weak solution of (9)–(17) in the sense of Definition 2. For any t ∈ [0, T ], as j → ∞, there holds 1
a. b. c. d.
1
ρε j → ρ0 in C 0, 29 ([0, T ]; L 2 (Ω)), ρε j (t) → ρ0 (t) in L q (Ω) for any q ∈ [1, p∗ ), ρε j (t) ρ0 (t) in L p∗ (Ω), vε j v0 in L 2 (0, T ; H 1 (Ω)n ).
4 Conclusions We presented, to the best of our knowledge, the first rigorous investigation of the sharp-interface limit of (1)–(6). Our main result (Theorem 1) is merely valid under the restrictive conditions (28) and (29). The question whether Theorem 1 also holds true under weaker assumptions remains open, cf. also the discussion of (28) in [8, Sect. 5.7].
References 1. H. Abels, Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities. Commun. Math. Phys. 289(1), 45–73 (2009) 2. L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems (Clarendon Press, Oxford, 2000)
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3. S. Benzoni-Gavage, R. Danchin, S. Descombes, Well-posedness of one-dimensional Korteweg models. Electron. J. Differ. Equ. 59, 1–35 (2006) 4. S. Benzoni-Gavage, R. Danchin, S. Descombes, On the well-posedness for the Euler–Korteweg model in several space dimensions. Indiana Univ. Math. J. 56(4), 1499–1579 (2007) 5. D. Bresch, B. Desjardins, C.K. Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems. Commun. Partial Differ. Equ. 28(3–4), 843–868 (2003) 6. X. Chen, Global asymptotic limit of solutions of the Cahn–Hilliard equation. J. Differ. Geom. 44(2), 262–311 (1996) 7. R. Danchin, B. Desjardins, Existence of solutions for compressible fluid models of Korteweg type. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 18(1), 97–133 (2001) 8. J. Daube, Sharp-Interface Limit for the Navier–Stokes–Korteweg Equations. Ph.D. thesis, University of Freiburg (2016). https://freidok.uni-freiburg.de/data/11679 9. R.J. DiPerna, P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547 (1989) 10. E. Feireisl, A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids (Springer Science & Business Media, Berlin, 2009) 11. B. Haspot, Existence of global weak solution for compressible fluid models of Korteweg type. J. Math. Fluid Mech. 13(2), 223–249 (2011) 12. H. Hattori, D. Li, The existence of global solutions to a fluid dynamic model for materials for Korteweg type. J. Partial Differ. Equ. 9(4), 323–342 (1996) 13. K. Hermsdörfer, C. Kraus, D. Kröner, Interface conditions for limits of the Navier–Stokes– Korteweg model. Interfaces Free Bound. 13(2), 239–254 (2011) 14. M. Köhne, J. Prüss, M. Wilke, Qualitative behaviour of solutions for the two-phase Navier– Stokes equations with surface tension. Math. Ann. 356(2), 737–792 (2013) 15. D.J. Korteweg, Sur la forme que prennent les équations du mouvement des fluides si l’on tient compte des forces capillaires causées par des variations de densité considérables mais continues et sur la théorie de la capillarité dans l’hypothèse d’une variation continue de la densité. Arch. Néerl. 2(6), 1–24 (1901) 16. M. Kotschote, Strong solutions for a compressible fluid model of Korteweg type. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25(4), 679–696 (2008) 17. M. Kotschote, Strong well-posedness for a Korteweg-type model for the dynamics of a compressible non-isothermal fluid. J. Math. Fluid Mech. 12(4), 473–484 (2010) 18. S. Luckhaus, L. Modica, The Gibbs-Thompson relation within the gradient theory of phase transitions. Arch. Ration. Mech. Anal. 107(1), 71–83 (1989) 19. J. Simon, Compact sets in the space L p (0, T ; B). Ann. Math. Pura Appl. 146(4), 65–96 (1987)
Asymptotic Behavior of a Solution of Relaxation System for Flow in Porous Media E. Abreu, A. Bustos and W. J. Lambert
Abstract We introduce a novel modeling of phase transitions in thermal flow in porous media by using hyperbolic system of balance laws, instead of system of conservation laws. We are interested in two different behaviors of the balance system: the long time behavior, in which we study the solution with fixed relaxation term and very large time; and the behavior of the solution when the relaxation term is taken to zero and the time is fixed. We also are interested in solving the question: “Does this balance system tend to the conservation system under equilibrium hypothesis?”. To answer this question, we introduce a projection technique for the wave groups appearing in the system of equations and we study the behavior of each group. For a particular Riemann datum, using the projection method, we show the existence of a decaying traveling profile supported by source terms and we analyze the behavior of this solution. We corroborate our analysis with numerical experiments. Keywords Balance laws · Asymptotic expansion · Non-equilibrium relaxation Riemann problem · Finite volume · Flow in porous media
E. Abreu thanks for financial support through grants FAPESP No. 2014/03204-9, CNPq No. 445758/2014-7 and UNICAMP/FAEPEX No. 519.292-0280/2014. W. Lambert was supported through FAPERJ grant No. E-26/110.241/2011. A. Bustos thanks FAPESP for a graduate fellowship through grant No. 2011/23628-0. E. Abreu (B) Department of Applied Mathematics, University of Campinas, Campinas, SP 13083-970, Brazil e-mail: [email protected] A. Bustos Universidad Sergio Arboleda, Cra. 13, numbers 74-64, Bogotá, Colombia e-mail: [email protected] W. J. Lambert ICT - UNIFAL, Cidade Universitária - BR 267 Km 533 Rodovia José Aurélio Vilela, Poços de Caldas, Alfenas, MG, Brazil e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_2
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1 Introduction Many models exhibiting phase transitions can be modeled with the aid of hyperbolic systems of conservation laws and equilibrium hypothesis, for which some class of variables are conserved in each one of these phases. For example, for the model of liquid water and steam (gaseous water) under thermodynamic equilibrium, we have three physical phases: liquid phase, gaseous phase, and transition phase (liquid and gaseous). Under thermodynamic equilibrium hypothesis, we can use the Gibbs phase rule [16], which is a law that states the number of degrees of freedom of the system f = c − p + 2,
(1)
where f is the number of degrees of freedom, c is the number of components, and p is the number of phases in thermodynamic equilibrium. Under these conditions, we model this phenomena by using three systems of conservation laws (one system for each physical phase) and connection conditions between these physical phases. In papers [13, 14], a novel formalism to deal with this class of models and equations modeling phase change in problems with phase transition phenomena has been proposed. Each distinct physical phase is considered to be under a thermodynamic equilibrium. We can generalize this definition only saying that each phase is under equilibrium to encompass more general problems, for instance, in biology, physics, and chemistry (see [6, 8, 9, 12, 15]). By assuming this equilibrium, the system is modeled by a class of conservation laws with unknowns called primary variables and some variables that are obtained as function of these primary unknowns, called trivial variables. In porous media flow models, we also consider the unknown Darcy speed, that we always denote as u in what follows. Each physical phase belongs to a more general phase space Ω and shares the boundary with at least one other physical phase. All physical phases together compose a phase space, which we will call by stratified variety, see Fig. 1a for an example of a stratified variety for the nitrogen and steam injection in a porous medium. Although we are able to give mathematical answers to many physical process in this problem, we need to handle with lack of uniqueness, mainly of the solution connecting states in different phases. The nonuniqueness is very common for weak solution for hyperbolic systems and this situation gives rise to several admissibility criteria, or entropy conditions, that select the unique solution from the multiplicity of solutions with respect to the Riemann problem that would otherwise exist, see [10] (and also [3–5]) on recent developments in hyperbolic conservation law problems and related issues. However, for phase transitions, see [8] or [12], the most common criteria or kinetic relations are not strong enough to single out solutions; see also [15] for the Riemann problem for fluid flow of real materials. Many artificial conditions are introduced in order to overcome this nonuniqueness; see [10] and references cited therein. However, these conditions are unnatural and can lead to non-adequate solution, i.e., the solution does not describe the correct physical phenomena. Thus, a natural question is: Is there an alternative modeling
Asymptotic Behavior of a Solution of Relaxation System …
17
process to account the “exact” physical problem along with uniqueness of solution for a given initial datum? We believe that a possible positive answer is by means of the use of system of balance laws with relaxation terms. This methodology is not new, it has appeared in other kinds of problems, mainly in elasticity, see [9], however for problems in multiphase flow, this methodology is not used. The general form of balance laws with relaxation term is first-order differential equations that can be written, in one dimensional space, as G(U )t + Fx (U ) = ε−1 Q(U ),
along with a relaxation time factor
> 0; (2)
where G, F and Q are smooth vector functions of U = U (x, t) : × + → n . Equations like (2) are multiscale systems, since they have two speeds (advection Fx (U ) and relaxation −1 Q(U )) with a particular behavior, the relaxation time has small values in comparison with the speed determined by the characteristic lines. When the relaxation time has small values, it is said that the balance law system is stiff. With this in mind, and after the modeling the phase transition by using relaxation systems, other questions have appeared: “How is the solution behavior when the relaxation time goes to zero?” and also “What is the long time behavior of solution?” Roughly speaking, we are interested in knowing if the relaxation solution tends to some equilibrium, which are the states U satisfying Q(U ) = 0. In this work, we consider this new insight to handle phase transition with relaxation systems. Indeed, by means of this novel approach we were able to analyze a particular Riemann initial datum, where we take two different states in the same physical phase under this relaxed condition. We point out that this analysis was done for a single physical phase. However, the technique used here can be generalized for two or more phases. For concreteness, in Sect. 2, we revisit a phase transition model for the injection of steam and nitrogen into a porous medium with water, proposing a system of relaxation balance laws. This physical phenomenon was first considered and analyzed in [6, 13, 14], in which associated 3 by 3 system of balance laws was taken under a thermodynamic equilibrium, where these equilibria composed an equilibrium stratified variety, see Fig. 1a. In Sect. 2.1, we introduce a methodology to deal with this class of problem, this methodology consists in studying the projection of wave groups into the equilibrium manifold. Here, for a particular Riemann datum, at least qualitatively, we have proved two behavior of solutions: we prove that the long time (late-time) behavior and the behavior of −→ 0 lead the solution of the balance system of equations to the equilibrium solution (thermodynamic equilibrium). In Sect. 3, we compare the qualitative analytical behavior of solution with the numerical solution, corroborating our analysis. Finally, in Sect. 4, we draw our conclusions and perspectives.
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2 Physical and Mathematical Models Here, we extend the system of conservation laws proposed in [6, 13, 14] in order to consider the nonequilibrium effects of phase transitions. The system of equations consists of balance laws modeling the injection of steam, nitrogen, and water into a homogeneous porous media, q ∂ ∂ g−→a,w ϕψgw ρgW sg + u f g ψgw ρgW = , ∂t ∂x ∂ ∂ ϕ ρW sw + ψgw ρgW sg + u ρW f w + ψgw ρgW f g = 0, (3) ∂t ∂x ∂ ∂ ϕψgn ρg N sg + u f g ψgn ρg N = 0, ∂t ∂x ∂ ϕ Hˆ r + h w ρW sw + (h gw ρgW ψgw + h gn ρg N ψgn )sg + ∂t ∂ u h w ρW f w + (h gw ρgW ψgw + h gn ρg N ψgn ) f g = 0, ∂x where sw and sg are the water and gaseous saturation; f w and f g are the fractional flux of water and gas, and ϕ is the constant porosity. We assume that sg + sw = 1 and thus f g + f w = 1, i.e., the system is completely filled with water, steam, and gaseous nitrogen. The other unknowns in the system are the temperature T (in Kelvin degrees); the fraction of steam, ψgw , and nitrogen, ψgn , in the gaseous phase. Since we assume that the mixture of steam and nitrogen is additive and that there is no change of volume in the mixture of both gases, then ψgw + ψgn = 1, we can utilize only one variable ψgw and ψgn = 1 − ψgw ; and the Darcy speed u. Moreover, ρW , ρgW and ρg N are the water, steam, and nitrogen densities, which are functions of T ; Hr is the rock enthalpy and h gw and h gn are the steam and nitrogen enthalpy per mass unity and these functions are assumed be depending on T . It is useful to define Hw = h w ρW and Hg = ψgn h gn ρg N + ψgw h gw ρgW . All other variables are functions depending on (sg , T, ψgw ). In the system (3), the first equation denotes the mass balance of steam in the water phase, conservation of water (in liquid and gaseous phases); the second denotes the conservation of nitrogen; the third denotes the energy conservation. The term qg−→a,w is the rate of condensation (or evaporation) of water. This term is empirical one and measures the nonequilibrium effects of phase change. To obtain this term, first we admit that we have some states on thermodynamic equilibrium. For the physical situation for which water, steam, and nitrogen coexist, denoted as t p physical situations, if we admit thermodynamic equilibrium, we use the Gibbs phase rule (1). Since we have two phase p = 2 and two components c = 2, the number of degrees of freedom is two. Since we assume that the flow is under fixed pressure, the only degree of freedom is the temperature, T . Thus the steam composition ψgw (and ψgn = 1 − ψgw ) are written as function of temperature T . Thus the t p physical situation under thermodynamic equilibrium is formed by the states satisfying, see Fig. 1a,
Asymptotic Behavior of a Solution of Relaxation System …
19
t p = {(sg , T, ψgw ), such that 0 ≤ sg ≤ 1; 293 ≤ T ≤ 373.15; ψgw = ρgw (T )/ρgW (T )}.
(4) Since the term qg−→a,w measures how the system is far from equilibrium, we propose the following expression: qg−→a,w =
ρgw (T ) − ψgw . ρgW (T )
(5)
Although in [7], it was introduced a general function for qa,w→g , here we will just use the simple model (5). Notice, from (4), that the kernel of qa,w→g is the region t p. Moreover, this function is an attractor in the sense that states far from equilibrium t p tends to t p when −→ 0. Indeed, it is also possible to obtain this condition when t −→ ∞. Besides, on the t p situation, system (3) reduces to ∂ ∂ ϕ ρW sw + ρgw sg + u ρW f w + ρgw f g = 0, ∂t ∂x ∂ ∂ ϕρgn sg + u f g ρgn = 0, ∂t ∂x ∂ ˆ ϕ Hr + h w ρW sw + (h gw ρgw + h gn ρgn )sg + ∂t ∂ u h w ρW f w + (h gw ρgw + h gn ρgn ) f g = 0. ∂x
(6)
Notice that (6) is the same system in [6, 13, 14]. We also can recover the system (6), if we use asymptotic expansion on the variable U = (sg , ψgw , T ). The zero-order approximation of expansion of system (3) recovers (6), see [7]. In the next sections, we are interested in analyzing, for a single Riemann datum on the t p situation, if solution of system (3) tends to the solution of system (6) under two different regimes: long time (late-time) behavior, i.e., ε fixed and t −→ ∞; and the regime for which t is fixed and ε −→ 0. This analysis is important, because several models admit instantaneous thermodynamic equilibrium and the dynamics is modeled by conservation laws, however, all physical phenomena of phase transitions exhibit a relaxation time for which the system is far from equilibrium.
2.1 Mathematical Analysis We are interested in drawing the qualitative behavior and we choose a Riemann datum on the equilibrium sheet, i.e., for states on t p satisfying (4). Then we consider the particular Riemann datum, L = (s L , TL , ψgw (TL ), u L ) = (0.1, 320, 0.104897, 1) and R = (s R , TR , ψgw (TR ), u R ) = (0.4, 360, 0.61013, ·), see Fig. 1a. In [13], it was shown that it is not necessary to prescribe the Riemann Darcy speed on the right side u R , such that u R is obtained from the Riemann solution. Moreover in [13], it
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was also shown that it is possible to obtain the Riemann solution only on the space of variables, without considering u. Here we describe our analysis for this solution. To analyze the solution, we need to study the wave speeds appearing in the solution. We utilize a projection algorithm of waves on the equilibrium manifold. Thus, our strategy to study this problem is as follows: 1. In the first step, we obtain the wave sequence connecting the left and right Riemann data by admitting that we do not have source terms, i.e., we consider the waves of the system (3) setting qa,w→g = 0 with Riemann datum on t p. We denote the wave sequence as ω and each wave in this sequence as ωk , which is a shock or a rarefaction. The constant states are denoted as Uk . In the space U = (sg , T, ψgw ), the wave ωk defines a curve γk = ((sg )k , Tk , (ψgw ))k . Thus, we can write the Riemann solution as ω1
ω2
ω3
ωn
→ U1 − → U2 − → ··· − → UR , UL −
(7)
in which n is the number of different waves in the Riemann solution. We can use the theory for existence and uniqueness of this Riemann solution, using techniques for conservation laws and prove that the wave sequence ω is unique. 2. In the second step, for each curve γk and constant state Uk , we study the interaction of these waves and states with the the source term qa,w→g . Notice that the source term modifies only the states and curves that are far from equilibrium. For each group of curve and states, the system of equations (3) reduces to a simpler one. We are interested in analyzing for theses groups the behavior for t −→ ∞ and ε fixed; and the behavior of t fixed and ε −→ 0. 3. After we obtain these solutions and behaviors, we compare the solution with the solution of the equilibrium system (6) for the same Riemann datum. Applying the previous algorithm for the system (3), we need first to identify all waves. In [7], it was proved that the system (3) has three different eigenvalues, λB L =
u fg u ∂ fg , λc = , ϕ ∂sg ϕsg
and
λe =
f g + Hw ρgw ρg N u , ϕ sg + Hˆ r + Hw ρgW ρg N
(8)
in which = −Hg ρgW ρg N + (Hg − Hw )ρgW ρg N and the symbol represents the derivative with respect to T . It was also found that the eigenpair associated to λc is a contact discontinuity, where only the composition ψgw changes. We denote this wave group as Sc ; the eigenpair associated to λ B L is a Buckley–Leverett type, where only the saturation sw changes. Associated to this waves, we have shocks (denoted as SBL ) and rarefactions (denoted as RBL ) and the eigenvector as r B L . For the eigenpair associated to λe , all variables change. This eigenpair is called evaporation wave. Associated to this waves we have shocks (denoted as Se ) and rarefactions (denoted as Re ) and the eigenvector re . To obtain the Riemann solution, we use the geometrical compatibility, i.e., the waves go from the slowest to fastest one. Here, we have three wave groups with
Asymptotic Behavior of a Solution of Relaxation System …
21
different speeds. In the region considered for the Riemann datum, it was shown in [7] that the speed of the wave associated to evaporation family is the slowest one, after that there is a Buckley–Leverett shock SBL and, then a contact discontinuity Sc . The Riemann solution comprises first of an Re , i.e., a rarefaction associated to λe , since ∇λe · re > 0. Since the temperature changes only on this wave, the rarefaction is obtained from L state to the state M = (s M , TM , (ψgw ) M ) in Re , in which TM = TR = 360. Since for this state M, we have λe (M) < λ B L (M), there is a constant state. Since the gas saturation changes only on the evaporation wave and on the Buckley–Leverett, the saturation sg changes from s M to s R , then this BuckleyLeverett wave leads M to the state M " = (s R , TR , (ψgw ) M ). Since only sg varies, we can project all analysis in the plane (sg , f g ), from Oleinik construction using convex hull, there is a Buckley–Leverett shock connecting M to (s R , TR , (ψgw ) M ) u ( f g (s R , TR ) − f g (s M , TR )) with speed v s = . Since it is possible to verify that v s < ϕ sR − sM λc (s R , TR , (ψgw ) M ), there is a constant state and finally there is a contact discontinuity connecting (s R , TR , (ψgw ) M ) to right state R. This solution is summarized in Fig. 1b in the phase space and in Fig. 2a in the space V × x for a fixed t. Using (7), we can write the solution as Sc Re S BL (9) L −→ M −−→ M −→ R. Before we study the interaction of each wave group with the source term qg−→a,w , we obtain the Riemann solution for the datum L = (s L , TL , ψgw (TL ), u L ) = (0.1, 320, 1) and R = (s R , TR , ψgw (TR ), u R ) = (0.4, 360, ·) for the system 6. In [7], it was proved that the system (6) has two eigenvalues: the first eigenvalue is λ B L given by (8) and eigenvector r B L = (1, 0, 0) and the eigenvalue λT is given by − ρgn β)(ρW α − Hw β) − (αρgw − α θ )ρgn ρW ) f g ((ρgn ρgw u , − ρ β)(ρ α − H β) − (αρ − α θ )ρ ρ ) − H ˆ r ρW ρgn β ϕ sg ((ρgn ρgw W w gn W gn gw (10) for which α = Hg − Hw , β = ρgw − ρW and θ = ρgw − ρgn . The eigenvector associated to this eigenvalue can be obtained in [14]. For this field, both temperature and saturation change, thus this field is called thermal field. The rarefaction for this field is denoted by RT and the shock is ST . Also in [14], we find the complete Riemann solution for system (6), which encompasses the particular Riemann datum L and R. For these states, in t p, first we have a rarefaction RT connecting the state L to a intermediary state 1 = (s1 , TR ). From 1, there is a Buckley–Leverett shock from s1 to s R with constant temperature. Using (7), we can write the solution as RT S BL (11) L −→ 1 −−→ R.
λT =
Both solutions are summarized in Fig. 1. Now, we are able to analyze the decaying behavior of this solution. Our strategy here is to split the analysis for each wave. The first wave in the Riemann solution is a Re . This waves leads an equilibrium state to a nonequilibrium state M. Notice that the
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(a)
(b)
(c)
Fig. 1 a The space t p, given by (4), in the phase space (sg , T, ψgw ). b Two family of solutions. The wave sequence connecting L, M, M " and R is the solution of system (3), for particular Riemann datum, where we disregard the effects of qa,w−→g . The wave sequence connecting L, 1 and R are the wave sequence, where we assume thermodynamic equilibrium, see [14]. c Shock connecting s M to s R in the plane (sg , f g )
equilibrium will reach when ψgw = ρgW (TR )/ρgw (TR ). If we consider the relaxation term, we conjecture that the only variable changing is ψgw . This conjecture dues to several facts. The first one is because of the form of qa,w→g that depends only on ψgw and an constant number ρgW (TR )/ρgw (TR ). Moreover, there are two conservation laws: one for the mass of water and other for energy. Since the temperature is constant after TR , we can disregard the energy equation and we have only two equations. Then in the state between the waves Re and SBL , since sg and T are constant, the main evolution equation reduces to ∂ψgw ∂ψgw 1 ψgW (TR ) + uρgW f g = − ψgw ∂t ∂x ε ψgw (TR ) u f g ∂ψgw ψgW (TR ) 1 + = − ψgw . ϕs M ∂ x ϕs M ρgW ε ψgw (TR )
ϕρgW s M ∂ψgw ∂t
−→ (12)
Asymptotic Behavior of a Solution of Relaxation System …
23
where f g = f g (s M , TR ), ρgW = ρgW (TR ). Here, we are disregarding variations in the variable u, since we are interested only in the projection of the solution in the space (T, sg , ψgw ). Notice that we have only an approximation of the original system. This analysis is corroborated by our numerical solutions; see Fig. 2 and [7] for more details. The solution of (12), via characteristic waves, is ψgw =
ρgW (TR ) ρgW (TR ) 1 ∗ ex p − − + ψgw t , ρgw (TR ) ρgw (TR ) ϕs M ρgW ε
(13)
u fg t = constant. Here, we are taking as iniϕs M ∗ tial condition any constant state for ψgw , we are interested only in the qualitative behavior. From Eq. (13), we can see that the behavior for large times and fixed ε or for fixed time and small ε are the same, in both cases ψgw tends to the equilib∗ . We call this wave Oe , see Fig. 2b. Notice that if there rium, independently on ψgw is no Buckley–Leverett between the states on the wave Oe to the equilibrium state ψgw (TR ) = ρgW (TR )/ρgw (TR ), the decaying behavior would not change. However, the Buckley–Leverett shock interferes with the wave sequence introducing another behavior. Here we need to separate our analysis by assuming that T is constant, then we disregard the energy equation. We also fix u constant (in the splitting spirit of analysis). Then we have two equations. An equation for mass conservation of water (second equation of (3)) and an equation with relaxation term (first equation of (3)). The first of all is associated with the equation for mass conservation of water. In the modeling of system (3), we disregard diffusive effects. However, in the shocks they are important to select shocks with viscous profile. Numerically, this viscous profile is even more evident, such that the diffusive term can be relatively large when compared with physical ones. For the equation for mass conservation, this Buckley–Leverett shock Se exhibits a traveling profile supported by the diffusive terms that we disregard. These traveling profiles travel in the coordinate system η = x − v s t. Several authors consider the existence of traveling profiles supported by relaxation terms (see, e.g., [9]). On this way, on this coordinate system, we assume that sg (x, t) = sg (η) and on this traveling profile, we assume that ψgw = ψgw (η). Then, from the first equation of (3), and after we apply the chain rule, we obtain on each characteristic of form x −
u fg qa,w→g dψgw u ∂ f g dsg s s + − v sg = −→ ψ gw −v + ϕ ∂sg dη ϕ dη ερgW ∂f qa,w→g ds − ψ gw −v s + ϕu ∂sgg dηg ερgW dψgw . = u fg dη − vs s ϕ
(14)
g
We are also interested only on the qualitative behavior. Notice, from this wave that dsg > 0, because sg increases in this wave; we also can see Fig. 2c, for a geometrical dη
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u ∂ fg . Indeed, notice that the second term in the last inϕ ∂sg equality is the Buckley–Leverett eigenvalue that is the speed of a rarefaction wave. From Fig. 1 we can see that v s is faster than λ B L in S M . From the same picture, we can see that the denominator of (14) is negative. Since qa,w−→g > 0 because ρgw (TR )/ρgW (TR ) > ψgw , joining all information together, we can see that the right side of (14) is negative. Moreover, if we have the qualitative behavior satisfying ψgw < 0, we have a decreasing wave as we see in Fig. 2, which in turn was obtained dη by a numerical scheme [1, 2]. On the other hand, if for some state we have that ψgw > 0, we would have ψgw increasing, and then in any case the wave is stable. dη We remark that this wave is not a traveling wave in the sense that it connects two equilibrium, here the existence of this profile is due only to the fact that the saturation wave crosses the decaying wave Oe . After the saturation wave crosses the profile,
inspection, that v s >
Fig. 2 A pictorial illustration of solution paths for a Riemann problem related to system (3) in an injection of steam and nitrogen in a porous medium (in all column are shown from top to bottom, the temperature, the saturation, and the fraction of steam). In details we have left column (with thermodynamic equilibrium), middle column (without thermodynamic equilibrium or with relaxation time when ε tends to zero), and right column (the corresponding numerical solutions). Indeed, from these profiles, we have numerical evidence that our approximate numerical solution bears a resemblance to the expected solution from the relaxation system (see next frames for Temperature, Gas saturation and Steam composition as function of the spatial domain)
Asymptotic Behavior of a Solution of Relaxation System …
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DIFFERENT APPROXIMATIONS FOR ε
TEMPERATURE T
360
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340 EQ ε =0.01 ε =0.05 ε =0.1 ε =0.5 ε =1 ε =1.5
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320 0
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DIFFERENT APPROXIMATIONS FOR ε EQ ε =0.01 ε =0.05 ε =0.1 ε =0.5 ε =1 ε =1.5
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DIFFERENT APPROXIMATIONS FOR ε
STEAM COMPOSITION ψgw
0.6 0.5 0.4 EQ ε =0.01 ε =0.05 ε =0.1 ε =0.5 ε =1 ε =1.5
0.3 0.2 0.1 0
1
Fig. 2 (continued)
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we obtain again a constant state for TR and s R with a system of form (12) where we substitute s M by s R and f g = f g (S R , TR ). The solution is obtained in the similar way by using (13). Then we have another decaying wave of form Oe . This wave occurs at the contact discontinuity Sc to the right state R. The above qualitative analysis corroborates the numerical solutions and shows, at least for this particular Riemann problem, that the solution is stable and tends to the equilibrium profile. We intend to use this methodology for solving the complete Riemann problem for states on the tp region and perform similar analysis to justify the existence of equilibrium and decaying profiles. From our analyses, we can see that the particular solution of the system (3) is stable for two regimes: long time behavior and for fixed time and epsilon −→ 0. Moreover, we show that both solutions tend in the phase space to equilibrium solution. In the next section, we study the behavior of numerical solution for fixed time and different relaxation terms ε tending to zero.
3 Numerical Experiments In short, balance laws models of type (3) are solved as follows (see [1, 2, 7] and references therein for more details). In a first step, the inhomogeneous term is removed and the Riemann problem for the resulting homogeneous system is solved and sampled. In the second step, the associated system of ordinary differential equations obtained by removing the convection terms is solved, using the solution from the first step as initial data. The advantages of this procedure are its simplicity and robustness. However, for certain applications [1, 2, 7], this method requires that the mesh size be quite small to obtain reasonable accuracy (see also [11]). It is worth mentioning that a characteristic feature of hyperbolic balances law is the inherent existence of nontrivial equilibrium solutions. Then, in order to design a scheme for inhomogeneous conservation laws, a key point is how to discretize the flux function Fx (U ) (or the advection term) and the source term ε−1 Q(U ) (or the relaxation term), keeping the proper balance a discrete analogue of this balance linked to equilibrium steady state solutions. In the recent years, prominent well-balanced and asymptotic preserving schemes were developed for solving balance laws; see [11] for a modern and deep discussion as well as for an excellent survey on this subject. A cheap unsplitting finite volume scheme was employed to carried out all numerical calculations efficiently for solving the model system of balance laws (3), in which numerical solutions are depicted in Fig. 2; the interested reader is referred to [1, 2] or [7] for further information about the numerical algorithm. In our numerical experiments, we consider fixed time t = 1 and ε −→ 0. These experiments corroborate the projection technique and show that for the particular Riemann datum, the solution of system (3) tends to equilibrium solution (6).
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4 Concluding Remarks We study a particular Riemann datum for a system of balance equations modeling flow of steam and nitrogen in a porous medium. We introduce a projection method to deal with stiff solutions, for which we split the solution into each wave group. Moreover, we analyze the decaying behavior of solution that we obtain for two regimes: the long time (late-time) behavior and the regime for which ε tends to zero and the time is fixed. These analyses show that both behaviors are quite similar, moreover, we show that the solution of system (3) tends to equilibrium solution of (6) for both regimes. We corroborate our analysis with convincing numerical results that support our findings.
References 1. E. Abreu, A. Bustos, W. Lambert, Non-monotonic traveling wave and computational solutions for gas dynamics Euler equations with stiff relaxation source terms. Comput. Math. Appl. 70, 2155–2176 (2015) 2. E. Abreu, A. Bustos, W. Lambert, A unsplitting finite volume method for models with stiff relaxation source term. Bull. Braz. Math. Soc. 47, 5–20 (2016) 3. E. Abreu, M. Colombeau, E.Y. Panov, Weak asymptotic methods for scalar equations and systems. J. Math. Anal. Appl. 444, 1203–1232 (2016) 4. B. Andreianov, K.H. Karlsen, N.H. Risebro, A theory of L1-dissipative solvers for scalar conservation laws with discontinuous flux. Arch. Ration. Mech. Anal. 201(1), 27–86 (2011) 5. B. Andreianov, D. Mitrovi´c, Entropy conditions for scalar conservation laws with discontinuous flux revisited. Ann. de l’Institut Henri Poincare Non Linear Anal. 32(6), 1307–1335 (2015) 6. J. Bruining, D. Marchesin, Analysis of nitrogen and steam injection in a porous medium with water. Transp. Porous Media 62(3), 251–281 (2006) 7. A. Bustos, Ph.D. Thesis, Institute of Mathematics, Statistics and Scientific Computing, University of Campinas, Brazil, 2015 8. R.M. Colombo, Andrea Corli, Continuous dependence in conservation laws with phase transitions. SIAM J. Math. Anal. 31(1), 34–62 (1999) 9. G.-Q. Chen, A.E. Tzavaras, Remarks on the contributions of constantine M. Dafermos to the subject of conservation laws acta math. Scientia 32B, 3–14 (2012) 10. C.M. Dafermos, in Hyperbolic Conservation Laws in Continuum Physics, vol. 325 (Grundlehren der Mathematischen Wissenschaften), Fundamental Principles of Mathematical Sciences (Springer, Berlin, 2016), XXXVIII, p. 826 11. L. Gosse, Computing Qualitatively Correct Approximations of Balance Laws (Exponential-Fit, Well-Balanced and Asymptotic-Preserving), vol 2, XIX (Springer, Mailand, SIMAI Springer Series, 2013), p. 341 12. H. Hattori, The Riemann problem for thermoelastic materials with phase change. J. Differ. Equ. 205, 229–252 (2004) 13. W. Lambert, D. Marchesin, The Riemann problem for multiphase flows in porous media with mass transfer between phases. J. Hyperbolic Differ. Equ. 81, 725–751 (2009) 14. W. Lambert, J. Bruining, D. Marchesin, The Riemann solution for the injection of steam and nitrogen in a porous medium. Transp. Porous Media 81, 505–526 (2010)
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15. R. Menikoff, B. Plohr, The Riemann problem for fluid flow of real materials rev. Mod. Phys. 61, 75–130 (1989) 16. B.J. Merkel, B. Planer-Friedrich, Groundwater Geochemistry (A Practical Guide to Modeling of Natural and Contaminated Aquatic Systems), ed. by D.K. Nordstrom (Springer, Berlin, 2008), p. 221
Optimal Control of Level Sets Generated by the Normal Flow Equation Angelo Alessandri, Patrizia Bagnerini, Roberto Cianci and Mauro Gaggero
Abstract The goal of this work is the optimal control of level sets generated by the normal flow equation. The problem consists in finding the normal velocity that minimizes a given performance index expressed by means of a cost functional. In this perspective, a sufficient condition of optimality requiring the solution of a system of partial differential equations is derived. As in general it is difficult to solve such a system, an approximation scheme based on the extended Ritz method is proposed to find suboptimal solutions. The control law is forced to take on a fixed structure depending nonlinearly on a finite number of parameters to be suitably chosen. The selection of the parameters is accomplished by using a gradient-based technique. To this end, the adjoint equation is derived to compute the gradient of the cost functional with respect to the parameters of the control law. The effectiveness of the proposed approach is shown by means of simulations. Keywords Level set methods · Normal flow · Optimal control Approximation · Extended Ritz method · Adjoint equation
This work has been partially supported by the AFOSR with grant no. FA9550-15-1-0530, by INDAM-GNCS and by INDAM-GNFM. A. Alessandri · P. Bagnerini (B) · R. Cianci Department of Mechanical Engineering, University of Genoa, Via All’Opera Pia 15, I-16145 Genoa, Italy e-mail: [email protected] A. Alessandri e-mail: [email protected] R. Cianci e-mail: [email protected] M. Gaggero National Research Council of Italy, Via De Marini 6, I-16149 Genoa, Italy e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_3
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1 Introduction Many engineering and scientific problems are characterized by a number of different regions interacting and depending on various factors, such as physical laws and geometry. A lot of examples exist in cross-disciplinary contexts, like fluid dynamics, material science, computational biology, biomedicine, land protection, and energy engineering. The literature presents a variety of methods to simulate front propagation. Among the various alternatives, level set methods are widely used to propagate fronts in two- or three-dimensional domains in many fields, such as image processing, detonation and deflagration waves, seismic analysis, optimal path planning, and shape design [4, 5, 7, 13]. In level set methods, the front is implicitly represented at each time by a certain level set (typically, the zero level) of a given function [14]. They have a number of advantages over Lagrangian approaches. In fact, they rely on typical geometric quantities that can be easily computed, such as the curvature or the normal to the front. Furthermore, changes of topology are handled (the surfaces can merge, divide, etc.), and the extension to three or more dimensions is straightforward. The different level set methods depend on the choice of the velocity field in the dynamic equation. Two examples used in many applications are the normal flow, where the velocity field is along the normal direction to the front, and the mean curvature flow, where the velocity is proportional to the curvature at each point of the front. Based on the preliminary results of [1, 2], the aim of this work is to optimally drive a moving front using the normal flow (NF) equation, in order to overcome the computational difficulties that have prevented to attack this problem up to now. Though there exists a vast literature concerning the control of systems described by partial differential equations (PDEs), very little has been done on the control of level sets. The lack of contribution may be due to the difficulty in treating the problem, both theoretically and numerically, and to the poor awareness of the potential application of the resulting control paradigm, which only in very recent times has emerged to some extent. Among the few available results, in [10] a prey–predator model inspired by biology is presented. The control of level sets resulting from the two-phase Stefan problem is the topic addressed in [8, 11], where the solution is searched for by using gradient-based methods. In this context, the present work focuses on the optimal control of the normal flow equation by regarding the speed as a control action. Unfortunately, finding a solution to such a problem is almost impossible since it would require to solve analytically a system of PDEs. Thus, we rely on suitable finite-dimensional approximations based on the extended Ritz method (ERIM) [16, 17, 21]. The ERIM can be applied to any functional optimization problem, even in high-dimensional settings [22]. It is based on the idea of constraining the control law to take on a fixed structure, where a finite number of free parameters can be suitably chosen. Then, the original problem is turned into a mathematical programming one that consists in optimizing the parameters. We exploit the ERIM to trade among complexity of the approximating structure, performance, and computational effort required to solve optimal control problems of fronts generated by level sets methods. The optimal parameters of the
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control policies are chosen by means of gradient-descent methods that rely on either a numerically-approximate gradient based on finite differences or the exact gradient obtained by solving the related adjoint equations [1, 2]. Simulations show that the latter approach ensures great savings in the computational time as compared to the former one with the same approximation accuracy. A similar use of adjoint methods to compute the gradient of cost functionals for the purpose of identification is described in [19, 20]. This chapter is organized as follows. In Sect. 2, the problem of the optimal level set control is formulated. The approximate solution method based on the ERIM is described in Sect. 3. The adjoint equation for the computation of the gradient of the cost functional is presented in Sect. 4. Simulation results are shown in Sect. 5. Conclusions are drawn in Sect. 6.
2 Optimal Control of Level Set Dynamics Level set methods are based on a very intuitive idea. Let Ω ⊂ Rq and t ≥ 0 be a space domain and the time, respectively. Consider a front or boundary in two or three dimensions separating two regions (two phases). Suppose to move this front with a known speed. Level set methods consist in considering the front represented at each time t as the zero level set of a function φ : Ω × [0, T ] → R, where T > 0 is a given time horizon. The front x(t, s) is then given at time t by the set of points such that φ(x(t, s), t) = 0, where s is the arc-length parameter of the initial curve x(0, s) (see Fig. 1). Differentiating with respect to t, we obtain the Hamilton–Jacobi (HJ) equation φt (x, t) + v(x, t) · ∇φ(x, t) = 0 in Ω × [0, T ],
(1)
where v(x, t) := dtd x(t, s), i.e., the Lagrangian material particle velocity, gives the direction of propagation of the front at the point x(t, s) and ∇ denotes the spatial gradient. Specifically, we focus on the choice of a velocity v(x, t) that is proportional to the normal to the front, i.e., v(x, t) = u ∇φ(x, t)/|∇φ(x, t)|, where u represents the propagation speed. By replacing the expression of v in (1), we obtain the NF equation (2) φt (x, t) + u |∇φ(x, t)| = 0 in Ω × [0, T ]. Of course, we need to fix the initial conditions φ0 : Ω → R, i.e., φ(x, 0) = φ0 (x), for all x ∈ Ω. Usually, φ0 is the signed distance to the initial front. Based on the aforesaid, one can control the time evolution of (2) via a suitable choice of u, which plays the role of a control action. It can be chosen either as a function of time or both of time and space. In the former case we have u : [0, T ] → R, whereas in the latter we deal with u : Ω × [0, T ] → R. The level set control of (2) consists in choosing either u(t) (space-independent policy) or u(x, t)
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φ(x, t) φ(x, t) Ω
Ω
{x ∈ Ω : φ(x, t) = 0}
{x ∈ Ω : φ(x, t) = 0} Fig. 1 Examples of fronts described by zero level sets
(space-dependent policy) in such a way to move the front associated with some level set of φ(x, t) as desired. From now on, let us denote the set-valued mapping associated with the level set l ∈ R of the function φ by Γl : [0, T ] ⇒ C , where Γl (t) := {x ∈ Ω : φ(x, t) = l}. Moreover, we denote by U the set of admissible control functions like, for instance, the set of measurable bounded functions. This set is made of the smooth functions t → u(t) or (x, t) → u(x, t) such that there exist solutions to the Cauchy problem for (2) in a space denoted by F . The evolution of the level set over time may be associated with a performance index depending either on the boundary or on the interior of the shape. In particular, let us focus on the following cost: J (t0 , φ, u) :=
T
L(φ, u)(t) dt + K (φ),
t0
with t0 ∈ [0, T ], where L and K are functionals defined on F × U and F , respectively. For example, consider L(φ, u)(t) = K (φ) =
Ω Ω
g(φ(x, t), u(x, t), t)H (φ(x, t)) d x g(φ(x, ¯ T ))H (φ(x, T )) d x,
where g¯ : R → R is a final penalty term. Since we search for the control action u ∈ U that minimizes J (t0 , φ, u), we define V (t0 , φ) := inf J (t0 , φ, u). u∈U
(3)
Following [9, Theorem 1, p. 6], the solution of (3) can be obtained by finding a smooth-enough V (t, φ) that solves the system of PDEs
Optimal Control of Level Sets Generated by the Normal Flow Equation
Vt + Vφ φt + L(φ, u ◦ ) + u ◦ Vφ |∇φ| = 0 φt + u ◦ |∇φ| = 0,
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(4)
where u ◦ ∈ arg min L(φ, u) + u Vφ |∇φ| . u∈U
Unfortunately, it is not easy to find the analytic solution of (4). This motivates the use of numerical methods to search for an approximate solution to the level set optimal control problem (3). From now on, with a little abuse of notation, we will simply write J (φ, u) instead of J (0, φ, u).
3 Approximate Solutions with the Extended Ritz Method We propose an approach based on the ERIM to find an approximate solution to problem (3). The basic idea consists in constraining the optimal control law u ◦ (x, t) to take on a fixed structure, denoted by γ , where a finite number of parameters to be suitably tuned are inserted. This approach was originally proposed for the optimal control of lumped parameter systems [16, 17]. Recently, it has been used also for the optimal control of systems described by PDEs [3]. In this context, it may be regarded as a compromise between the classical paradigms “discretize-thenoptimize” and “optimize-then-discretize” [18]. Thus, it appears well suited also to treating the optimal control of level sets. The considered fixed-structure functions γ are the following: γ (·, w) =
n
ci ψ (·, κi ) + b, ci ∈ R, b ∈ R, κi ∈ Rl ,
(5)
i=1
where ψ(·, κi ) is a parametrized basis function, n is the overall number of basis functions, and all the parameters are collected in the vector w :=(c, b, κ), where κ :=(κ1 , κ2 , . . . , κl ), with the dimension of w equal to O(n) = n(1 + l) + 1. Most of commonly used approximating functions can be written as in (5), such as feedforward neural networks, free-node splines, radial-basis-function networks with adjustable centers and widths, and trigonometric polynomials with free frequencies and phases. Approximating structures like (5) are known to be endowed with better approximation capabilities than classical linear combinations of fixed basis functions, corresponding to the Ritz method of the calculus of variations (see, e.g., [6, 12]). Here the goal is to approximate the mapping (x, t) → u ◦ (x, t) that solves problem (3) by imposing the control law u(x, t) = γ (x, t, w)
(6)
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in the NF equation (2), where w ∈ R O(n) . Both φ and u are functions of w, hence also the cost functional depends on w. From now on, according to the context, we will highlight the dependence of J either on φ and u or simply on w. Thus, problem (3) reduces to find the optimal weights wo that minimize the cost J , i.e., wo ∈ arg min J (w).
(7)
w∈R O(n)
In the following, we will address the problem of finding a solution of (7). As proposed in [1, 2], we will use gradient-based methods to solve (7) subject to (2).
4 Adjoint Equation and Optimization Procedure As discussed in the previous section, we need to compute an analytic form of the gradient of the cost J with respect to the parameters w to solve (7) and therefore find the optimal weights of the approximating function γ in (5). However, due to the nonlinear dependence of J on the parameters, this is usually a difficult task. For the sake of generality, let us consider the generic cost functional
T
J (φ, u) = 0
Ω
h(φ(x, t), u(x, t), t) d x dt +
Ω
¯ h(φ(x, T )) d x,
(8)
where h : R × R × [0, T ] → R and h¯ : R → R, which we want to minimize subject to the NF equation (2). Clearly, the use of (6) in (2) and (8) makes the cost J depend on w both directly and indirectly since also the solution of (2) is affected by w. To reduce the notational burden, from now on we will drop the dependence on x and t and write ˜ explicitly the dependence on w. Moreover, let φ(w, w) ˜ := φ(w + w) ˜ − φ(w) ∈ F , where w˜ ∈ R O(n) . Of course, if w˜ → 0, also φ˜ tends to zero. Since the problem is of functional type, we need to deal with the Fréchet derivative ˜ w). of the cost functional along the direction (φ, ˜ To this end, from (2) we have ˜ + γ (w + w) ˜ |∇φ(w + w)| ˜ =0 φt (w + w)
(9a)
φt (w) + γ (w) |∇φ(w)| = 0 .
(9b)
After replacing γ (w + w) ˜ with a Taylor expansion of the first order centered in w and using the same approximation for the norm of the gradient of φ, i.e., |∇φ(w + w)| ˜ = |∇φ(w)| +
∇φ(w) (φ˜ x , φ˜ y ) + r0 , |∇φ(w)|
where w˜ → r0 (w, w) ˜ is a remainder of order higher than one, it follows from (9) that φ˜ t + |∇φ| ∇w γ w˜ + γ F · Φ˜ + r1 = 0,
(10)
Optimal Control of Level Sets Generated by the Normal Flow Equation
35
where, adopting the same notation proposed by [19], we let F :=(F1 , F2 ) with ˜ accounts for all F1 := φx /|∇φ|, F2 := φ y /|∇φ|, Φ˜ :=(φ˜ x , φ˜ y ), and w˜ → r1 (w, w) the remainders of order higher than one. To compute the derivative of the cost functional in (φ, w) along the direction ˜ w), (φ, ˜ consider ˜ w + w) J (φ + φ, ˜ − J (φ, w) = +
T 0 Ω
Ω
h φ (φ, γ ) φ˜ d x dt +
T 0
Ω
h u (φ, γ ) ∇w γ w˜ d x dt
h¯ φ (φ) φ˜ d x + r2 ,
(11)
Algorithm 1 Pseudo-code of the algorithm for the selection of the optimal weights 1: procedure optimization 2: Inputs: 3: equations (2), (12), (13), and N 4: Outputs: 5: vector of optimal parameters w◦ 6: Main loop: 7: bestCost ← ∞ 8: for k from 1 to N do 9: generate a random initial choice of wk 10: while (stopping criterion is not satisfied) do 11: solve the NF equation (2) 12: solve the adjoint equation (12) 13: compute the gradient (13) 14: perform a step of the descent algorithm by using the gradient (13) to update wk 15: end while 16: compute the cost J (wk ) corresponding to wk 17: if (J (wk ) < bestCost) then 18: bestCost ← J (wk ) 19: w ◦ ← wk 20: end if 21: end for 22: end procedure
where we have highlighted the dependence of u on w, and w˜ → r2 (w, w) ˜ is a remainder of order higher than one. The goal is to find the first-order necessary conditions of optimality by using the first variation with (10) as a constraint. Introducing the Lagrange multiplier (x, t) → μ(x, t) and adding the product between μ(x, t) and (10) to the right-hand side of (11), after easy steps, we obtain the adjoint equation
−μt = (μγ F1 )x + (μγ F2 ) y − h φ (φ, γ ) in Ω × [0, T ) μ(x, T ) = −h¯ φ (φ(x, T )) in Ω .
Thus, the gradient of the cost functional turns out to be
(12)
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A. Alessandri et al.
∇w J (φ, u) = 0
T
Ω
h u (φ, γ ) + μ |∇φ| ∇w γ d x dt .
(13)
When performing the search for the optimal weights, the initial guess of w may give rise to different final results due to local minima that may affect (7). In order to mitigate this risk, one may consider multistart techniques, by randomly choosing N initial weights, as described in Algorithm 1, where we take as stopping criterion the standard conditions for descent methods. The approximation of the solution of the optimal control problem requires the use of efficient numerical techniques for the solution of both the forward HJ equation and the backward adjoint equation. Clearly, the schemes for the forward and backward equations cannot be the same, as the structure of the adjoint equation is often very different from its counterpart in the forward system. It is known that the use of adjoint equations in the control of PDEs drastically decreases the computational time, but often at the cost of a considerable increase in the effort to stabilize the adjoint equation. In our case, both (2) and (12) are of hyperbolic type. Therefore, we have used high-order finite-difference schemes for hyperbolic PDEs. Moreover, imposing the correct boundary conditions is not a trivial task for the adjoint equation (12), as the source term h φ (φ, γ (w)) may have a large variability for values of w far from the global optimum, thus creating spurious and nonphysical reflections at the boundary.
5 Simulation Results In this section, we present the numerical results in 2D and 3D obtained in two different simulation examples. In the first one, the goal is inducing level sets to vanish as fast as possible, whereas in the second one, we want to track a given reference curve over time. For the sake of brevity, in the following, we will refer to the two examples with the terms “vanishing” and “tracking,” respectively. All the simulations were performed on a personal computer with a 2.6 GHz Intel Xeon CPU with 64 GB of RAM. We used the routine fmincon of the Matlab Optimization Toolbox to compute the optimal values of the parameters. The NF equation (2) and the adjoint equation (12) were solved numerically by using the Matlab toolbox for HJ equations developed by Mitchell [15]. In more details, we used an upwind second-order essentially non-oscillatory (ENO) scheme [13, chap. 3] in space and a second-order total variation diminishing Runge–Kutta scheme in time. It is worth noting that the convective flux terms in the adjoint equation (12) depend explicitly on x. Since Mitchell’s toolbox does not take into account this dependency, we modified the numerical scheme by deriving the flux and adding a source term. In all the examples, we focused on the nonlinear approximating structure (5) for the control law, using one-hidden-layer feedforward neural networks with sigmoidal activation functions. As it will be detailed later on, we considered different values for the number n of basis functions, i.e., we chose n = 5, 10, 15, and 20.
Optimal Control of Level Sets Generated by the Normal Flow Equation
0.5
level set - step 1 x2
0.5
0
level set - step 5 x2
0.5
0
x1
-0.5 -0.5
0
0
x1
-0.5 0.5
level set - step 10 x2
-0.5
0
level set - step 15 x2
0
x1
-0.5 0.5
0.5
37
-0.5
0
level set - step 20 x2
0
x1
-0.5 0.5
0.5
-0.5
0
x1
-0.5 0.5
-0.5
0
0.5
Fig. 2 Simulation results for the 2D “shape 1” vanishing example with n = 5 basis functions 1
level set - step 1 x2
1
0.5
0 0
level set - step 4 x2
1
0.5
x1 0.5
1
1.5
0 0
level set - step 8 x2
1
0.5
x1 0.5
1
1.5
0 0
level set - step 12 x2
0.5
x1 0.5
1
1.5
0 0
1
level set - step 15 x2
0.5
x1 0.5
1
1.5
0 0
x1 0.5
1
1.5
Fig. 3 Simulation results for the 2D “shape 2” vanishing example with n = 5 basis functions
As said, the first example deals with making level sets to vanish as fast as possible. In this case, we adopted the following cost functional to minimize: J=
T 0
Ω
Hˆ (φ(x, t))2 d x dt,
z 1 1 Hˆ (z) = + tanh 2 2 τ
where Hˆ is a smooth approximation of the Heaviside step function H (·) and τ is a positive constant [19]. In all the simulations, τ was fixed equal to 10−2 . We focused on two different shapes for the level sets, denoted as “shape 1” and “shape 2,” respectively. In both cases, the final time instant T was fixed to 1.5, the regular grid was composed of 50 × 75 nodes, and the interval [0, T ] was sampled with 50 subintervals of length Δt = 0.03. We chose suitable lower and upper bounds for the control input u(x, t), fixed to −0.5 and +0.5, respectively. Figures 2 and 3 show the results at selected time steps. In the first case, the initial ellipse vanishes at the time step 20, whereas in the second one, there is a change of topology of the front, with the level set vanishing at the time step 15. The second example concerns the tracking of a given reference curve over time. In other words, the goal is to find a control policy u(x, t) such that the zero level set of the function φ(x, t) tracks a reference front φref (x, t). Likewise in [1, 2], we considered the following cost function measuring the difference between the reference and actual level sets: T 2 Hˆ (φ(x, t))− Hˆ (φref (x, t)) d x dt, (14) J= 0
Ω
where Hˆ (·) is the previously-introduced approximation of the Heaviside step function. As in the vanishing example, in the 2D case we focused on two different shapes, denoted again with a little abuse of notation as “shape 1” and “shape 2,” respectively. In both cases, the final time instant T was fixed equal to 1.5, and a sampling time
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ref. level set - step 1 ref. level set - step 13 ref. level set - step 25 ref. level set - step 37 ref. level set - step 50 x2 0.5 x2 0.5 x2 0.5 x2 0.5 x2
0.5 0
0
x1
-0.5 -0.5
0.5
0
0
x1
-0.5 0.5
level set - step 1 x2
-0.5
0.5
0
0
0.5
level set - step 13 x2
0
x1
-0.5 -0.5
0
x1
-0.5
0.5
0
x1
-0.5
0
0.5
level set - step 25 x2
x1
-0.5
0.5
0
x1
-0.5
0
0.5
level set - step 37 x2
-0.5
0.5
0
0.5
level set - step 50 x2
0
x1
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x1
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0
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0
-0.5
0
-0.5 0.5
0
-0.5
-0.5
0
x1
-0.5 0.5
-0.5
0
0.5
Fig. 4 Simulation results for the 2D “shape 1” tracking example with n = 5 basis functions ref. level set - step 1 ref. level set - step 13 ref. level set - step 25 ref. level set - step 37 ref. level set - step 50 0.5 0.5 0.5 0.5 x2 x2 x2 x2 x2
0.5
0
0
-0.5 -0.5 0.5
x1 0
0.5
level set step 1 x2
-0.5 -0.5
-0.5 -0.5 0.5
0
0
x1 0
0.5
level set - step 13 x2
0
x1 0
0.5
-0.5 -0.5
0
-0.5 -0.5 0.5
x1 0
0.5
level set - step 25 x2
0
x1 0
0.5
-0.5 -0.5
0
-0.5 -0.5 0.5
x1 0
0.5
level set - step 37 x2
0
x1 0
0.5
-0.5 -0.5
-0.5 -0.5 0.5
x1 0
0.5
level set - step 50 x2
0
x1 0
0.5
-0.5 -0.5
x1 0
0.5
Fig. 5 Simulation results for the 2D “shape 2” tracking example with n = 5 basis functions
Δt = 0.03 was adopted. The mesh used for “shape 1” was a regular grid made up by 75 × 50 nodes, while for the “shape 2” we used 50 × 50 nodes. We constrained u(x, t) to lie in the ranges [0, 0.9] and [0, 0.3] for the “shape 1” and “shape 2,” respectively. Concerning the 3D case, we chose a final time T equal to 1.5, a regular grid of 90 × 60 × 60 points, and a sampling time Δt = 0.05. The lower and upper bounds for the control inputs were chosen equal to 0 and 0.2, respectively. Figures 4, 5, and 6 show the results. Notice that a change of topology occurs in the 2D “shape 2” example, as two ellipses join into a unique curve. In both the vanishing and tracking examples, we applied the ERIM with different numbers of basis functions, i.e., n = 5, 10, 15, and 20, to find approximate solutions by using both the exact adjoint-based gradient of the cost functionals and the gradient obtained by means of finite-difference approximations. The accuracy of the approximation was almost the same for all the considered cases, thus showing that n = 5 or n = 10 are convenient choices since they ensure a sufficient accuracy with quite simple approximating structures. Notice that the accuracy of the solutions using either the adjoint-based gradient and the numerical approximate one is quite similar. However, the use of the finite-difference gradient requires much more computational time. To this end, the simulation times in all the considered examples are reported in Table 1. The time required to perform the simulations using the adjoint-based gradient is about ten times lower than the corresponding time needed by the finite-difference gradient.
Optimal Control of Level Sets Generated by the Normal Flow Equation
39
Fig. 6 Simulation results for the 3D tracking example with n = 5 basis functions
Table 1 Summary of the mean simulation times (in seconds) n Adjoint-based gradient Finite-difference gradient Vanishing 2D shape 1
Vanishing 2D shape 2
Tracking 2D shape 1
Tracking 2D shape 2
Tracking 3D
5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20
1.48·102 1.98·102 2.55·102 3.27·102 1.28·102 1.51·102 1.89·102 2.90·102 5.65·101 1.48·102 1.78·102 2.24·102 7.44·102 4.38·102 1.26·103 1.04·103 2.39·103 2.60·103 2.35·103 2.05·103
1.46·103 3.54·103 6.29·103 7.94·103 1.08·103 2.68·103 5.01·103 7.29·103 3.80·103 7.38·103 1.66·104 2.09·104 7.67·103 5.44·104 1.32·104 8.73·104 6.35·104 6.37·104 7.47·104 7.95·104
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6 Conclusions Since the optimal control of fronts generated by NF level set methods cannot be solved analytically, an approach based on the ERIM has been developed to find computationally-tractable approximate solutions. The optimization of the cost functional has been performed subject to the NF dynamics as a constraint. The parameters of the approximate controller have been selected by descent techniques requiring the computation of the gradient of the cost with respect to the parameters of the control law, obtained by solving the related adjoint equations backwards in time. The proposed approach enables to solve problems not yet attacked because of both theoretical and practical difficulties. As a future work, we plan to improve the approach by reducing the optimization deficiency due to local minima trapping. Another direction of research will be the investigation of a cascade of dynamics governed by PDEs connected to front propagation, i.e., one or more equations describing a physical phenomenon in addition to the normal flow one.
References 1. A. Alessandri, P. Bagnerini, M. Gaggero, Optimal control of level sets dynamics in Proceedings of the American Control Conference (2014), pp. 2208–2213, 2. A. Alessandri, P. Bagnerini, M. Gaggero, A. Traverso, Further results on the optimal control of fronts generated by level set methods, in Proceedings of the American Control Conference (2016), pp. 5225–5230 3. A. Alessandri, M. Gaggero, R. Zoppoli, Feedback optimal control of distributed parameter systems by using finite-dimensional approximation schemes. IEEE Trans. Neural Netw. Learn. Syst. 23(6), 984–996 (2012) 4. L. Almeida, P. Bagnerini, A. Habbal, S. Noselli, F. Serman, A mathematical model for dorsal closure. J. Theor. Biol. 268(1), 105–119 (2011) 5. L. Almeida, P. Bagnerini, A. Habbal, Modeling actin cable contraction. Comput. Math. Appl. 64(3), 310–321 (2012) 6. A.R. Barron, Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. Inf. Theory 39(3), 930–945 (1993) 7. J.M. Berg, K. Holmström, On parameter estimation using level sets. SIAM J. Control Optim. 37(5), 1372–1393 (1999) 8. M.K. Bernauer, R. Herzog, Optimal control of the classical two-phase Stefan problem in level set formulation. SIAM J. Sci. Comput. 33(1), 342–363 (2011) 9. I. Ekeland, T. Turnbull, Infinite-Dimensional Optimization and Convexity (The University of Chicago Press, Chicago Lectures in Mathematics, 1983) 10. M. Haque, A. Rahmani, M. Egerstedt, A. Yezzi, Efficient foraging strategies in multi-agent systems through curve evolutions. IEEE Trans. Autom. Control 59(4), 1036–1041 (2014) 11. M. Hinze, S. Ziegenbalg, Optimal control of the free boundary in a two-phase Stefan problem. J. Comput. Phys. 223(2), 657–684 (2007) ◦ 12. V. Kurková, M. Sanguineti. Comparison of worst-case errors in linear and neural-network approximation. IEEE Trans. Inf. Theory 48(1), 264–275 (2002) 13. R. Kimmel, Numerical Geometry of Images (Springer, New York, 2004) 14. Level Set Methods and Fast Marching Methods, vol. 3, 2nd edn., Cambridge Monographs on Applied and Computational Mathematics (Cambridge University Press, Cambridge, 1999)
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15. I.M. Mitchell, The flexible, extensible and efficient toolbox of level set methods. J. Sci. Comput. 35(2–3), 300–329 (2008) 16. T. Parisini, R. Zoppoli, Neural networks for feedback feedforward nonlinear control systems. IEEE Trans. Neural Netw. 5(3), 436–449 (1994) 17. T. Parisini, R. Zoppoli, Neural approximations for multistage optimal control of nonlinear stochastic systems. IEEE Trans. Autom. Control 41(6), 889–895 (1996) 18. A. Quarteroni, Numerical Models for Differential Problems (Springer, Milan, Italy, 2009) 19. I. Yang, C.J. Tomlin, Identification of surface tension in mean curvature flow, in Proceedings of the American Control Conference (2013), pp. 3290–3295 20. I. Yang, C.J. Tomlin, Regularization-based identification for level set equations, in Proceedings of the IEEE Conference on Decision and Control (2013), pp. 1058–1064 21. R. Zoppoli, T. Parisini, Learning techniques and neural networks for the solution of N-stage nonlinear nonquadratic optimal control problems, in Systems, Models and Feedback: Theory and Applications, ed. by A. Isidori, T.J. Tarn (Birkhäuser, Boston, 1992), pp. 193–210 22. R. Zoppoli, M. Sanguineti, T. Parisini, Approximating networks and extended Ritz method for the solution of functional optimization problems. J. Optim. Theory Appl. 112(2), 403–440 (2002)
Emergent Dynamics for the Kinetic Kuramoto Equation Debora Amadori and Jinyeong Park
Abstract In this note, we study the emergent dynamics of the kinetic Kuramoto equation, which is a mean-field limit of the Kuramoto synchronization model. For this equation, also referred to as the Kuramoto–Sakaguchi equation Lancellotti (Transp Theory Stat Phys 34:523–535, 2005 [13]), we present two approaches for the analysis on its dynamics. First, for the system of identical oscillators, we apply a wave-fronttracking algorithm which is used for scalar conservation laws. This method gives a quantitative estimate on the approximate BV solution to the kinetic model Amadori et al. (J Differ Equ 262:978–1022, 2017, [2]). Second, we study the emergence of phase concentration phenomena by directly analyzing the dynamics of the order parameters. This can show the asymptotic behavior of the system from generic initial data Ha et al. (J Park, 2016, [8]). Keywords Kinetic Kuramoto model · Approximate BV solutions Large time behavior
1 Introduction In this section, we illustrate the Kuramoto model and its mean-field limit, the Kuramoto–Sakaguchi equation, and study their basic properties. We refer to [11, 12] for the original formulation and to [1] for a general review of the Kuramoto model.
D. Amadori (B) University of L’Aquila, L’Aquila, Italy e-mail: [email protected] J. Park Universidad de Granada, Granada, Spain e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_4
43
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1.1 The Kuramoto Model for N Oscillators We consider a set of N coupled oscillators of the same amplitude, and represent each oscillator by using polar coordinates, i.e., by its phase θi with i = 1, . . . , N . In the Kuramoto model [11, 12], the dynamics of oscillators are governed by the following first-order system of ODEs: θ˙i = Ωi +
N K sin(θj − θi ), for i = 1, · · · , N N j=1
(1)
where Ωi is a natural frequency of the i-th oscillator and K > 0 is a coupling strength. When the oscillators are decoupled, i.e., K = 0, each oscillator runs along the unit circle individually with the frequency Ωi . By taking summation on both sides of (1), we can attain N N θ˙i = Ωi i=1
i=1
due to the odd interaction function. Thus, when the synchronization occurs, i.e., all of the frequencies become to coincide, the synchronized frequency is determined to be the average of natural frequencies. With out loss of generality, we may assume that the average of the natural frequencies is zero, Ni=1 Ωi = 0. Consider the positions of N oscillators on the unit circle, zj = eiθj . The Kuramoto order parameters are defined by means of the centroid of oscillators: Reiφ :=
N 1 iθj e . N j=1
(2)
Here R and φ represent, respectively, amplitude and phase of the average position of the N oscillators. We multiply (2) by e−iθi on both sides and compare the imaginary parts to attain N 1 R sin(φ − θi ) = sin(θj − θi ). (3) N j=1 Using the relation (3), the Kuramoto model can be rewritten into the following form: θ˙i = Ωi + KR sin(φ − θi ) , i = 1, · · · , N .
(4)
Under suitable assumptions on the coupling strength K, the Ωi and the initial configuration for the θi , in [10] the authors proved the emergence of synchronization for (4), that is, the following long-time behavior property: lim max |θ˙i − θ˙j | = 0 .
t→∞ 1≤i,j≤N
Emergent Dynamics for the Kinetic Kuramoto Equation
45
The proofs are based on the dynamics of order parameters. We refer to [9] for a review on the recent progresses in the Kuramoto model and on other synchronization models.
1.2 The Kinetic Kuramoto Equation When the number N of oscillators is very large, it is efficient to describe the system with a kinetic equation. Let f = f (θ, Ω, t) be a probability density function. By the BBGKY hierarchy argument, one can “pass to the limit” as N → ∞ in system (1) and obtain its continuous version: ∂t f + ∂θ (w[f ]f ) = 0, w[f ] := Ω − K sin(θ − θ∗ )ρ(θ∗ , t) d θ∗ ,
(5)
T
where ρ(θ, t) := f (θ, Ω, t) d Ω , R f (θ, Ω, t) d θ = g(Ω) .
T
ρ(θ, t) d θ = 1 ,
T := R/2π Z ,
T
Here g(Ω) represents the density function for the natural frequencies. By the periodicity in θ of f , it follows that g is independent of time. The rigorous derivation of (5) is presented by Lancellotti [13] using Neunzert’s method [14], and in [7]. In [13], it is shown the existence of classical solution to (5). Similarly to (2), we define the order parameters R and φ for the kinetic Kuramoto equation by taking the centroid of the oscillators whose distribution over the unit circle is given by ρ(θ, t): Reiφ :=
T×R
eiθ f (θ, Ω, t) d θ d Ω =
T
eiθ ρ(θ, t) d θ .
(6)
With similar argument in (3), we attain R sin(φ − θ ) =
T×R
sin(θ∗ − θ )f (θ∗ , Ω∗ , t) d θ∗ d Ω∗ =
T
sin(θ∗ − θ)ρ(θ∗ , t) d θ∗ ,
with R = R(t), φ = φ(t). The Eq. (5) can be rewritten as ∂t f + ∂θ Ωf − KRf sin(φ − θ ) = 0.
(7)
We refer to [4] for a numerical study and a stability analysis of (5), and to [5, 6] for the analysis of measure-valued solutions.
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2 BV Solutions and the Front-Tracking Algorithm 2.1 Identical Oscillators In this section, we study the existence of BV solutions to (5) for identical oscillators g(Ω)d Ω = δ(Ω). In this situation, f has the following form: f (θ, Ω, t) = f¯ (θ, t) ⊗ δ(Ω) for some f¯ , with ρ = f¯ . The Kuramoto–Sakaguchi equation for identical oscillators reads finally as
∂t ρ − K∂θ (L[ρ]ρ) = 0, θ ∈ T, t > 0, L[ρ(·, t)](θ ) = T sin(θ − θ∗ )ρ(θ∗ , t) d θ∗ .
(8)
Alternatively, from (7) we obtain L[ρ](θ ) = R sin(θ − φ) ,
∂t ρ − ∂θ KRρ sin(θ − φ) = 0.
(9)
After a time rescaling, one can assume that K = 1. Another way to express the integral term L[ρ] is L[ρ](θ ) =
1 iθ e z¯ − e−iθ z , 2i
z=
T
eiθ ρ(θ, t) d θ .
As a consequence, T
ρ(θ, t)L[ρ](θ ) d θ =
1 (z¯z − z¯ z) = 0 . 2i
(10)
Front-tracking approximations. Here we present a procedure to define approximate solutions to (8), provided with a BV , 2π -periodic initial data ρ0 ≥ 0. Let (θ (t), ρ(t)) = (θ (t; θ¯ ), ρ(t; ρ)) ¯ be a forward characteristics associated with (8), that is, a solution to the characteristic system: θ˙ = −L[ρ], ρ˙ = ρ(∂θ L[ρ]), t > 0, ¯ ρ). (θ, ρ)(0) = (θ, ¯
(11)
In the sequel, we construct the approximate solution by means of the characteristic system (11). • Step A (Sampling of the initial data): Let N ∈ N and choose 0 ≤ θ01 < θ02 < . . . < θ0N < 2π .
(12)
Emergent Dynamics for the Kinetic Kuramoto Equation
47
We set ρ0j , the initial value of ρ in the interval (θ0j , θ0(j+1) ), as follows: ρ0j := ρ0 (θ0j +) ,
j = 1, . . . , N .
(13)
Finally we set θ0(N +1) = θ01 + 2π , θ00 = θ0N − 2π . For the accuracy of the approximation, we introduce a parameter ε > 0 and require that max ρj (0)|θj+1 (0) − θj (0)| < ε . (14) max |θj+1 (0) − θj (0)| < ε , j
j
The second condition implies the local accuracy of the L1 norm. • Step B (Approximate solutions): We define an approximate solution ρ N = ρ(θ, t) on a periodicity interval (θ0 (t), θN (t)) as a function of the form ρ(θ, t) =
N −1
ρj (t)χ(θj (t),θj+1 (t)) (θ ) ,
θ ∈ (θ0 (t), θN (t)) ,
j=0
where the ρj and θj satisfy the following discrete system of characteristic equations: θ˙j = −L[ρ](θj ), t > 0, L[ρ](θj+1 ) − L[ρ](θj ) ρ˙j = ρj θj+1 − θj
(15)
subject to initial data: θj (0) = θ0j ,
ρj (0) = ρ0j ,
j = 0, . . . , N − 1 .
Approximations that preserve stationary solutions. Here we give a definition of the ρ0j , alternative to the one in (13), that has the property of preserving the centroid of the initial datum ρ0 , see (6), and hence the order parameters R(0), φ(0). More precisely, we claim that for any integer N ≥ 2 there exists a piecewise constant function ρN such that ρN − ρ0 L1 (T) → 0 as N → ∞ and that
e ρ0 (θ ) d θ = iθ
T
T
eiθ ρN (θ ) d θ .
(16)
In particular, let ρ0 (θ ) = ρ(θ ) be a bounded, stationary solution to (8), with positive mass; from (8), this is equivalent to ρ(θ )L[ρ](θ ) = const. = 0, because of (10). Therefore, since ρ has no point masses, one has L[ρ](θ ) ≡ 0 and then R = 0. The approximate initial data ρN constructed below has the the same property, and hence L[ρN ] = 0. Therefore, from (15), ρ˙j (0) = θ˙j (0) = 0 and hence the approximate solution is stationary as well. To construct a piecewise constant initial data with the property (16), we proceed as follows:
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• Let ρ E (θ ) and ρ O (θ ) be the even and odd part of ρ, that is ρ(θ ) + ρ(−θ ) , 2
ρ E (θ ) =
ρ O (θ ) =
ρ(θ ) − ρ(−θ ) . 2
Let N ≥ 2 be a positive integer and θ0 = 0 < θ1 < . . . < θN = π be a partition of [0, π ], with π/2 belonging to the partition. Then define, for j = 0, . . . , N − 1: θj+1 ρjE
θj
=
cos θρ E (θ ) d θ
θj+1 θj
cos θ d θ
θj+1 ρjO
,
θj
=
sin θρ E (θ ) d θ
θj+1 θj
sin θ d θ
.
(17)
Since (θj , θj+1 ) is contained either in [0, π/2] or in [π/2, π ], the denominators above differ from 0. • The corresponding functions, N −1
N −1
ρjE χ(θj ,θj+1 ) (θ ) ,
j=0
ρjO χ(θj ,θj+1 ) (θ )
j=0
are extended on (−π, π ) as a even function ρNE and a odd function ρNO , respectively, and then by periodicity on R. Let ρN = ρNE + ρNO be their sum. Using the definition (17), one can find that
T
ρN (θ ) cos θ d θ = =2
N −1 j=0 π
T
ρjE
ρNE (θ ) cos θ d θ
θj+1 θj
cos θ d θ
ρ E (θ ) cos θ d θ =2 0 = ρ E (θ ) cos θ d θ = ρ(θ ) cos θ d θ ; T
T
similarly one has that T ρN (θ ) sin θ d θ = T ρ(θ ) sin θ d θ . Hence (16) holds. Moreover, if δN = maxj (θj+1 − θj ) → 0 as N → ∞, then the piecewise constant function ρN converges to ρ0 in L1 (T). The claim is proved. Properties of the approximations and convergence. Let I (t) be the L1 norm of ρ(·, t) on the periodic domain T, that is I (t) := ρ(·, t) L1 (T) =
N −1
ρj θj+1 − θj
j=0
We list some properties of the approximations in the next Proposition.
Emergent Dynamics for the Kinetic Kuramoto Equation
49
−1 Proposition 1 ([2], Lemma 3.1 and 3.3). Let {(ρj (t), θj (t))}Nj=0 be a solution to the characteristic system (15). Then, the following assertions hold:
1. The set of discontinuities satisfies θ0 (t) < θ1 (t) < θ2 (t) < · · · < θN (t), ∀ t > 0. 2. The local masses are conserved: ρj (t) θj+1 (t) − θj (t) = ρj (0) θj+1 (0) − θj (0) , j = 0, . . . , N − 1 and hence the total mass is conserved: I (t) = I (0) . 3. The following lower and upper bounds hold, ρj (0)e−I (0)t ≤ ρj (t) ≤ ρj (0)eI (0)t ,
∀j, t ≥ 0.
In particular, if minj ρj (0) > 0, then minθ ρ(θ, t) > 0 for every t > 0. 4. The following BV estimate holds, d TVρ(·, t) ≤ b(t) (TVρ(·, t) + 2) , dt
b(t) := min{I (0), ρ(·, t) −
1 1 }. 2π L (T)
Theorem 1 ([2]). Let ρ0 : R → [0, ∞) ∈ BVloc , 2π -periodic, with T ρ0 d θ = 1. Then the Cauchy problem for (8) with initial data ρ0 admits a unique weak solution ρ(θ, t) ≥ 0, with ρ(·, t) ∈ BVloc , 2π -periodic, T ρ(θ, t) d θ = 1. The BV seminorm TVρ(·, t) can increase exponentially as t → ∞. The solution ρ is obtained as the strong limit of front-tracking approximations. Indeed one can prove the following. Let εn be any sequence → 0+ and assume that N = N (εn ), the θ0j in (12), and the ρ0j satisfy (14) with ε = εn . Then the corresponding sequence of approximates solutions, constructed by the previous algorithm, converges in L1loc to the weak solution ρ, by using a compactness argument in BV (for finite intervals of time) combined with the uniqueness of the limit.
2.2 Nonidentical Oscillators In this section, we consider Eq. (5) for a bimodal distribution function for g(Ω)d Ω, as considered in [3, Sect. 3]: g(Ω)d Ω = m− δ−Ω¯ + m+ δΩ¯ ,
¯ m± > 0, m− + m+ = 1. Ω,
In this case, recalling that the distribution gd Ω is preserved in time, the distribution function f takes the form
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f (θ, Ω, t) = m− f − (θ, t) ⊗ δ−Ω¯ + m+ f + (θ, t) ⊗ δΩ¯ ,
T
f ± (θ, t) d θ = 1 ,
and the density ρ(θ, t) is expressed by ρ(θ, t) =
R
f (θ, Ω, t)d Ω = m− f − (θ, t) + m+ f + (θ, t)
= ρ − (θ, t) + ρ + (θ, t) where ρ ± are the partial densities ρ − (θ, t) := m− f − (θ, t),
ρ + (θ, t) := m+ f + (θ, t) .
¯ Eq. (5) reduces to For Ω = ±Ω, ¯ θ f ± − K∂θ (L[ρ]f ± ) = 0 , ∂t f ± ± Ω∂ with L[ρ] being the convolution term given in (8). By the linearity of the operator L, one has L[ρ(·, t)] = m+ L[f + ] + m− L[f − ] = L[ρ + ] + L[ρ − ] . ¯ K as By a rescaling of time, we can set the parameters Ω, = Ω
Ω¯ > 0, K
= 1. K
Hence, the local density functions ρ ± satisfy the coupled 2 × 2 system: θ ρ − − ∂θ (L[ρ]ρ − ) = 0, (θ, t) ∈ T × R+ , ∂t ρ − − Ω∂ θ ρ + − ∂θ (L[ρ]ρ + ) = 0 , ∂t ρ + + Ω∂ the two equations being coupled through the term L[ρ] = L[ρ + ] + L[ρ − ]. Initial data ρ0± are provided for ρ ± . In this context, front-tracking approximations are suitably defined as follows. As in Step A for identical oscillators, one chooses two partitions of [0, 2π ), {θ0j± }Nj=1 ⊂ [0, 2π ) ,
± θ0j± < θ0(j+1) ,
± ± θ0(N +1) = θ01 + 2π ,
for N ∈ N, and piecewise constant approximations of the initial data ρj± (0) ≥ 0, so that the approximate initial data is represented by ρ ± (θ, 0) =
N j=1
ρj± (0)χ θ ± ,θ ± 0j
0(j+1)
(θ ) ,
± ± θ ∈ θ01 , θ01 + 2π .
Emergent Dynamics for the Kinetic Kuramoto Equation
51
The approximate solution, on a periodicity interval (θ1± (t), θ1± (t) + 2π ), writes as ρ ± (θ, t) =
N
± ρj± (t)χIj± (θ ) , Ij± = θj± (t), θj+1 (t) ,
(18)
j=1
which is extended by periodicity to every θ ∈ R, and L[ρ] is set by L[ρ](θ ) = L[ρ + ](θ ) + L[ρ − ](θ ), with L[ρ ± ](θ ) = sin(θ − θ∗ )ρ ± (θ∗ , t) d θ∗ . T
Here, θj± and ρj± satisfy the dynamical system − L[ρ](θj+ ), θ˙j+ = Ω − L[ρ](θj− ), θ˙j− = −Ω ρ˙j+ = ρj+ ρ˙j− = ρj−
+ L[ρ](θj+1 ) − L[ρ](θj+ ) + θj+1 − θj+ − L[ρ](θj+1 ) − L[ρ](θj− ) − θj+1 − θj−
,
(19)
.
As a immediate consequence, a solution of (19) satisfy the local mass conservation: ρ˙j± = −ρj±
± θ˙j+1 − θ˙j± ± θj+1 − θj±
⇒
d ± ± ρj θj+1 − θj± = 0 . dt
Notice that the mass in between two consecutive θj± s can be zero, and this occurs if ρj± (0) = 0 (= ρj± (t)). As for the identical case, the location of the discontinuities of ρ + and ρ − does not intersect for any positive time, that is − (t) , θj− (t) < θj+1
+ θj+ (t) < θj+1 (t) ,
j = 1,... ,N ,
and the ρj± (t)s satisfy a time-exponential upper and lower bound ([3], Lemmas 3 and 4). In the following Lemma, we establish a bound on the total variation of ρ for every bounded time interval. Lemma 1. Let (ρi± (t), θi± (t)) be a solution to system (19) and ρ ± be given in (18). Let ρ = ρ + + ρ − . Then, for t ≥ 0 and I0 = ρ(0) L1 (T) , we have following estimate: TV ρ(·, t) ≤ eI0 t TV ρ(·, 0) + eI0 t − 1 .
(20)
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Proof. Let’s define ±
±
T (t) := TV {ρ (·, t); T} =
N
± |ρi+1 − ρi± | .
i=1
To estimate the derivative of T ± , we introduce the notation ΔL L[ρ](θ ) − L[ρ](θ˜ ) (θ, θ˜ ) := , Δθ θ − θ˜
θ = θ˜
and use the mean value theorem to define ξi+ , ΔL + (θ , θ + ) = ∂θ L[ρ](ξi+ ) , Δθ i+1 i
+ . ξi+ ∈ θi+ , θi+1
Now, let’s compute + ρ˙i+1 − ρ˙i+ + + = ρi+1 · ∂θ L[ρ](ξi+1 ) − ρi+ · ∂θ L[ρ](ξi+ ) + + = ρi+1 − ρi+ ∂θ L[ρ](θi+1 ) (21) + + + + + + + ρi+1 · ∂θ L[ρ](ξi+1 ) − ∂θ L[ρ](θi+1 ) − ρi · ∂θ L[ρ](ξi ) − ∂θ L[ρ](θi+1 ) . (22)
By definition of L, (8), one has that max ∂θk L[ρ](θ ) ≤ θ
T
ρ(θ, t) d θ =: I0 = I0+ + I0− ,
k = 1, 2, . . .
where I0± = ρ ± (0) L1 (T) . Then we can evaluate the terms in (21)–(22) and get: + + + (23) − ρi+ ) ρ˙i+1 − ρ˙i+ ≤ I0 ρi+1 − ρi+ sgn(ρi+1 + + + + + + + ρi+1 · ξi+1 − θi+1 + ρi · ξi − θi+1 . By summing up the terms in (23), we find that + d + + T (t) = sgn(ρi+1 − ρi+ ) ρ˙i+1 − ρ˙i+ dt i=1 N
≤ I0 T + (t) +
N i=1
+ + + ξi − θi+ ρi+ · ξi+ − θi+1
Emergent Dynamics for the Kinetic Kuramoto Equation
= I0 T + (t) + +
= I0 T (t) +
N
53
+ ρi+ · θi+1 − θi+
i=1 I0+ .
A completely analogous estimate holds for T − . Summing up, we obtain d + T (t) + T − (t) ≤ I0 T + (t) + T − (t) + I0 , dt so that TV ρ(·, t) ≤ T + (t) + T − (t) ≤ eI0 t T + (0) + T − (0) + eI0 t − 1 . Finally, possibly slightly changing the location of the θ0j± to be all distinct, we have T + (0) + T − (0) = TV ρ(·, 0) and hence (20). This completes the proof.
3 Phase Concentration of the Kuramoto–Sakaguchi Equation In this section, we present the phase concentration phenomena by using the dynamics of the order parameters defined in (6).
3.1 Identical Oscillators In this part, we study the emergence of point attractor for the identical Kuramoto– Sakaguchi equation by analyzing the dynamics of order parameters. We take time derivatives on (6) and apply (9) with integration by parts. Then, we attain the following dynamics for the order parameters. Lemma 2 ([8]). Let ρ be a solution to (9) and let R and φ be the order parameters defined by the relation (6). Then, R and φ satisfy (i) R˙ = KR
T 2
sin2 (θ − φ)ρ(θ, t) d θ, φ˙ = −
˙ (R) ˙ 2 − 2(KR)2 + 2R(φ) (ii) R¨ = R
T
K 2
T
sin (2(θ − φ)) ρ(θ, t) d θ.
sin2 (θ − φ) cos(θ − φ)ρ(θ, t) d θ.
It is easy to see that R˙ ≥ 0 for any initial data. We next divide the spatial domain T into disjoint intervals and observe the change of mass in each intervals. For a positive constant δ > 0, we define a time varying set Jδ := Jδ+ ∪ Jδ− (see Fig. 1):
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Fig. 1 Geometric descriptions of Jδ+ and Jδ−
Jδ+ (t) := {θ ∈ T : |θ − φ(t)| < δ}, Jδ− (t) := {θ ∈ T : |θ − (φ(t) + π )| < δ}. Theorem 2 ([8]). Suppose that the coupling strength, the density function g and the initial datum satisfy K > 0, g(Ω)d Ω = δ(Ω), ρ0 ∈ C , 1
T
ρ0 d θ = 1 and R0 := R(0) > 0.
Then, for a classical solution ρ : T × R+ → R to (8), the mass concentrates around the average phase φ(t) asymptotically. More precisely, for any δ > 0, there exists T1 = T1 (δ) ≥ 0, such that ρ(θ, t) d θ = 1 and lim t→∞ J + (t) δ R(0)(cos δ) |ρ(θ, t)|2 d θ ≤ e− 2 K(t−T1 ) |ρ(θ, T1 )|2 d θ ∀ t ≥ T1 . Jδ− (t)
Jδ− (T1 )
Here, we provide a brief sketch of the proof for Theorem 2 which consists of two steps: First, we show that the asymptotic mass concentration on the interval Jδ , i.e., for any δ ∈ (0, π/2), we attain lim
t→∞ J δ
ρ(θ, t)d θ = 1.
(24)
From Lemma 2, we have R˙ → 0 as t → 0 (see Proposition 4.1 in [8]) and, for any ε > 0, there exists a finite time t∗ > 0 such that sin2 (θ − φ(t))ρ(θ, t) d θ ≤ (sin δ)2 ε, t ≥ t∗ . T
Emergent Dynamics for the Kinetic Kuramoto Equation
55
Since | sin(θ − φ(t))| > sin δ for θ ∈ T \ Jδ , we obtain the following estimate:
(sin δ)2
T\Jδ
ρ(θ, t) d θ < ≤
T\Jδ
T
sin2 (θ − φ(t))ρ(θ, t) d θ
sin2 (θ − φ(t))ρ(θ, t) d θ ≤ (sin δ)2 ε, t ≥ t∗ ,
which proves (24). Second, we show the exponential decay of mass in the interval Jδ− which is antipodal to φ. We next introduce the Lyapunov functional:
Λ(t) :=
Jδ− (t)
|ρ(θ, t)|2 d θ =
φ+π+δ
|ρ(θ, t)|2 d θ.
φ+π−δ
Then, we can attain that the functional Λ(t) satisfies Gronwall’s inequality: d Λ(t) ≤ −R(0) cos δKΛ(t), t ≥ T1 , dt which shows the exponential decay of mass in the interval Jδ− . We refer [8] for the detailed computation. Concentration of mass around φ has been proved in [5, Theorem 3.1] by a different argument without the exponential decay estimate of mass in the interval Jδ− .
3.2 Nonidentical Oscillators In this part, we study the dynamics of nonidentical oscillators. By the similar calculations with identical case, we can attain the following dynamics for the order parameters. Lemma 3 ([8]). Let f = f (θ, ω, t) be a solution to (5), and (R, φ) be order parameters defined by (6). Then, we have R˙ = − φ˙ =
1 R
T×R
sin(θ − φ)Ωf (θ, Ω, t) d θ d Ω + KR
K cos(θ − φ)Ωf (θ, Ω, t) d θ d Ω − 2 T×R
T
T
sin2 (θ − φ)ρ(θ, t) d θ. sin (2(θ − φ)) ρ(θ, t) d θ.
For the nonidentical case, we define time varying interval Lγ (t) around φ such that
π π Lγ (t) := φ(t) − + γ , φ(t) + − γ = J π+−γ (t) 2 2 2 for a positive constant γ (Fig. 2).
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Fig. 2 Geometric descriptions of Lγ (t)
To show the emergence of phase concentration, we provide two steps of arguments: First, we show the preservation of mass concentration when it is already confined initially in an interval. Second, we will attain the transition from generic to C 1 initial configuration
√
confinement into an interval. 0 , we set For ε0 ∈ 0, 3 4 3 − 1 and π3 ≤ γ0 < arcsin 1 − 2√2ε3+1 M∗ (ε0 , γ0 ) :=
2 + ε0 + cos γ0 . (1 + sin γ0 )(1 + cos γ0 )
The following result explains that the dynamics preserves the concentration if the initial mass in this interval is greater than M∗ (ε0 , γ0 ). Theorem 3 ([8]). Suppose that the following conditions hold. 1. The frequency density function g = g(Ω) and coupling strength K satisfies M 1 supp g(Ω) ⊂ (−M , M ), K > 1+ . ε0 ε0 2. Suppose that initial datum f0 satisfies (i) f0 (θ, Ω) = 0 in T × (R\[−M , M ]), ||f0 ||L∞ < ∞, (ii) f (θ, Ω, 0) d θ d Ω ≥ M∗ (ε0 , γ0 ), Lγ0 (0)×R
Then, for any C 1 -solution to (5), there exists a time-dependent interval Lγ0 (t) ⊂ T centered around φ(t), with fixed width such that d dt
Lγ0 (t)×R
f (θ, Ω, t) d θ d Ω ≥ 0 ,
(25)
Emergent Dynamics for the Kinetic Kuramoto Equation
57
Lγ0 (t)
|f (θ, Ω, t)|2 d θ ≥ CeCt ,
Ω ∈ supp g(Ω).
(26)
The first result of Theorem 3, (25), shows the non-decreasing mode of mass concentration in the interval Lγ0 (t). The second result, (26), implies the emergence of spike-like distribution for each conditional distribution in the interval Lγ0 (t). The next theorem shows the emergence of phase concentration for generic initial configuration. Theorem 4 ([8]). Let f : T × R × R+ → R be a classical solution to (5). Suppose g is supported on the interval [−M , M ], R(0) := R0 > 0, and K is sufficiently large (depending on the support of g and 1/R0 ). Then, lim inf R(t) ≥ R∞ t→∞
M − := 1 + K
M2 M +4 2 K K
and lim ft 1T\L∞ (t) L∞ (T×R) = 0.
t→∞
Here, L∞ (t) ⊂ T is a time-dependent interval, centered at φ(t) with the constant width M (1 + R∞ ) 1 − R∞ 2 + ε, 1− + arccos K R2∞ R∞ where ε is an arbitrary constant in (0, 1). Notice that as K → ∞ the width of L∞ can be made arbitrarily small and R∞ tends to 1. The large coupling strength K controls the amplitude R to be away from zero, and this leads the phase transition into the interval. Since the proof of Theorem 4 needs complicated steps, we refer [8] for the detailed description.
References 1. J.A. Acebron, L.L. Bonilla, C.J.P. Pérez Vicente, F. Ritort, R. Spigler, The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137–185 (2005) 2. D. Amadori, S.-Y. Ha, J. Park, On the global well-posedness of BV weak solutions for the Kuramoto–Sakaguchi equation. J. Differ. Equ. 262, 978–1022 (2017) 3. D. Amadori, S.-Y. Ha, J. Park, in Innovative Algorithms and Analysis A nonlocal version of wave-front tracking motivated by the Kuramoto–Sakaguchi equation (Springer INdAM Series, 2017) 4. N.J. Balmforth, R. Sassi, A shocking display of synchrony. Phys. D 143, 21–55 (2000) 5. D. Benedetto, E. Caglioti, U. Montemagno, On the complete phase synchronization for the Kuramoto model in the mean-field limit. Commun. Math. Sci. 13, 1775–1786 (2015) 6. J.A. Carrillo, Y.-P. Choi, S.-Y. Ha, M.-J. Kang, Y. Kim, Contractivity of transport distances for the kinetic Kuramoto equation. J. Stat. Phys. 156, 395–415 (2014)
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7. H. Chiba, Continuous limit of the moments system for the globally coupled phase oscillators. Discret. Contin. Dyn. Syst. 33, 1891–903 (2013) 8. S.-Y. Ha, Y.-H. Kim, J. Morales, J. Park, Emergence of phase concentration for the Kuramoto– Sakaguchi equation, https://arxiv.org/abs/1610.01703 (submitted) 9. S.-Y. Ha, D. Ko, J. Park, X. Zhang, Collective synchronization of classical and quantum oscillators. EMS Surv. Math. Sci. 3(2), 209–267 (2016) 10. S.-Y. Ha, H.K. Kim, J. Park, Remarks on the complete synchronization of Kuramoto oscillators. Nonlinearity 28, 1441–1462 (2015) 11. Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer, Berlin, 1984) 12. Y. Kuramoto, International Symposium on Mathematical Problems in Mathematical Physics, Lecture notes in theoretical physics, vol. 30 (1975), p. 420 13. C. Lancellotti, On the Vlasov limit for systems of nonlinearly coupled oscillators without noise. Transp. Theory Stat. Phys. 34, 523–535 (2005) 14. H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, in Kinetic theories and the Boltzmann equation. Lecture Notes in Mathematics, vol. 1048 (Springer, Berlin)
A Hyperbolic Model of Nonequilibrium Phase Change at a Sharp Liquid–Vapor Interface Matthieu Ancellin, Laurent Brosset and Jean-Michel Ghidaglia
Abstract Our aim is to simulate numerically the impact of a breaking wave on a wall when phase change between liquid and gas happens (Ancellin et al. Proceedings of the 26th International Offshore and Polar Engineering Conference 886–893, 2016 [2]). We thus need to model two compressible phases separated by a sharp interface allowing mass exchange. It leads us to extend the Euler conservation equations to a hyperbolic system of balance laws including nonequilibrium phase change. For real fluids, when the value of the latent heat is high, there might be no solution to the Riemann problem. However we are able to discretize it in the finite volume framework using the Roe-type scheme of Ghidaglia et al. (Eur J Mech B/Fluids 20:841–867, 2001 [4]), the numerical diffusion playing the stabilizing role of the thermal diffusion. Keywords Two-phase flow · Phase change Hyperbolic System of Conservation Laws · Non conservative product · Sloshing
1 Context For the transportation of Liquefied Natural Gas (LNG) in tanks on floating structures, such as LNG carriers, one of the main technologies today is to store the LNG at ambient pressure and cryogenic temperature using membrane containment system completely covering the inner sides of the tank walls. For strong see states, the motion of the ship may cause significant motions of the liquid in the tank, which may M. Ancellin (B) · L. Brosset GTT (Gaztransport & Technigaz), Saint-Rémy-lès-Chevreuse, France e-mail: [email protected] L. Brosset e-mail: [email protected] M. Ancellin · J.-M. Ghidaglia CMLA ENS Cachan CNRS Université Paris-Saclay, Cachan, France e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_5
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induce violent impacts on the containment system. While designing such containment systems, GTT has to ensure that these impacts may not cause any damage during the whole life of the floating structure. The typical approach for a sloshing assessment is based on sloshing model tests, with a model tank, typically at scale 1:40, filled with water and a heavy gas and placed on the platform of a hexapod. However part of the physics of a real wave impact is not modeled in such an experiment. In particular, the LNG in the tank is at thermodynamic equilibrium with its vapor phase, and thus it may condensate or evaporate during an impact. The experimental or numerical studies taking this effect into account are seldom. The final goal of this work is to add a phase change model inside numerical simulations of a breaking wave impact to better understand its influence. Modeling Framework A real impact may involve a complex flow of bubbles, droplets, and free surface instabilities. However, in the current state of the art for the numerical simulation of breaking wave impacts, they are not taken into account. The breaking wave is modeled as a relatively smooth sharp interface separating two pure phases (see for instance [3, 5]). For this work, we restrict ourselves to this framework. Since the liquid impact is a brief and violent phenomenon (typically of the order of the millisecond), thermodynamic equilibrium hypotheses at the liquid–vapor interface might not be relevant. Thus, our objective is the numerical simulation of nonequilibrium phase change between two compressible fluids at a sharp liquid–vapor interface. This paper will present some details on our approach to cope with this challenge.
2 Mathematical Model In this section, we will discuss mathematical modeling of the two-phase flow with interfacial phase change.
2.1 Conservation Equations We have chosen to describe the two phases with the help of an order parameter χ (x, t) describing the local phase (for instance, χ = 1 for the gas and χ = 0 for the liquid). The conservation equations for two nonmiscible species separated by a permeable free surface can be written as [1]:
A Hyperbolic Model of Nonequilibrium Phase…
61
∂t ρ + ∇ · (ρu) = 0,
(1a)
∂t (ρu) + ∇ · (ρu ⊗ u + p) = 0, ∂t (ρ E) + ∇ · ((ρ E + p) u) = 0, ∂t (χρ) + ∇ · (χρu) − J · ∇χ = 0,
(1b) (1c) (1d)
where ρ, u, p, and E = e + |u|2 /2 denote respectively the density, the velocity, the pressure, and the specific total energy of the fluid. Equation (1d) is the gas mass balance law. The phase change is modeled by its last term J · ∇χ , where J is the interfacial surfacic mass flux oriented by the normal vector at the interface. Equation (1d) can be rewritten with the help of (1a) to get
J ∂t χ + u − ρ
· ∇χ = 0.
(2)
The order parameter χ is the solution of an advection equation with velocity u − ρJ . In case of phase change (J = 0), the position of the free surface does not move according to the material velocity u but is also shifted by the local change of phase. An equation of state is needed to close the system. For instance, it can take the form p = p(ρ, e, χ ). (3) We have introduced one more variable in the equations and thus we need one more closure relation. It can take the form of an expression for J relating the interfacial mass flux to the local thermodynamical state: J = Jl→g ( pg , Tg , pl , Tl ) νl→g ,
(4)
where Jl→g is an expression for the evaporation mass flux as a function of the pressures and temperatures pg , Tg , pl and Tl (respectively in the gas and the liquid). The local unit normal vector at the interface νl→g is oriented from the liquid side to the gas side. The expression for Jl→g needs to be compliant with the Second Principle of Thermodynamics to ensure the physical relevance of the model (and thus the good behavior of the numerical code). This point will not be discussed in more depth in this paper.
2.2 Hyperbolicity Theorem 1. For an equation of state such that
∂p ∂ρ s,χ
> 0, the system (1) is
hyperbolic. Proof. For the sake of readability, only the 1D case will be presented here. The general case can be found in [1].
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The system can be written as ∂t v + A(v)∂x v = 0. The advection matrix A reads ⎛ ⎞ 0 1 0 0 ⎜ ⎟ ⎜ ⎟ ⎜ c2 − u 2 − (H − u 2 )Γ − χ pχ (2 − Γ )u Γ pχ ⎟ ⎜ ⎟ ⎟, A=⎜ ⎜ 2 ⎟ 2 2 ⎜(c − H − (H − u )Γ − χ pχ )u H − Γ u (1 + Γ ) u pχ u ⎟ ⎜ ⎟ ⎝ J⎠ J χ 0 u− −χ u− ρ ρ
(5)
where c and Γ are respectively the speed of sound and the Grüneisen coefficient defined as ∂p 1 ∂p 2 c = , Γ = , ∂ρ s,χ ρT ∂s ρ,χ and pχ is defined as 1 pχ = ρ
∂p ∂χ
ρ,e
.
Using a symbolic computation software, the eigenvalues and eigenvectors of A can be computed. The eigenvalues are u − c, u, u − ρJ and u + c. A matrix of right eigenvectors reads ⎞
⎛ 1
1
pχ
1
⎟ ⎜ ⎟ ⎜ J ⎜ u−c u+c ⎟ u pχ u − ⎟ ⎜ ρ ⎟ ⎜ ⎟. ⎜ R=⎜ 2 ⎟ c J ⎜ H − uc H − pχ H − u H + uc⎟ ⎟ ⎜ Γ ρ ⎟ ⎜ 2 ⎠ ⎝ J 2 χ χ + χ pχ − c χ 2 ρ
(6)
Riemann Problem for Constant Mass Flux By making the hypothesis of a constant phase change mass flux J across the interface, the system can be rewritten as a usual system of conservation laws. The Riemann problem for this system can then easily be studied. Its solutions can be sketched as in Fig. 1. The sonic waves associated with the eigenvalues u ± c and the contact discontinuity associated with the eigenvalue u are similar to the waves of the usual Riemann
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Fig. 1 Scheme of the solution of the Riemann problem with phase change (here condensation). The liquid–vapor interface is not following the contact discontinuity with velocity u. The fluid between the interface and the contact discontinuity is the fluid having just changed of phase. Due to the density ratio between the phases, the fluid changing phase must contract or expand to keep a reasonable pressure, thus generating the two sonic waves
problem for the Euler equations without phase change. The wave associated with the eigenvalue u − ρJ is the interface through which a mass flow rate J changes phase.
2.3 Comparison with Other Models This approach to simulate interfacial liquid–vapor phase change is new (to our knowledge). But it can be related to existing families of models. From a certain point of view, it is similar to the models described for instance in [8] or [10]. Like our model, these works use an order parameter to describe the twofluid flow. This parameter evolves due to phase change through a source term which is nonzero only at the interface. Our nonconservative phase change term J · ∇χ can also be seen as a source term which is nonzero only at the interface. But unlike a source term of order 0, the first-order phase change term does not depend on the diffusion of the interface. From another point of view, our model is similar to the “ad hoc Riemann solution” model described for instance in [6] or [9]. In these works, the phase change is modeled by computing nonclassical solutions to the Riemann problem for the Euler equations for a van der Waals-type equation of state. These nonclassical solutions are similar to the solution described in the previous section and on Fig. 1. The dynamic of phase change is determined by a kinetic relation which plays the same role as (4). The technical difficulty of computing nonclassical solutions to the Riemann problem has been replaced in our model by the difficulty of defining and computing the nonconservative term J · ∇χ . We prefer the latter approach: having an explicit system of conservation laws including phase change will make easy the use of a Roe-type scheme. The exact solution to the Riemann problem may not be physical anyway in the non-isothermal case, as we will discuss in the next section.
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2.4 Limitation To model a sharp liquid–vapor interface with phase change (far from the critical point), two difficulties have to be overcome: The two phases have very different densities. Changing only the phase (i.e., changing the local order parameter χ from 0 to 1 or 1 to 0) of the fluid leads to liquid with the density of the gas or vice versa. The fluid needs to expand or contract while changing phase to avoid these nonphysical states. The (approximate) Riemann solvers that we mentioned previously offer a satisfying answer to this problem, by coupling the phase change with the mechanics. Other techniques, such as the diffuse source terms of [7], can also be used to circumvent this difficulty. The two phases have very different energies. Due to the latent heat, changing only the phase of the fluid without heat exchange leads to nonphysical temperature variation. In the solution to the Riemann problem, such as the one of Fig. 1, the Rankine–Hugoniot condition across the liquid–vapor interface imposes a temperature jump of the order of magnitude of |Tl − Tg |
L 200 K, Cp
(7)
where L is the latent heat and C p is the isobaric thermal capacity (of the liquid or of the gas). The huge value of the latent heat means that the problem cannot be studied only as a hyperbolic mechanical problem: the thermal diffusion should not be neglected. For a realistic fluid in a state far from the critical point (for instance LNG in cryogenic tank), the exact solution to the Riemann problem for the hyperbolic system might not exist. However, the numerical diffusion of our numerical resolution with an approximate Riemann solver will be able to stabilize the system. We can thus plan to decouple the resolution of the hyperbolic system (1) from the parabolic heat equation, for instance with a time splitting approach.
3 Discretization and Test Cases This section is dedicated to the numerical resolution of the model presented previously. After a short summary of the method, some simple test cases will be presented.
3.1 Discretization Strategy Our strategy to deal with the nonconservative term will be to linearize it locally, as we did for the Riemann problem of the previous section. For instance at time step n,
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around the interface between the cells j and j + 1, we set
n χ , J · ∇χ ∇ J j+1/2
(8)
n is an approximation of the local surface mass flux. The system can thus where J j+1/2 be rewritten as a system of conservation equations. This system is then discretized using the Roe-type scheme of [4]. This approach has been used with two “proof of concept” codes using different descriptions of the liquid–vapor interface.
• In the first one, a mixture model is introduced to describe the cell containing the interface. A one-velocity hypothesis is used to simplify the equations. Only the isothermal case has been studied in depth for the moment. • The second one is based on the interface reconstruction scheme of [3]. The reconstructed sharp interface evolves according to an ALE discretization of (1) in the frame of the interface evolving at velocity u − ρJ . In the following paragraphs, some simple test cases will be used to validate these discretizations and illustrate the physics of phase change.
3.2 Numerical Results To our knowledge, no reference test case for the treatment of the difficulties presented in Sect. 2.4 exists in the literature. As a validation of the model and its discretization, we will thus present two original test cases. More results can be found in [1].
3.2.1
Test Case 1: Forced Isothermal Phase Change
The first test case focuses on the mechanical aspect of the problem, i.e., the evolution of pressure and velocity when mass is exchanged between two phases with low density ratio, where the density ratio is defined as DR =
ρg , ρl
where ρg is the density of the gas and ρl is the density of the liquid. Description We consider a closed 1D container filled with liquid and gas as displayed in Fig. 2. The two fluids follow simple isothermal equations of state for ideal and stiffened gases: ρg ∝ p, ρl − ρl,0 ∝ p − p0 . (9)
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Fig. 2 Initial conditions of both test cases. The 1D domain is limited by walls on both sides
At the interface a mass flux is imposed of the form J (t) = A sin(ωt),
(10)
where A and ω are constant parameters. Assuming a homogeneous pressure in the domain, an analytic solution for the pressure evolution can easily be derived. Results and Comments We have chosen equations of state such that DR = 10−3 ,
ρg,0 cg2 ρl,0 cl2
= 4 × 10−5
where cg and cl are, respectively, the speed of sound of the gas and the liquid. These values are of the order of magnitude for water at 373 K or methane at 111 K. The coefficients of (10) have been chosen as A = 0.1 ρg,0 cg ,
ω = 0.05
cg , z g,0
where z g,0 is the initial size of the gas domain. This choice is motivated by the wish to have a non-negligible phase change while keeping flow subsonic and the pressure homogeneous in the domain. In Fig. 3, the evolution of the pressure is compared to the analytic solution. The results match well. In Fig. 4, the velocity profile has been plotted during evaporation. Both discretizations of the interface show a clean and stable profile. It might not have been the case for a time splitting resolution separating the mechanics from phase change, as mentioned in Sect. 2.4.
3.2.2
Test Case 2: Return to Thermodynamic Equilibrium
The purpose of this second test case is to illustrate the role of the thermal diffusion in non-isothermal problems. Here, only the effect of the numerical diffusion is presented. It is at the same time necessary for the stability of the solution and biasing the effect of the physical diffusion.
Dimensionless pressure p/p0
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5 4 3 2 Exact solution Diffuse int. Int. reconstruction
1 0 0
50
100
150
200
Dimensionless time t cg,0 /
250
300
g,0
Fig. 3 Evolution of the pressure at the center of the domain as a function of time for the two discretizations of the model. They are compared with the exact solution derived assuming an homogeneous pressure
Dimensionless velocity u/cg,0
0.035 0.030
Diff.int. Int.reconst.
0.025 0.020 0.015 0.010 0.005 0.000 −0.005 0.0
0.5
1.0
1.5
2.0
Dimensionless abscissa Fig. 4 Profile of velocity at dimensionless time t cg,0 /z g,0 155, during the evaporation phase of the second period. The gas (on the right) escapes from the interface, its velocity is positive. The velocity of the liquid (on the left) is almost zero (actually slightly negative due to liquid compressibility)
Description The initial conditions are similar to those of the previous test case. The pressure and the temperature are initially homogeneous in the domain, but out of equilibrium p(t = 0) = p sat (T (t = 0)). The gas and the liquid are modeled respectively with (non-isothermal) ideal gas and stiffened gas equations of state. The mass flux at the interface is evaluated by an Hertz–Knudsen-type relation of the form
J ∝ p − p sat (T ) .
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Total entropy (J K−1 )
Total gas mass (kg)
68 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 100 80 60
40 cells 60 cells 80 cells
40 20 0 0.00
0.05
0.10
0.15
0.20
0.25
Time (s) Fig. 5 Top: evolution of the total mass of gas as a function of time for three different meshes. Bottom: evolution of the total entropy in the domain as a function of time for the same simulations
This problem will be solved with the interface reconstruction scheme mentioned previously. Results and Comments Only the condensation case ( p(0) > p sat (T (0))) is presented here. In Fig. 5, it can be checked that the total gas mass diminishes and reach some constant state while the total entropy rises. However, due to the lack of physical thermal diffusion here, the final equilibrium state is here only a local equilibrium state at the liquid–vapor interface. This final state depends on the size of the mesh: the finer the mesh, the smaller is the boundary layer which has to go back to equilibrium, thus the smaller the amount of condensation needed. A realistic boundary layer would be determined by the physical thermal diffusion. Here we have checked that the model and its discretization allow an entropic return to equilibrium, although this return is strongly influenced by the numerical diffusion. The next step is the coupling of this model with the heat equation to get more physically meaningful results. First results on this way are presented in [1].
4 Conclusion A mathematical model for the description of interfacial phase change across a sharp interface has been presented. It takes the form of a hyperbolic system of balance laws including the mass flux across the interface. This form allows an easy discretization
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using a Roe-type scheme. It has been checked numerically that the discrete model is able to deal with realistic density ratio and latent heat. The importance of a coupling with the heat equation has been discussed. This role played by the thermal diffusion might be critical for the studied application. First results [1] seem to indicate that during a wave impact, the thermal boundary layer effectively returning to equilibrium at the interface might be so small that the effect of phase change is almost negligible. However, as mentioned in the first section, we have only studied the scenario of a “clean” wave without foam, bubbles, and droplets. These phenomena will be strongly coupled ith phase change and they might amplify its effect by increasing the total interfacial area. On the other hand, LNG is actually not a pure phase but a mixture of methane, nitrogen, and some other hydrocarbons. The next step is thus to include the physics of multispecies phase change into the model presented in this paper.
References 1. M. Ancellin, Sur la modélisation physique et numérique du changement de phase interfacial lors d’impacts de vagues, PhD thesis, École Normale Supérieure Paris-Saclay, 2017 2. M. Ancellin, L. Brosset, J.-M. Ghidaglia, Preliminary numerical results on the influence of phase change on wave impacts loads, in Proceedings of the 26th International Offshore and Polar Engineering Conference, vol. 3 (ISOPE, 2016) pp. 886–893 3. J.-P. Braeunig, L. Brosset, F. Dias, J.-M. Ghidaglia, Phenomenological study of liquid impacts through 2D compressible two-fluid numerical simulations, in Proceedings of the 19th International Offshore and Polar Engineering Conference, vol. 3 (ISOPE, 2009), pp. 21–29 4. J.-M. Ghidaglia, A. Kumbaro, G. Le Coq, On the numerical solution to two fluid models via a cell centered finite volume method. Eur. J. Mech. B/Fluids 20(6), 841–867 (2001) 5. P.-M. Guilcher, Y. Jus, N. Couty, L. Brosset, Y.-M. Scolan, D. Le Touzé. 2D simulations of breaking wave impacts on a flat rigid wall – part 1: influence of the wave shape, in Proceedings of the 24th International Offshore and Polar Engineering Conference, vol. 3 (ISOPE, 2014), pp. 232–245 6. M. Hantke, W. Dreyer, G. Warnecke, Exact solutions to the Riemann problem for compressible isothermal Euler equations for two-phase flows with and without phase transition. Q. Appl. Math. 71(3), 509–540 (2013) 7. S. Hardt, F. Wondra, Evaporation model for interfacial flows based on a continuum-field representation of the source terms. J. Comput. Phys. 227(11), 5871–5895 (2008) 8. S. Le Martelot, R. Saurel, B. Nkonga, Towards the direct numerical simulation of nucleate boiling flows. Int. J. Multiph. Flow 66, 62–78 (2014) 9. C. Rohde, C. Zeiler, A relaxation riemann solver for compressible two-phase flow with phase transition and surface tension. Appl. Numer. Math. 95, 267–279 (2014) 10. R. Saurel, F. Petitpas, R. Abgrall, Modelling phase transition in metastable liquids: application to cavitating and flashing flows. J. Fluid Mech. 607, 313–350 (2008)
The Cauchy Problem for the Maxwell–Schrödinger System with a Power-Type Nonlinearity Paolo Antonelli, Michele D’Amico and Pierangelo Marcati
Abstract We discuss some results on the Maxwell–Schrödinger system with a nonlinear power-like potential. We prove the local well-posedness in H 2 (R3 ) × H 3/2 (R3 ) and the global existence of finite energy weak solutions. Then we apply these results to the analysis of finite energy weak solutions for Quantum Magnetohydrodynamic systems. Our interest in this problem is motivated by some models arising in quantum plasma dynamics. Keywords Nonlinear Maxwell-Schrödinger · Quantum Magnetohydrodynamics Finite energy solutions
1 Introduction We investigate the existence of local and global in time solutions to the following Cauchy problem for the nonlinear Maxwell–Schrödinger system in the threedimensional space: ⎧ ⎨ i∂t u = − 21 Δ A u + φu + |u|2(γ −1) u (1) A = PJ (u, A) ⎩ u(0) = u 0 , A(0) = A0 , ∂t A(0) = A1 ,
P. Antonelli · M. D’Amico (B) Gran Sasso Science Institute, L’Aquila, Italy e-mail: [email protected] P. Antonelli e-mail: [email protected] P. Marcati Gran Sasso Science Institute, Department of Information Engineering, Computer Science and Mathematics, University of L’Aquila, L’Aquila, Italy e-mail: [email protected]; [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_6
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where Δ A = (∇ − i A)2 denotes the magnetic Laplacian, φ = φ(ρ) = (−Δ)−1 ρ, with ρ: = |u|2 , represents the Hartree-type electrostatic potential, while the power nonlinearity describes the self-consistent interaction potential. J (u, A) = I m(u(∇ ¯ − i A)u) is the electric current density and P = I − ∇divΔ−1 denotes the Leray– Helmholtz projection operator onto divergence free vector fields. All the physical constants are normalized to 1. The Maxwell–Schrödinger system, without the power-type nonlinearity, describes the dynamics of a charged nonrelativistic quantum particle interacting with its selfgenerated (classical) electromagnetic field, see for instance [6, 16]. It is well known to be invariant under the gauge transformations; for sake of simplicity we decide to work in the Coulomb gauge, i.e., we assume div A = 0. It is straightforward to verify that also power-type nonlinearities appearing in (1) are gauge invariant. Our interest in (1) is motivated by the possibility to develop a general theory for quantum fluids in the presence of self-induced electromagnetic interacting fields. The Quantum Magnetohydrodynamic (QMHD) systems arise in the description of quantum plasmas, for example in astrophysics, where magnetic fields and quantum effects are non-negligible, see [8, 9, 17, 18] and the references therein. In particular, we want to establish a suitable theory in the space of energy for the following system: ⎧ ⎪ ⎨ ∂t ρ + div J = 0 √ , Δ ρ 1 J⊗J ⎪ + ∇ P(ρ) = ρ E + J ∧ B + ρ∇ √ , ⎩ ∂t J + div ρ 2 ρ
(2)
where ρ and J denote the charge and current densities of the quantum plasma respectively. The pressure term P(ρ) is assumed to be isentropic of the form P(ρ) = γ γ−1 ρ γ , 1 < γ < 3. The hydrodynamical system above is complemented by the Maxwell equations for the electromagnetic fields E and B
divE = ρ, ∇ ∧ E = −∂t B divB = 0, ∇ ∧ B = J + ∂t E.
(3)
The Maxwell–Schrödinger system has been widely studied in the mathematical literature, under the various choice of gauges. In [7], the global existence of finite energy weak solutions has been investigated by using the method of vanishing viscosity. However no uniqueness in proved there. In [14, 15], the authors obtain the global well-posedness in a wide class of high regularity Sobolev spaces by using the semigroup associated to the magnetic Laplacian, following Kato’s theory [10, 11], and by means of a fixed point argument. However their treatment is not adequate to deal with solutions in the energy space. More recently, a global well-posedness result in the energy space has been proven in [4] by using the analysis of a short time wave packet parametrix for the magnetic Schrödinger equation and the related linear, bilinear, and trilinear estimates. More precisely they prove the existence of a unique global sufficiently regular solution; then for initial data in the energy space they show that the global solution is the unique strong limit of the regular solutions obtained before.
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In the present paper, we focus on the Cauchy problem for the Maxwell– Schrödinger system with a power-type nonlinearity (no one of the previous references deals with this kind of problem). We state in the sequel the two main results we prove. The first one regards the local 3 well-posedness theory for (1) in H 2 (R3 ) × H 2 (R3 ). More precisely let us denote by
X : = (u 0 , A0 , A1 ) ∈ H 2 (R3 ) × H 3/2 (R3 ) × H 1/2 (R3 ) s.t. div A0 = div A1 = 0 . Theorem 1 (Local wellposedness). Let γ > 23 . For all (u 0 , A0 , A1 ) ∈ X there exists a maximal time 0 < Tmax ≤ ∞ and a unique (maximal) solution (u, A) to (1) such that • u ∈ C([0, Tmax ); H 2 (R3 )), 3 1 • A ∈ C([0, Tmax ); H 2 (R3 ) ∩ C 1 ([0, Tmax ); H 2 (R3 )), div A = 0 • Let (u 0 , A0 , A1 ) = (u(·), A(·), ∂t A(·)), then ∈ C(X ; C([0, T ]; X )), for any 0 < T < Tmax The following blowup alternative holds: either Tmax = ∞ or Tmax < ∞ and we have lim ( u(t) H 2 + A(t) H 23 + ∂t A(t) H 21 ) = ∞ .
t→Tmax
Our proof plays on the the construction of the evolution operator associated to the magnetic Laplacian, based on Kato’s approach [10, 11], then we perform a fixed point argument to approximate the solutions to the Maxwell–Schrödinger system by the classical Picard iteration scheme. Differently from [15], in our case the solutions obtained by this method cannot be extended globally in time, indeed the power-type nonlinearity does not lead to a Gronwall-type inequality capable to bound the higher order norms of the solution at any time. To overcome this difficulty, we regularize the system (1) by making use of the Yosida approximations of the identity, hence we are able to get the global wellposedness for the approximating system in H 2 (R3 ) × H 3/2 (R3 ). Moreover, by using the uniform bounds provided by the higher order energy, defined by the norm of X , we prove the existence of a finite energy weak solution to (1). This is established by the following theorem. Theorem 2 (Global Weak Solutions). Let 1 < γ < 3, (u 0 , A0 , A1 ) ∈ X , then there exists, globally in time, a finite energy weak solution (u, A) to (1), such that u ∈ L ∞ (R+ ; H 1 (R3 )), A ∈ L ∞ (R+ ; H 1 (R3 )) ∩ W 1,∞ (R+ ; L 2 (R3 )). The same results can be rephrased in a straightforward way in any other choice of the gauge condition. Moreover, it is possible to include a Hartree nonlinear term of the form (| · |−α ∗ |u|2 )u, with 0 < α < 3. With those results at hand, we want to develop a suitable theory in the energy space for the QMHD system in (2). The major obstacle in this direction, which is
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also the major difference with respect to the usual QHD theory, regards the possibility to give sense to the nonlinear term related to the Lorenz force ρ E + J ∧ B. The standard energy estimates on the Maxwell–Schrödinger system (1) lead √ 2 ∞ 2 ∞ 3/2 ∞ 2 ∞ 1 ∞ to √Jρ ∈ L ∞ t Lx , ∇ ρ ∈ Lt Lx , J ∈ Lt Lx , B ∈ Lt Lx , ∇ ∧ J ∈ Lt Lx ∩ Lt −1,3/2
Wx . Unfortunately, these bounds are not sufficient to apply the compensated compactness of Tartar [12, 13, 19] and in particular the argument in the Lecture 40 / L 3x (the boundedness in at least one of these norms of [20], indeed J ∈ / L 2x and B ∈ would be sufficient). Therefore, the analysis of the Lorenz force for finite energy solutions needs still to be better understood. It is impossible to use the result of [7] for the study of (2) and even the solutions obtained in [4] do not allow to well define the Lorentz force term. The results of [14, 15], without the nonlinear potential, combined with the methods of [2, 3], allow instead to analyze the pressureless QMHD case. The additional difficulty introduced by the power nonlinearity in the Maxwell– Schrödinger system (1) in 3–D, namely a nonlinear pressure term in the QMHD system, cannot be easily managed. Usually, the proof of well-posedness for the NLS in higher regularity spaces, combines higher order energy estimates with the use of sharp Strichartz estimates. However, to our knowledge, there are not intrinsic Strichartz estimates for (1); actually there are many Strichartz estimates available in the literature for the Schrödinger equations with a prescribed magnetic potential, but our solution to (1) does not fall in that class. The lack of such Strichartz estimates would lead to a superlinear Gronwall inequality and hence into an upper bound which blows up in finite time. The paper is organized as follows. In Sect. 2, we prove Theorem 1. In Sect. 3, we investigate the problem of global existence of solutions. In Sect. 4, we apply our main results to the analysis of the QMHD system.
2 Proof of Theorem 1 In this section, we are going to prove the local well-posedness result stated in Theorem 1, by using a fixed point argument. We denote with U A (t, s) the propagator associated to he magnetic Laplacian Δ A , constructed by using Kato’s approach. In the following lemma, we collect the fundamental estimates, we will use for the iteration scheme. Lemma 1. Let A ∈ H 1 (R3 ) and u ∈ H 2 (R3 ), then the following estimates hold (∇ − i A)u H 1 (R3 ) (1 + A H 1 (R3 ) ) u H 2 (R3 ) , PJ (u, A) H 21 (R3 ) u H 1 (R3 ) u H 2 (R3 ) + A H 1 (R3 ) u 2H 2 (R3 ) , Δ A u L 2 u H 2 + A 4H 1 u L 2 , u H 2 Δ A u L 2 + A 4H 1 u L 2 , (∇ + i A)u L 6 u H 2 + A 4H 1 u L 2 ,
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∂ A e t L 1t L 3x . U A (t, s)u H 2 ≤ u H 2 1 + A 4L ∞ 1 H t x First of all, let us define the space 3
1
X T : = {(u, A) s.t. u ∈ C([0, T ], H 2 (R3 )), A ∈ C([0, T ], H 2 (R3 )) ∩ C 1 ([0, T ], H 2 (R3 )), div A = 0, u L ∞ 2 3 ≤ R1 , A t Hx (R )
3
3 L∞ t H 2 (R )
+ ∂t A
1
2 3 L∞ t Hx (R )
≤ R2 } ,
(4) where R1 , R2 , T > 0 will be chosen later. It is straightforward to see that X T , endowed with the distance 2 3 , A 1 − A 2 4 4 d((u 1 , A1 ), (u 2 , A2 )) = max{ u 1 − u 2 L ∞ L t L x (R3 ) } , t L x (R )
(5)
is a complete metric space. Let (u 0 , A0 , A1 ) ∈ X , we define the map Φ:X T → X T , (v, B) = Φ(u, A), (u, A) ∈ X T , where
t
U A (t, s)(φu + |u|2(γ −1) u)(s)ds √ √
t √ sin(t −Δ) sin(t −Δ) A1 + PJ (u, A)(s)ds B(t) = cos(t −Δ)A0 + √ √ −Δ −Δ 0 v(t) = U A (t, 0)u 0 − i
0
Since γ > 23 the function z → |z|2(γ −1) z is C 2 (C; C), then by the Sobolev embedding H 2 → L ∞ we have 2γ −1
2γ −1
2 u ∞ 2 R |u|2(γ −1) u L ∞ 1 t Hx Lt H x
2 φu L ∞ t Hx
,
2 u 2L ∞ H 3/4 u L ∞ t Hx x t
R13 ,
Lemma 1 gives 2γ −1 4 3 2 2 v L ∞ ≤ C (1 + R ) exp(T R ) u + T R + T R 1 2 0 H 2 1 . 1 t Hx By using again Lemma 1, we have the estimate 2 1/2 R (1 + R1 ), PJ L ∞ 1 t Hx
which, combined with the Strichartz estimates for the wave equations, leads to 2 3/2 + ∂t B ∞ 1/2 ≤ C 2 (1 + T ) A 0 H 3/2 + A 1 H 1/2 + T R (1 + R1 ) . B L ∞ 1 L t Hx t Hx Now by choosing R1 , R2 and T in a suitable way, we see that Φ maps X T into itself. Proceeding in a similar way, we can also prove that by possibly choosing a smaller T > 0, the map Φ is indeed a contraction on X T . This proves that for any initial data (u 0 , A0 , A1 ) ∈ X , there exists a unique local solution (u, A) to (1) such
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that u ∈ C([0, T ]; H 2 (R3 )), A ∈ C([0, T ]; H 3/2 (R3 )) ∩ C 1 ([0, T ]; H 1/2 (R3 )). By a standard argument, it is straightforward to show that it may be extended to a maximal solution (u, A), with u ∈ C([0, Tmax ); H 2 (R3 )), A ∈ C([0, Tmax ); H 3/2 (R3 )) ∩ C 1 ([0, Tmax ); H 1/2 (R3 )) and that the blowup alternative holds true, namely Tmax < ∞ if and only if lim ( u(t) H 2 + A(t) H 3/2 + ∂t A(t) H 1/2 ) = ∞.
− t→Tmax
In order to complete the proof of Theorem 1, it remains to prove the continuous dependence with respect to the initial data. The proof follows this strategy: first we prove the continuous dependence for more regular solutions, then by an approximation argument, we extend it to arbitrary solutions (u, A) ∈ X . We refer to [1] for more details.
3 Global Existence In the previous section, we proved the local well-posedness of (1) in H 2 × H 3/2 . However, the presence of the power-type nonlinearity in (1) prevents us from obtaining a global bound for (u(t), A(t), ∂t A(t)) X . This is different from what can be proven in [15]: indeed, while in the case of Hartree nonlinearity, it is possible to use the following estimate: (−Δ)−1 (|u|2 )u H 2 u 2H 3/4 u H 2 , which is linear in the higher order norm, in our case with a power-type nonlinearity one has 2(γ −1) (6) |u|2(γ −1) u H 2 (R3 ) u L ∞ (R3 ) u H 2 (R3 ) , which requires to bound u in H s (R3 ), with s > 23 . Therefore it follows that the related Gronwall-type inequality becomes superlinear in the higher order norm, hence it blows up in finite time. Our strategy to investigate global in time existence will be based on the regularization of the nonlinear terms, provided by the classical Yosida approximations of the identity. We then consider the following approximating system: ⎧ ⎪ ⎪ ⎨
1 iu t = − Δ A u + φ u + N (u ) 2 = J PJ A ⎪ ⎪ ⎩ u (0) = u 0 , A (0) = A0 , ∂t A (0) = A1 ,
(7)
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where J = (I − Δ)−1 , A = J A , N (u ) = J |J u |2(γ −1) J u + , J = J (u , A ), φ = φ(|u |2 ) and we denote ∇ A = ∇ − i A . The total energy of this approximating system is given by
E=
1 1 1 1 |∇ A˜ u |2 + |∇φ |2 + |∇ A |2 + |∂t A |2 + |J u |2γ d x 2 2 2 γ R3
(8)
which is conserved along the flow of solutions. A local well-posedness result, analogous to Theorem 1, can be proved for the system (7) in a straightforward way. Proposition 1. For all (u 0 , A0 , A1 ) ∈ X , there exists Tmax > 0 and a unique maximal solution (u , A ) to (7) such that
); H 2 (R3 )), A ∈ C([0, T ); H 3/2 (R3 )) ∩ C 1 ([0, T ); H 1/2 (R3 )) u ∈ C([0, Tmax max max
and the usual blowup alternative holds true. Moreover, the solution depends continuously on the initial data. The regularization of the nonlinear terms yields indeed the global existence of solutions, so we have the following proposition. Proposition 2. The solution obtained in Proposition (1) exists globally in time, namely (u (t), A (t), ∂t A (t)) X is finite for any t ∈ R. The proof on Proposition 2 is based on the following estimate: u (t) H 2 ≤ C( u 0 L 2 , E)e
t ∂t A
1/2 L∞ t Hx
.
(9)
The key ingredient for the proof of (9) is that now the estimate for the H 2 -norm of the regularized nonlinearity N (u ) is linear with respect to the higher order norm, 2(γ −1)
N (u ) H 2 |J u |2(γ −1) J u H 2 u H 1
u H 2 ,
see Lemma 4.3 in [1] for details. It follows from (9) that, in order to get a bound on the H 2 norm of the approximating solution u , it is sufficient to control ∂t A ∞ 21 . Using the energy estimates for the wave equation
L t Hx
3/2 + ∂t A ∞ 1/2 C(T ) A 0 H 3/2 + A 1 H 1/2 + J PJ ∞ 1/2 A L ∞ L t Hx L t Hx t Hx and, by exploiting the Yosida regularization, we get 1/2 PJ ∞ −1/2 J ∞ 3/2 ≤ C(E). J PJ L ∞ L t Hx Lt Lx t Hx
It follows that A (t) H 3/2 + ∂t A (t) H 1/2 is uniformly bounded on compact time intervals and, consequently by (9), also u (t) H 2 is finite. Hence, by the blowup
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alternative, the solution (u , A ) to (7) exists globally in time. In order to conclude the proof of Theorem 2, we need to show that (u , A ) converges to a solution of (1), when → 0. By exploiting the apriori bounds given by the conservation of mass and energy 1,∞ 2 1 ∞ 1 Lx, we obtain that, up to subsequences, there exist u ∈ L ∞ t Hx , A ∈ L t Hx ∩ Wt such that ∗
1 3 u u in L ∞ t Hx (R × R ) ∗
1 3 A A in L ∞ t Hx (R × R )
∗
∂t A ∂t A in
2 L∞ t L x (R
× R ). 3
(10) (11) (12)
Proposition 3. The weak limit (u, A) in (10), (11) is a finite energy weak solution to the Cauchy problem (1), with initial datum (u 0 , A0 , A1 ). The proof of Proposition 3 relies on a standard compactness argument (see [1] for more details). Moreover we can prove that the initial condition for (1) is satisfied.
4 Quantum Magnetohydrodynamics The quantum magnetohydrodynamic equations are the governing nonlinear equations for the electromagnetic waves in dense magnetoplasmas. They consist of the continuity equation and the electron and ion momentum equations (here i =ions and e =electrons) ∂n e,i + ∇ · (n e,i ue,i ) = 0, ∂t ∂ 1 + ue · ∇ ue = −n e e E + ue × B − ∇ Pe + n e F Qe , ne m e ∂t c ni m i
∂ 1 + ui · ∇ ui = Z i en i E + ui × B , ∂t c
the Faraday law c∇ × E = −
∂B , ∂t
and the Maxwell equation including the magnetization spin current ∇ ×B=
1 ∂E 4π J p + Jm + , c c ∂t
where n j and m j are the number density and the mass of the particle species j respectively, u j is the particle fluid velocity, Z i is the ion charge state, J p = −n e eue + Z i n i eui is the plasma current density, and Jm = ∇ × M is the electron magnetization
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spin current density, with M = (n e μ2B /k B TFe )B, where μ B = e/2m e c is the Bohr magneton (magnetic moment of an electron caused by either its orbital or spin angular momentum). The term F Qe is the sum of the quantum Bohm potential and intrinsic angular momentum spin forces F Qe
√ Δ ne μ2B − =∇ √ ∇ B, ne k B TFe
where B = |B|. The pressure for degenerate (close to zero temperature) electrons is given by ([5]) 3/2
4eB(2m e )1/2 E F Pe = 3(2π )2 2 c
1+2
n max n L =1
n L ω B 1− EF
3/2 ,
where p F and E F are the Fermi momentum and the Fermi energy respectively. If p2F < ω B (strong Landau quantization condition) one has the following density 2m scaling laws: 1/3
2/3
5/3
• If B = 0 then p F ≈ n e , E F ≈ n e , Pe ≈ n e • If B > 0 then p F ≈ n e , E F ≈ n 2e , Pe ≈ n 3e
As we can see, under certain conditions, the pressure term can be approximated by a power law and this motivates the introduction of the nonlinear power-like potentials in (1). The above equations are written for a two species charged particle system (bipolar quantum fluid model). As a simplification, we focus the attention on a one-species charged quantum plasma, whose dynamics is described by (2). First of all, let us point out the relation between the nonlinear Maxwell–Schrödinger system (1) and quantum magnetohydrodynamic (QMHD) system (2). We can formally see the equivalence of the two systems in the following way. Let (u, A) be a solution of (1); by using the Madelung transformations, we can define the associated hydrodynamical quantities ¯ − i A)u). We are interested in the balance laws satisfied ρ := |u|2 and J := I m(u(∇ by ρ and J . So, by differentiating in time the expression for ρ, we get ∂t ρ + div J = 0, which is exactly the first line of (2). By doing the same with J , we have 1 ∂t J + div Re{(∇ − i A)u ⊗ (∇ − i A)u} + ∇ P(ρ) = = ρ E + J ∧ B + ∇Δρ. 4 Now we want to express the quadratic term inside the divergence in terms of the hydrodynamical quantities. It is straightforward to see that
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J⊗J √ √ . Re{(∇ − i A)u ⊗ (∇ − i A)u} = ∇ ρ ⊗ ∇ ρ + ρ
It follows that ∂t J + div
J ⊗ J 1 √ √ + ∇ P(ρ) = ρ E + J ∧ B + ∇Δρ − div(∇ ρ ⊗ ∇ ρ) ρ 4
By noting that 1 ρ∇ 2
√ Δ ρ 1 √ √ = ∇Δρ − div(∇ ρ ⊗ ∇ ρ), √ ρ 4
(13)
we recover exactly the second equation of (2). At this point, we stress again that the presence of the nonlinear power-like potential in (1) is fundamental to recover the nontrivial pressure term in the QMHD. As already said, the above computations are just formal. They can be made rigorous by means of a polar factorization technique. Given any complex valued function u ∈ H 1 (R3 ), we may define the set of its polar factors as P(u) := {ϕ ∈ L ∞ (R3 ) : ϕ L ∞ ≤ 1, u =
√ ρϕ a.e. in R3 },
√ √ where ρ := |u|. Thus, for any ϕ ∈ P(u), we have |ϕ| = 1 ρ d x a.e. in R3 and ϕ √ is uniquely defined ρ d x a.e. in R3 . Clearly the polar factor is not uniquely defined in the nodal regions, i.e., in the set {ρ = 0}. √ Lemma 2. (Polar factorization). Let u ∈ H 1 (R3 ), A ∈ L 3 (R3 ), and let ρ := |u|, ϕ ∈ P(u). Let us define := Re(ϕ(−i∇ ¯ − A)u) ∈ L 2 (R3 ), then we have √ √ • ρ ∈ H 1 (R3 ) and ∇ ρ = Re(ϕ∇u); ¯ • the following identity holds a.e. in R3 , √ √ Re{(−i∇ − A)u ⊗ (−i∇ − A)u} = ∇ ρ ⊗ ∇ ρ + ⊗ .
(14)
Moreover, let {u n } ⊂ H 1 (R3 ), {An } ⊂ L 3 (R3 ) be such that u n converges strongly to u in H 1 and An converges strongly to A in L 3 , then we have √ √ ∇ ρn → ∇ ρ, n → , in L 2 (R3 ), where
√ ρn : = |u n |, n : = Re(ϕ¯n (−i∇ − An )u n ).
Proof. See [1] 1 Definition 1. Let ρ0 , J0 , E 0 , B0 ∈ L loc (R3 ), then a finite energy weak solution to system (2)–(3) in the space-time slab [0, T ) × R3 is given by a quadruple √ ( ρ, , φ, A) such that
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√ ρ ∈ L ∞ ([0, T ); H 1 (R3 )), ∈ L ∞ ([0, T ); L 2 (R3 )), φ∈L ∞ ([0, T ); H 1 (R3 )), A ∈ L ∞ ([0, T ); H 1 (R3 )) ∩ W 1,∞ ([0, T ); L 2 (R3 )); √ √ 2. ρ: = ( ρ)2 , J : = ρ, E: = −∂t A − ∇φ, B: = ∇ ∧ A; 2 2 3. J ∈ L ([0, T ); L loc (R3 )); 4. ∀ η ∈ Cc∞ ([0, T ) × R3 ),
1.
0
T
R3
ρ∂t η + J · ∇η d xdt +
R3
ρ0 (x)η(0, x) d x = 0;
5. ∀ζ ∈ Cc∞ ([0, T ) × R3 ; R3 ),
T 0
R3
J · ∂t ζ + ⊗ :∇ζ + P(ρ)divζ + ρ E · ζ + (J ∧ B) · ζ
1 √ √ + ∇ ρ ⊗ ∇ ρ:∇ζ + ρΔdivζ d xdt + J0 (x) · ζ (0, X ) d x = 0; 4 R3
6. E, B satisfy (3) in [0, T ) × R3 in the sense of distributions; 7. (finite energy) the total mass and energy defined by
M(t): =
R3
ρ(t, x) d x,
(15)
1 √ 2 1 2 1 1 1 |∇ ρ| + || + f (ρ) + |∂t A|2 + |∇ A|2 + |∇φ|2 d x 2 2 2 2 2 (16) respectively, are finite for every t ∈ [0, T ). Here f (ρ) = γ1 ρ γ . E(t) =
R3
Proposition 4. Let (ρ0 , J0 , B0 , E 0 ) be such that ρ0 : = |u 0 |2 , J0 : = Re(u¯ 0 (−i∇ − B0 : = ∇ ∧ A0 , E 0 : = −A1 − ∇φ0 , φ0 : = (−Δ)−1 |u 0 |2 A0 )u 0 ), √ for some (u 0 , A0 , A1 ) ∈ X , then there exists Tmax > 0 such that ( ρ, , φ, A) is a finite energy weak solution to (2)–(3) with initial data (ρ0 , J0 , B0 , E 0 ) in the spacetime slab [0, Tmax ) × R3 . Moreover, the energy is conserved for all t ∈ [0, Tmax ). In view of Lemma 2 we can now prove Proposition 4. Let (u 0 , A0 , A1 ) ∈ X be given, then by our main Theorem 1 there exists a unique solution (u, A) to (1) in [0, Tmax ) × R3 such that u ∈ C([0, Tmax ); H 2 (R3 )), A ∈ C([0, Tmax ); H 3/2 (R3 )) ∩ √ ¯ + A)u), C 1 ([0, Tmax ); H 1/2 (R3 )). Let us now define ρ: = |u|, : = Re(ϕ(−i∇ where ϕ is a polar factor for u, and let φ: = (−Δ)−1 ρ. √ By differentiating ρ and J = ρ with respect to time we get i = −div J, ∂t ρ =2Re u¯ − (−i∇ − A)2 u − iφu − i|u|2(γ −1) u 2
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and, after some tedious but rather straightforward calculations, we get ∂t J + div Re{(−i∇ − A)u ⊗ (−i∇ − A)u} + ∇ P(ρ) = 1 ρ(−∂t A − ∇φ) + J ∧ (∇ ∧ A) + ∇Δρ. 4 We now use the polar factorization Lemma 2 to infer that √ √ Re{(−i∇ − A)u ⊗ (−i∇ − A)u} = ∇ ρ ⊗ ∇ ρ + ⊗ and consequently we get 1 √ √ ∂t J + div( ⊗ ) + ∇ P(ρ) = ρ E + J ∧ B + ∇Δρ − div(∇ ρ ⊗ ∇ ρ). 4 By recalling identity (13) we see that this is the equation for the current density in the QMHD system (2). Actually the computations are rigorous only when (u, A) are sufficiently regular. However for solutions to (1) considered in Theorem 1 they can be rigorously justified in the weak sense, namely in the sense of Definition 1 by regularizing the initial data and by exploiting the continuous dependence and the H 1 −stability of the polar factorization stated in Lemma 2. It only remains to prove that E, B satisfy the Maxwell equations, but this comes in a straightforward way from the wave equation in (1) and the definitions E = −∂t A − ∇φ, B = ∇ ∧ A.
References 1. P. Antonelli, M. D’Amico, P. Marcati, Nonlinear Maxwell–Schrödinger system and quantum magneto-hydrodynamics in 3-D, Accepted Comm. Math. Sci 2. P. Antonelli, P. Marcati, On the finite energy weak solutions to a system in quantum fluid dynamics. Commun. Math. Phys. 287(2), 657–686 (2009) 3. P. Antonelli, P. Marcati, The quantum hydrodynamics system in two space dimensions. Arch. Ration. Mech. Anal. 203, 499–527 (2012) 4. I. Bejenaru, D. Tataru, Global well-posedness in the energy space for the Maxwell–Schrödinger system. Commun. Math. Phys. 288(1), 145–198 (2009) 5. S. Eliezer, P. Norreys, J.T. Mendona, Effects of Landau quantization on the equations of state in intense laser plasma interactions with strong magnetic fields. Phys. Plasmas 12, 052115 (2005) 6. R.P. Feynman, R.B. Leighton, M. Sands, The Schrödinger equation in a classical context: a seminar on superconductivity (Chapter 21), in The Feynman Lectures on Physics, Vol III Quantum Mechanics (Addison-Wesley Publishing Co., Inc, Reading, Mass. London, 1995) 7. Y. Guo, K. Nakamitsu, W. Strauss, Global finite-energy solutions to the Maxwell–Schrödinger system. Commun. Math. Phys. 170, 181–196 (1995) 8. F. Haas, A magnetohydrodynamic model for quantum plasmas. Phys. Plasmas 12, 062117 (2005) 9. F. Haas, Quantum Plasmas: An hydrodynamic Approach (Springer, New York)
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10. T. Kato, Linear evolution equations of “hyperbolic” type. J. Fac. Sci. Univ. Tokyo Sect. I(17), 241–258 (1970) 11. T. Kato, Linear evolution equations of “hyperbolic" type II. J. Math. Soc. Japan 25, 648–666 (1973) 12. F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5(3), 489–507 (1978) 13. F. Murat, Compacité par compensation: condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8(1), 69–102 (1981) 14. M. Nakamura, T. Wada, Local well-posedness for the Maxwell–Schrödinger equation. Math. Ann. 332(3), 565–604 (2005) 15. M. Nakamura, T. Wada, Global existence and uniqueness of solutions to the Maxwell– Schrödinger equations. Commun. Math. Phys. 276, 315–339 (2007) 16. L.I. Schiff, Quantum Mechanics, 2nd edn. (McGraw-Hill, New-York, 1955) 17. P.K. Shukla, B. Eliasson, Nonlinear aspects of quantum plasma physics. Phys. Usp. 53, 51–76 (2010) 18. P.K. Shukla, B. Eliasson, Novel attractive force between ions in quantum plasmas. Phys. Rev. Lett. 108, 165007 (2012) 19. L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics: Heriot–Watt Symposium, vol. IV, Research Notes in Mathematics, vol. 39, Pitman, Boston, Mass-London (1979), pp. 136–212 20. L. Tartar, in An Introduction to Navier–Stokes Equation and Oceanography. Lecture Notes of the Unione Matematica Italiana, vol. 1 (Springer, Berlin, UMI, Bologna, 2006)
Construction and Approximation of the Polyatomic Bitemperature Euler System Denise Aregba-Driollet and Stéphane Brull
Abstract This paper is devoted to the study of the bitemperature Euler system in a polyatomic setting. Physically, this model describes a mixture of one species of ions and one species of electrons in the quasi-neutral regime. We first derive the model starting from a kinetic polyatomic model and performing next a fluid limit. This kinetic model is shown to satisfy fundamental properties. Finally, a numerical scheme is derived and tested. Keywords Non conservative hyperbolic system · BGK model · Fluid limit Plasma · Kinetic scheme
1 Introduction This work is devoted to the nonconservative bitemperature Euler system in the context of plasma physics. Physically, this model describes the interaction of one species of ions and one species of electrons, under the quasi-neutrality assumption. The aim of this paper is more precisely to provide a construction and an approximation of the Euler bitemperature system. This derivation is based on a hydrodynamic limit performed on an underlying polyatomic kinetic model. In [1], this model has been recovered starting from a kinetic monoatomic model. Here such a procedure will lead to a general γ law. More precisely, the kinetic model is a polyatomic BGK model based on one energy variable [3] coupled with Ampère and Poisson equations.
D. Aregba-Driollet · S. Brull (B) Institut de Mathématiques de Bordeaux, 351, cours de la Libération, 33405 Talence cedex, France e-mail: [email protected] D. Aregba-Driollet e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_7
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One advantage of this kinetic model is its conservative form. Hence, starting from a standard semi-discretization of this model, the Chapman–Enskog procedure leads to a numerical scheme for the bitemperature model. The plan of the paper is the following. Section 2 deals with the different models that are used in this paper. In Sect. 3, the fluid model is obtained starting from the polyatomic model. The last part is devoted to the numerical approximation of the fluid model. A kinetic scheme is first derived and next tested.
2 The Mathematical Models 2.1 The Bitemperature Euler System The bitemperature Euler system writes ⎧ ⎪ ⎪ ⎪ ⎨
= 0, ∂t ρ + ∂x (ρu) ∂t (ρu) + ∂x (ρu 2 + pe + pi ) = 0, ∂t (ρe εe + 21 ρe u 2 ) + ∂x (u(ρe εe + 21 ρe u 2 + pe )) − u(ci ∂x pe − ce ∂x pi ) = νei (Ti − Te ), ⎪ ⎪ ⎪ ⎩ ∂t (ρi εi + 21 ρi u 2 ) + ∂x (u(ρi εi + 21 ρi u 2 + pi )) + u(ci ∂x pe − ce ∂x pi ) = −νei (Ti − Te ),
(1) where ρ = ρe + ρi ≥ 0 represents the total density of the plasma and u the average velocity of the plasma. Te and Ti represent the electronic and the ionic temperatures. ρe = n e m e , ρi = n i m i are electronic and ionic densities, where electronic and ionic concentrations n e and n i are assumed to be linked by Z = n e /n i ≥ 1. In this paper, Z will be considered as constant. This assumption means that the model is situated at the quasi-neutral regime. m e et m i represent the electronic and ionic masses. The mass fractions ρα cα = , α = e, i ρ are then constant and write ce =
Z me , ci = 1 − ce . mi + Z me
The electronic and ionic pressures and temperatures are related by pe = n e k B Te and pi = n i k B Ti . The electronic and ionic internal energies are given by εe =
1 n e k B Te , γe − 1
εi =
1 n i k B Ti , γi − 1
where γe , γi are constant lying in [1, 3], and k B is the Boltzmann constant. In the present case, we consider the general case of γ law contrary to [1], where only the monoatomic case has been considered.
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2.2 The Kinetic Model 2.2.1
Notations
Kinetic models are described by the distribution function f β of species β depending on time t ∈ R+ , space x ∈ R3 , velocity v ∈ R3 and on internal energy I ∈ R+ . Hydrodynamic quantities of species (β = e, i) are defined by nβ =
R3 ×R+
f β I αβ dvd I, u β =
1 nβ
R3 ×R+
v f β I αβ dvd I, Eβ =
R3 ×R+
(m β
v2 + I ) f β I αβ dvd I. 2
Velocities and temperatures of the mixture u et T are defined by u=
1 (ρe u e + ρi u i ) ρ 1
(2)
5 5 1 1 1 2 2 2 + α + α ρ ρ ρu ( . T = 5 )n k T + ( )n k T + u + u − e e B e i i B i e i e i 2 2 2 2 2 ( 2 + αe )n e k B + ( 25 + αi )n i k B
(3)
The parameter αβ is related to γ by the formula γ = diatomic case, we have αβ = 0 and γ =
2.2.2
7 . 5
1 5 2 +α
+ 1. For example in the
A Polyatomic BGK Model
In this section, we consider the following polyatomic kinetic model for β ∈ {e, i}, ∂t f β (t, x, v, I ) + v · ∇ f β (t, x, v, I ) +
φβ 1 E · ∇ f β (t, x, v, I ) = Mβ − f β (t, x, v, I ) mβ τβ 1 + Mβ ( f β , f γ ) − f β (t, x, v, I ) , τβγ
(4) with ⎛ ⎞ 2 (v − u 1 ) I β ⎠, exp ⎝− Mβ = − 3 T k B Tβ (2π mk Bβ Tβ ) 2 φβ (Tβ ) 2k B mββ (v − u # )2 1 nβ I exp − Mβ ( f β , f γ ) = − , 3 kB T # 2k B mTβ (2π mk B T # ) 2 φβ (T # ) nβ
β
with
+∞
φβ (T ) = 0
I αβ exp(−
I ) d I, kB T
(5)
(6)
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and u# = T# =
1 τei
ρe u e + 1 τei
kB τei
ρe +
1 τie ρi u i 1 τie ρi
,
( 25 + αe )n e Te +
(7) kB 5 τie ( 2
+ αi )n i Ti + 21 ( τ1ei ρe u 2e + kB τei
( 25 + αe )n e +
kB 5 τie ( 2
1 2 τie ρi u i
− ( τ1ei ρe +
+ αi )n i
1 # 2 τie ρi )(u ) )
.
(8) In the present BGK model, we consider as suggested in [4], that τei = τie , because of the molecular mass discrepancies between electrons and ions. Hence, in the present paper, we generalize the situation of [1] to a polyatomic setting. In the case where τei = τie , we recover u # = u and T # = T . The model (4)–(8) is coupled to Ampère and Poisson equations through the electric field E as j ∂t E = − , ε0 ∇x · E =
ρ . ε0
j represents the plasma current, ρ the total charge and ε0 is the permittivity. j and ρ are defined by ρ= j=
R3
R3
(qe f e I αe + qi f i I αi ) dvd I = n e qe + n i qi , v(qe f e I αe + qi f i I αi ) dvd I = n e qe u e + n i qi u i .
We define the entropy of the mixture by H ( f e , f i ) = Hs ( f e ) + Hs ( f i ), with H ( f β ) =
2.2.3
R3
( f β ln( f β ) − f β )I αβ dvd I. (9)
Properties of the Model
Proposition 1. The model (4)–(8) conserves the mass per species, the total impulsion and the total energy. The proof is straighforward and based on the definition of the fictitious quantities (7, 8). We refer to [1] for the proof of this proposition in the monoatomic setting. Theorem 1. The model (4)–(8) satisfies a H theorem.
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The model satisfies an entropic inequality
+
1 τei
1 τe
R3 ×R+
(Me ( f e ) − f e ) ln( f e ) I αe dvd I +
(M e ( f e , f i ) − f e ) ln( f e ) I αe dvd I +
R3 ×R+
1 τie
1 τi
R3 ×R+
(Mi ( f i ) − f i ) ln( f i ) I αi dvd I
(M i ( f e , f i ) − f i ) ln( f i ) I αi dvd I ≤ 0.
R3 ×R+
The equality holds in the above equation if and only if there exists (n β , u, T ) ∈ R+ × R3 × R+ such that (v − u)2 1 I exp − − Mβ = . 3 kB T 2k B mTβ (2π mk Bβ T ) 2 φβ (T ) nβ
An important feature of this polyatomic model: it satisfies entropy dissipation properties which are compatible with macroscopic ones. The entropy has already been obtained in [1] starting directly from the fluid system. But in the present paper, we can show that it is compatible with the Boltzmann entropy (9) in the polyatomic case.
3 Construction of the Fluid Model 3.1 Scaling on the One Dimensional BGK Model Suppose that the system is space homogeneous in two directions. Hence, the distribution function f β of species β depends on time t ∈ R+ , space x ∈ R, velocity v1 ∈ R and on the energy variable I ∈ R+ . The model (4)–(8) can be rewritten in this case ⎧ qβ 1 1 ⎪ ⎨ ∂t f β + v1 ∂x f β + m β E∂v1 f β = ε (Mβ − f β ) + τβ,γ (Mβ − f β ), β = γ (10) = − εj2 , ∂t E ⎪ ⎩ ρ ∂x E = ε2 , where ε is a positive parameter proportional to the Knudsen number. In that case, the Maxwellian distributions (5, 6) become ⎛ ⎞ (v1 − u β )2 + v22 + v33 1 I ⎠, exp ⎝− Mβ = − 3 T kB k B Tβ 2 φβ (Tβ ) 2k B mββ (2π m T ) β β ⎛ ⎞ (v1 − u β )2 + v22 + v32 nβ 1 I ⎠. exp ⎝− Mβ ( f β , f γ ) = − 3 kB T 2k B T (2π k B T ) 2 φβ (T ) nβ
mβ
mβ
(11)
(12)
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3.2 Hydrodynamic Limit Proposition 2. The system (8)–(12) converges formally to the non conservative bitemperature Euler system where E is given by Ohm’s law 1 1 n e qe n i qi ρ ρ ∂x pe − ∂x pi = ( − )E = n e qe E = − n i qi E ρe ρi ρe ρi ρe ρi ρe ρi and νei =
k B ( 25 + αe )( 25 + αi )n e n i 1 . ( 25 + αe )n e + ( 25 + αi )n i τei
(13)
(14)
Proof. Performing a Chapman–Enskog expansion on species β, it comes that f β = Mβ + εgβ ,
(15)
with the constraints gβ I αβ dvd I = 0, R3 ×R+ v1 (m i gi I αi + m e ge I αe ) dvd I = 0, R3 ×R+
1 2 R3 ×R+ ( 2 m β v
+ I )I αβ dvd I = 0.
(16)
Moreover, Gauss equation (10) implies n i qi = n e qe + O(ε2 ). So, n i = Z n e + O(ε2 ). Plugging (15) into equation (4) leads to ∂t Mβ + v1 ∂x Mβ +
qβ 1 E∂v1 Mβ = −gβ + (M β − Mβ ) + O(ε), mβ τei
β ∈ {e; i}.
Mass conservation equation is obtained by integrating w.r.t v and I . Next, Ampère equation leads to 1 (qe m e v1 Me I αe + qi m i v1 Mi I αi ) dvd I ε2 R3 ×R+ 1 (qe m e v1 ge I αe + qi m i v1 gi I αi ) dvd I + O(1). + ε R3 ×R+
j=
So, the right-hand side of the previous equation gives R3 ×R+
(qe ge v1 Me I αe + qi gi v1 Mi I αi ) dvd I = O(ε).
So, combining the previous equation with (16), we get R3 ×R
qβ gβ v1 Mβ I αβ dvd I = O(ε), +
β ∈ {e; i}.
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The equation of the conservation of the impulsion is recovered. Moreover, proceeding as in [1], we get Ohm’s law (13). Next the energy equation on the electrons writes
qe 1 1 (∂t Me + v1 ∂x Me )( m e v 2 + I )I αe dvd I + E∂v1 Me ( m e v 2 + I )I αe dvd I 2 2 R3 ×R+ R3 ×R+ m β 1 1 = (M e − Me ) ( m e v 2 + I )I αe dvd I. τei R3 ×R+ 2
Moreover, a straightforward computation gives
1 5 (M e − Me ) ( m e v2 + I )I αe dvd I = ( + αe )n e k B (T − Te ). 2 2 R3 ×R+
So, according to the relation (3) defining T , we get
( 5 + αe )n e ( 25 + αi )n i 1 (M e − Me ) ( m e v2 + I )I αe dvd I = 5 2 k B (Ti − Te ) 2 ( 2 + αe )n e + ( 25 + αi )n i R3 ×R+
and νei is given by (14).
4 Numerical Approximation In this section, we derive a numerical scheme starting from a semi-discretization of the kinetic model. In [1], this approach has been developed for a monoatomic gas mixture. Moreover by using the formalism of discrete BGK models [2], an analogous scheme can be obtained including the polyatomic case.
4.1 Derivation of the Numerical Scheme The spacial discretization is defined by the step Δx and the cells C j =]x j− 21 , x j+ 21 [. We consider that Δx is constant whereas the time step Δt: t0 = 0, tn+1 = tn + Δt can be variable. We use a finite volume approach: for any unknown V (x, t), we look for the approximations V jn of the average of V at time tn on the cell C j . The initial condition U 0 being given, for β = e, i, we set ρβ0 = ρ 0 cβ , n 0β = ρβ0 /m β , and u 0β = u 0 . Suppose that at time tn an approximate solution U n = (ρ n , (ρu)n , Een , Ein ) is known and that we are able to define Uen and Uin such that ρβn = ρ n cβ , u nβ = u n , β = e, i.
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So, n nβ = ρβn /m β is also well defined for β = e, i and f en (v, I ), f in (v, I ) can be computed according to f βn (v, I ) = Mβ (Uβn ), β = e, i. For Pβ defined by ⎛
⎞ mβ ⎝ ⎠ I αβ f β I αβ dvd I. mβ v Pβ ( f β ) = 2 3 R (m β v2 + I )
Pβ ( f βn ) = Uβn ,
Pβ (v1 Mβ (Uβn )) = Fβ (Uβn ),
where Fβ is the flux for Euler equations for any given γ . Definition 1. For any value of v and I , the numerical flux is defined h β, j+ 21 (v, I ) = h β ( f β, j (v, I ), f β, j+1 (v, I ), v, I ) where forall v, I , h β (., ., v, I ) is locally Lipschitz and for any f : h β ( f, f, v, I ) = v1 f. For β = e, i, we set Fβ, j+ 21 = Fβ (Uβ, j , Uβ, j+1 ),
Fβ (Uβ , Vβ ) = Pβ (h β (Mβ (Uβ ), Mβ (Vβ ), ·, ·)). (17)
As in [1], we get δ nj+ 1 = −ci Fe,n j+ 1 ,2 + ce Fi,n j+ 1 ,2 2
2
(18)
2
and we define F j+ 21 by ⎛
F j+ 21
⎞ Fe, j+ 21 ,1 + Fi, j+ 21 ,1 ⎜F 1 + Fi, j+ 21 ,2 ⎟ ⎜ ⎟ = ⎜ e, j+ 2 ,2 ⎟. Fe, j+ 21 ,3 ⎝ ⎠ Fi, j+ 21 ,3
(19)
Proposition 3. We get a scheme that is consistent with Euler system (1): for n ≥ 0 if U n = {U jn } j∈Z is the approach solution of the system (1) at time tn , we set n n n n Uβ, j = (cβ ρ j , cβ ρ j u j , Eβ ), β = e, i.
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A kinetic flux h β is chosen as in Definition 1. Define the numerical fluxes Fβ, j+ 21 , F j+ 21 and δ j+ 21 by (17)–(19). The numerical solution at time tn+1 is given by the implicit scheme ⎧ Δt n n ⎪ ⎪ = ρ nj − − F , F j+ ρ n+1 1 1 ⎪ j j− 2 ,1 ⎪ Δx 2 ,1 ⎪ ⎪ ⎪ ⎪ ⎪ Δt ⎪ n+1 n+1 n n n n ⎪ ⎪ ⎨ ρ j u j = ρ j u j − Δx F j+ 21 ,2 − F j− 21 ,2 , ⎪ Δt ⎪ n+1 n+1 Δt n+1 n n n n n ⎪ E = E − − F − δ − u + Δtνei (Ti,n+1 F δ ⎪ 1 1 1 1 e, j j j − Te, j ), ⎪ e, j e, j+ 2 ,3 e, j− 2 ,3 j+ 2 j− 2 ⎪ Δx Δx ⎪ ⎪ ⎪ ⎪ ⎪ Δt Δt n+1 ⎪ n ⎩ Ei,n+1 Fi,n j+ 1 ,3 − Fi,n j− 1 ,3 + u n+1 δ nj+ 1 − δ nj− 1 − Δtνei (Ti,n+1 j = Ei, j − j j − Te, j ). Δx Δx 2 2 2 2
In the present paper, we specify the following kinetic flux h β ( f, g, v, I ) = v1
λ1 λ1 λ3 λ3 f (v, I ) − g(v, I ) − (g(v, I ) − f (v, I )), λ3 − λ1 λ3 − λ1 λ3 − λ1
(20) where λ1 and λ3 are constants that have to be fixed. This flux gives rise to a HLL scheme that is computed in the following subsection.
4.2 Numerical Results In the following test cases, the kinetic HLL scheme corresponding to the kinetic flux (20) is implemented and tested.
4.2.1
Stationary Shock Test Case
The left and right states of the Riemann problem are the following: ρ L = 1.001, ρ R = 3.640330609,
u L = 10, Te,L = 1, u R = 2.749750250, Te,R = 3.,
Ti,L = 1, Ti,R = 17.5060240977.
Then the solution of the 3 × 3 Euler system is a stationary shock. The solutions are computed for x ∈ [0, 1] at time t = 0.05 with 1000 cells for γe = γi = 53 . In this case, we compare the results obtained with the kinetic scheme with the results given by a relaxation method and when it is possible with an analytical solution. The results are depicted Figs. 1 and 2. For νei = 0, we observe a contact discontinuity propagating at velocity u R = 2.749750250, see Fig. 2 left. For the 3 × 3 Euler equations, u is a single eigenvalue and the left and right values of a contact discontinuity lie on an integral
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HLL electronic HLL ionic Relaxation electronic Relaxation ionic
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HLL electronic HLL ionic Relaxation electronic Relaxation ionic
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Fig. 2 Electronic and ionic temperatures computed with HLL and relaxation schemes for the stationary shock problem. Left : νei = 0, right : νei = 100
curve of the eigenvector (−ρ, 0, ε), in (ρ, u, ε) variables. Thus, a contact discontinuity must involve a jump of ρ. For the Euler bitemperature system, the eigenvalue u is double and the eigenvectors are r2 = (0, 0, −(γi − 1)ci , (γe − 1)ce ) and r3 = (−ρ, 0, εe , εi ). The observed contact discontinuity lies on an integral curve of r2 : only electronic and ionic temperatures jump. We also observe that the intermediate values of those temperatures differ with respect with the scheme. This is due to the fact that each scheme has a different viscosity, and therefore converges to a different interpretation of the nonconservative products in the equations. For νei = 100, both schemes give similar results. For all x, density, velocity, and total energy remain the same and are not depicted again. Electronic and ionic temperatures are represented in Fig. 2 right. For x < 0.5, as Ti = Te , the value of νei does not influence the solution. For x > tu R , by finite propagation speed, one can compute the value of Te and Ti . For x ∈ [0, tu R ], the result is qualitatively in coherence with the physical behavior of the plasma, as predicted in Zeldovitch [5]. In particular, we observe a high ionic temperature at the shock, and then a decrease, while the electronic temperature increases.
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Exact solution HLL solution
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4.2.2
Analytical Test Case
We consider now a purely polyatomic case, where for νei = 0 we have an analytical solution. It is possible to choose any values of γe and γi , but in the present case we take γe = 75 and γi = 53 . This means that the electrons are considered as a diatomic gas whereas ions are a monoatomic gas. The initial conditions are chosen such as ρ = 1, u = 3, on [0, 1]. The initial condition for the electronic temperature is chosen as
T e0 = 1 if x ≤ 0.5, T e0 (x) = 1 + exp −200(x − 0.5)2 , x ≥ 0.5 The initial condition on ionic temperature is chosen in order to satisfy p = pe + pi = 2. So, Ti0 is defined Ti0 = kmB ρi i (2 − n e k B Te0 ). In that case, an analytical solution can be computed. Indeed ρ and u are constant given by ρ = 1 and u = 3. Moreover, the analytical solution for the electronic temperature is written as T e(t, x) = T e0 (x − ut) +
p . nk B
The solution is computed at time t = 0.5 on the space interval [0, 1] with 2000 cells. The ionic and the electronic temperatures are displayed in Fig. 3.
References 1. D. Aregba-Driollet, J. Breil, S. Brull, B. Dubroca, E. Estibals, Modelling and numerical approximation for the non conservative bitemperature Euler system, Appear in M2AN Math. Model. Numer. Anal
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2. D. Aregba-Driollet, R. Natalini, Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J. Numer. Anal. 37(6), 1973–2004 (2000) 3. L. Desvillettes, R. Monaco, F. Salvarani, A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions. Eur. J. Mech. B/Fluids 2, 219–236 (2005) 4. J.M. Greene, Improved Bhatnaga-Gross-Krook model of electron-ion collisions. Phys. Fluids 16, 2022–2023 (1973) 5. B. Zel’dovich, P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena (Academic Press, Dublin, 1966)
An Implicit–Explicit Scheme Accurate at Low Mach Numbers for the Wave Equation System K. R. Arun, A. J. Das Gupta and S. Samantaray
Abstract We design and analyse a first-order accurate implicit–explicit (IMEX) scheme for the two-dimensional wave equation system in the low Mach number limit. It has been shown by Dellacherie [S. Dellacherie, Analysis of Godunov-type schemes applied to the compressible Euler system at low Mach number. J. Comput. Phys., 229(4): 978–1016, 2010, [1]] that the standard Godunov-type numerical schemes suffer from a severe loss of accuracy at low Mach numbers. This inaccuracy arises due to the inability of schemes to preserve the incompressible space of constant densities and divergence-free velocities. Guided by this principle, we design an IMEX scheme which possess the invariance property, and analyse its stability. The proposed scheme has been shown to be stable under a usual CFL-like condition, independent of the Mach number. The results of numerical experiments confirm the accuracy and stability of the new scheme when applied to low Mach number problems. Keywords Weakly compressible flow · IMEX-Runge-Kutta schemes Low Mach number limit · Asymptotic preserving schemes Semi-implicit discretization
1 Introduction Many physical processes involve the transition of behaviour of a fluid from compressible to incompressible regime or vice versa. Most of the current flow solvers are inadequate for such problems, since they are designed to simulate either compressible K. R. Arun · A. J. Das Gupta (B) · S. Samantaray School of Mathematics, Indian Institute of Science Education and Research Thiruvananthapuram, Thiruvananthapuram 695016, India e-mail: [email protected] K. R. Arun e-mail: [email protected] S. Samantaray e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_8
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or incompressible flows. One way to build a robust numerical model which takes into account the change of flow nature is to extend a compressible flow solver for the incompressible regime. However, this requires a strong mathematical analysis of the governing equations and the knowledge of the change in behaviour of each flow variable in the zero Mach number limit. Asymptotic analysis is one of the most efficient methods of approach for such mathematical studies. Godunov-type schemes are well adapted to capture discontinuous solutions, such as those containing shock waves. Nonetheless, they are not accurate when applied to flow problems at low Mach numbers; see [2] for a detailed discussion. Thus, it is necessary to modify Godunov-type schemes in a suitable way before employing them for low Mach number computations. A key step towards such modifications is a thorough understanding of the origin of inaccuracies. Recently, such a study for the Euler equations has been carried out by Dellacherie [1]. In [1], Dellacherie explains the inaccuracies of Godunov-type schemes at low Mach numbers on a Cartesian mesh. His work is based on the invariance of a socalled well-prepared subspace for the linear wave equation, and the loss of accuracy of compressible flow solvers at low Mach numbers result from their inability to preserve this well-prepared subspace. He also proposes a sufficient condition to ensure the invariance so that no spurious waves are generated by the schemes at low Mach numbers. In this paper, we consider the wave equation system with periodic boundary conditions on a d-dimensional torus. However, for the sake of simplicity, we present the discretisation only in the 2-D case. We derive a first-order accurate implicit– explicit (IMEX) scheme, satisfying the sufficient condition given by Dellacherie. Hence, the proposed scheme does not suffer from any inaccuracies when the Mach number goes to zero. Further, an L 2 stability analysis of the scheme shows that it is stable under a usual CFL-like condition which is independent of the Mach number. The results of some numerical case studies involving low Mach number problems are presented to validate our claims.
2 Low Mach Number Limit of the Wave Equation We consider the two-dimensional (2-D) scaled wave equation system with advection: ¯ · u = 0, ∂t ρ + (u¯ · ∇)ρ + ρ∇
(1)
a¯ ∇ρ = 0, ρε ¯ 2
(2)
∂t u + (u¯ · ∇)u +
2
where the unknowns are the density ρ > 0 and the velocity u = (u 1 , u 2 )T ∈ R2 . The wave equation system (1)–(2) is derived by linearising the isentropic Euler equations of gas dynamics about a constant linearisation state ρ = ρ¯ and u = (u¯ 1 , u¯ 2 ) with the linearised sound speed denoted by a. ¯ Here, ε := u ref /aref denotes a reference Mach number which is the ratio of a reference fluid speed to a reference sound speed.
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It has been rigorously shown in [3, 5] that the solutions of the compressible Euler equations converge to that of the incompressible Euler equations in the limit ε → 0. On the other hand, it is well known from literature on numerical schemes for the Euler equations that the conventional Godunov-type schemes suffer from a severe loss of accuracy when applied to low Mach number problems; see e.g. [2, 4] for details. Recently, Dellacherie [1] has analysed this problem of inaccuracy by considering the linear wave equation (1)–(2) as a simple model of the Euler equations. The main cure for inaccuracy which has been proposed in [1] is that the numerical solution at any time is close to an incompressible solution if the initial data is close to an incompressible data. A sufficient condition which ensures the above requirement is the invariance of a so-called ‘well-prepared’ space by the solution operator. The goal of the present work is to derive and analyse an IMEX scheme which has this invariance property. Before proceeding further, we recall some of the results from [1] which are relevant for the present work. Let us introduce the notations ρ∇ ¯ ·u ρ ρ . (3) U := , H (U ) := (u¯ · ∇) , L(U ) := a¯ 2 u u ∇ρ ρε ¯ 2 We consider a boundary value problem for (1)–(2) with periodic boundary conditions, i.e. we take our spatial domain to be Ω = Td which is the d-dimensional torus. We define the spaces E := U ∈ L 2 (Td )1+d : ∇ρ = 0, ∇ · u = 0 ,
ρd x = 0, ∇ × u = 0 . E ⊥ := U ∈ L 2 (Td )1+d :
(4) (5)
Td
Note that the space E is the space of all spatially constant densities and divergencefree velocities. In other words, it is the incompressible subspace of L 2 (Td )1+d and hereafter, we designate it as the well-prepared space. The Hodge decomposition theorem states that any U ∈ L 2 (Td )1+d can be uniquely decomposed as U = U˘ + U ⊥ , where U˘ ∈ E and U ⊥ ∈ E ⊥ . In the sequel, P : L 2 (Td )1+d → E denotes the projection map from L 2 (Td )1+d to E . Following [1], the convergence theorem of [5] can be stated in the simpler form: Theorem 1. Let U (t, x) denotes a solution of the initial value problem ∂t U + H (U ) + L(U ) = 0, t > 0, x ∈ Td , U (0, x) = U0 (x), x ∈ Td .
(6)
Let U1 (t, x) be a solution of the incompressible system ∂t U1 + H (U1 ) = 0, t > 0, x ∈ Td U1 (0, x) = PU0 (x), x ∈ Td .
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Then, U1 = PU and there holds the estimate U0 − PU0 = O(ε) implies U (t) − PU (t) = O(ε), for all t > 0.
(8)
Theorem 1 is an easy consequence of the conservation of energy and the invariance of the spaces E and E ⊥ by the solutions of (6). Next, we pass onto the construction of a numerical scheme for the wave equation system which maintains the E -invariance property stated above. However, the standard upwind discretisations introduce some amount of numerical viscosity required for stability and hence the problem which is solved numerically will not be hyperbolic, but parabolic in nature. In order to address this, we consider the following theorem of [1]: Theorem 2. Let U (t, x) be a solution to the initial value problem ∂t U + L (U ) = 0, t > 0, x ∈ Td , U (0, x) = U0 (x), x ∈ Td ,
(9)
where L is a linear spatial differential operator. Assume that the above initial value problem (9) is well-posed in L ∞ ([0, ∞), L 2 (Td )1+d ). If the operator L has the invariance property: U0 ∈ E implies U (t) ∈ E for all t > 0,
(10)
then the solution U (t) satisfies the norm estimate U0 − PU0 = O(ε) implies U (t) − PU (t) = O(ε) for all t > 0.
(11)
3 An IMEX Scheme Accurate at Low Mach Number It follows from the discussion in Sect. 2 that the key property a discretisation should possess, to be accurate at low Mach numbers, is the E -invariance. At the same time, it is also essential that the stability characteristics of the scheme should not degenerate as ε → 0. Since E = ker(L), henceforth, we shall designate L(U ) as the stiff part of the wave equation and the convective part H (U ) will be called the non-stiff part.
3.1 Time Discretisation Let 0 = t 0 < t 1 < · · · < t n < · · · be an increasing sequence of times. Let f n (x) denote an approximation to the value of a function f (t, x) at time t n . Discretising the non-stiff part explicitly and the stiff part implicitly yields the semi-discrete scheme:
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ρ n+1 = ρ n − Δt (u¯ · ∇)ρ n − ρΔt∇ ¯ · u n+1 ,
(12)
a¯ Δt∇ρ n+1 . ρε ¯ 2
(13)
u n+1 = u n − Δt (u¯ · ∇)u n −
2
In order to solve (12)–(13) for ρ n+1 and u n+1 , we first eliminate u n+1 between them to get the elliptic equation: −
a¯ 2 2 n+1 Δt Δρ + ρ n+1 = Ψ (ρ n , u n ), ε2
(14)
where Ψ (ρ n , u n ) = ρ n − Δt (u¯ · ∇)ρ n − ρΔt∇ ¯ · u n + ρΔt ¯ 2 ∇ · (u¯ · ∇)u n
(15)
is a known expression. Solving the elliptic equation (14) gives the updated value ρ n+1 of density. The velocity update (13) can now be evaluated explicitly to get u n+1 . Hence, the scheme consists of only two steps: solution of the elliptic problem (14) which is followed by an explicit evaluation of (13). In what follows, we easily establish the E -invariance property of the semi-discrete scheme. Theorem 3. Suppose that the functions ρ n and u n at t n satisfy ρ n (x) = const and ∇ · u n (x) = 0 for all x ∈ Td . Then, the updated functions ρ n+1 and u n+1 determined from (12)–(13) satisfy ρ n+1 (x) = const, ∇ · u n+1 (x) = 0, for all x ∈ Td .
(16)
Proof. Since ρ n = const and ∇ · u n = 0, the right hand side function Ψ defined in (15) simplifies into Ψ (ρ n , u n ) = ρ n . A simple application of the strong maximum principle for elliptic equations to (14) then yields that ρ n+1 = ρ n = const. It follows easily from (12) that ∇ · u n+1 = 0.
3.2 Space Discretisation This section is devoted to the space discretisation of the semi-discrete scheme (12)– (13). Let the spatial domain Ω be discretised into rectangular mesh cells of lengths Δx1 and Δx2 , respectively in the x1 and x2 directions. Let λ1 := Δt/Δx1 and λ2 := Δt/Δx2 denote the fixed mesh ratios. Let us also introduce the finite difference and averaging operators, e.g. in the x1 direction, δx1 ωi, j := ωi+ 21 , j − ωi− 21 , j , μx1 ωi, j := with analogous notations in the x2 direction.
ωi+ 21 , j + ωi− 21 , j 2
,
(17)
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In the following, we use the convention that a repeated index always denote summation from 1 to 2. Introducing numerical fluxes to approximate the space derivatives appearing in (12)–(13) yields the fully discrete scheme: n ˘n ˜ n+1 Ui,n+1 j = Ui, j − λk δxk Fk i, j − λk δxk Fk i, j .
(18)
The numerical fluxes are defined by, e.g. in the x1 direction, 1 n 1 n n n n ˘ F˘1 i+ 1 , j = A˘ 1 Ui+1, j + Ui, j − D Ui+1, j − Ui, j , 2 2 2
1 1 n+1 n+1 n+1 n+1 n+1 , − U F˜1 i+ 1 , j = A˜ 1 Ui+1, j + Ui, j − D˜ Ui+1, j i, j 2 2 2 where
⎛ ⎞ ⎞ 0 ρ¯ 0 u¯ 1 0 0 a¯ 2 A˘ 1 = ⎝ 0 u¯ 1 0 ⎠ , A˜ 1 = ⎝ ρε 0 0⎠ ¯ 2 0 0 u¯ 1 0 00
(19) (20)
⎛
(21)
are respectively the Jacobians of H and L and ⎛ ⎞ ⎛ ⎞ ν˘ ρ 0 0 ν˜ ρ 0 0 D˘ = ⎝ 0 ν˘ 1 0 ⎠ , D˜ = ⎝ 0 ν˜ 1 0 ⎠ 0 0 ν˘ 2 0 0 ν˜ 2
(22)
are the numerical diffusion matrices. Analogous expressions hold also for the fluxes in x2 direction. Note that we apply the same numerical diffusion matrices D˘ and D˜ in both x1 and x2 directions for the sake of simplicity. Substituting the expressions for the numerical fluxes in (18), we get the fully discrete update formulae: n n ρi,n+1 j = ρi, j − λk u¯ k δx k μx k ρi, j + n n u 1 i,n+1 j = u 1 i, j − λk u¯ k δx k μx k u 1 i, j
ν˘ ρ ν˜ ρ ¯ k δxk μxk u k i,n+1 (23) λk δx2k ρi,n j − ρλ λk δx2k ρi,n+1 j , j + 2 2 ν˘ 1 a¯ 2 ν˜ 1 + λk δx2k u 1 i,n j − 2 λk δxk μxk ρi,n+1 λk δx2k u 1 i,n+1 j + j , 2 ρε ¯ 2
(24)
n n u 2 i,n+1 j = u 2 i, j − λk u¯ k δx k μx k u 2 i, j
ν˘ 2 a¯ 2 ν˜ 2 + λk δx2k u 2 i,n j − 2 λk δxk μxk ρi,n+1 λk δx2k u 2 i,n+1 j + j . 2 ρε ¯ 2
(25)
The above Eqs. (23)–(25) define a fully discrete scheme, provided the numerical diffusion matrices D˘ and D˜ are specified. Before proceeding further, we first obtain some sufficient conditions on the numerical diffusion coefficients so that the scheme (23)–(25) possess the E -invariance property. Theorem 4. Suppose that the matrices D˘ and D˜ are positive-definite and that the condition (26) ν˘ 1 + ν˜ 1 = ν˘ 2 + ν˜ 2
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is satisfied. Then, the fully discrete scheme (23)–(25) has the E -invariance property. Proof. Substituting the exact solution values (ρ, u) of the wave equation in the discrete formulae (23)–(25) and using the Taylor expansions yield the modified equations: ρ∇ ¯ ·u ρ ρ ρ 2 ∂t =B + (u¯ · ∇) , + a¯ u u u ∇ρ 2 ρε ¯
(27)
where the operator B is defined by ⎞ ⎛ 2 0 Δt ⎝ aε¯ 2 Δ − (u¯ · ∇)2 2 2 ˜ ⎠ + 1 ( D˘ + D)(Δx B := 1 ∂x1 + Δx 2 ∂x2 ). 2 a ¯ 2 2 2 0 ∇ ⊗ ∇ − ( u ¯ · ∇) I 2 2 ε
(28)
Since the initial value problem (6) is posed on the torus Td , we can work on the whole Rd by assuming periodic extension of the unknown variables. It is convenient to work using the Fourier variables as it simplifies the calculations. Taking the Fourier transform of (27) in space gives ρξ ¯ · uˆ ρˆ ρˆ ρˆ ∂t = Bˆ + i(u¯ · ξ ) , + i a¯ 2 uˆ uˆ uˆ ξ ρ ˆ 2 ρε ¯
(29)
where ⎞ ⎛ a¯ 2 ξ 2 − (u¯ · ξ )2 0 Δt 2 2 2 ˜ ⎠ − 1 ( D˘ + D)(Δx ⎝ε Bˆ := − 1 ξ1 + Δx 2 ξ2 ). a¯ 2 ξ ⊗ ξ − (u¯ · ξ )2 I 2 2 0 2 2 ε
(30)
Note that (ρ, u) ∈ E if, and only if, (ρ, ˆ u) ˆ ∈ Eˆ , where Eˆ is given by ˆ ξ ) = 0 and ξ · u(·, ˆ ξ ) = 0 for all ξ ∈ Rd }. (31) Eˆ := {(ρ, ˆ u) ˆ ∈ L 2 (Td )1+d : ρ(·, Hence, the quantities of interest are ρˆ and ξ · u. ˆ Since the first equation of (29) is already an equation for ρ, ˆ we take the dot product of the second equation for uˆ in (29) with ξ to get ˆ + i(u¯ · ξ )(ξ · u) ˆ +i ∂t (ξ · u)
a¯ 2 Δt a¯ 2 2 2 2 (ξ · u) ˆ ( u ¯ · ξ ) ρξ ˆ = − ξ ρε ¯ 2 2 ε2 (32)
1 + (Δx1 ξ12 + Δx2 ξ22 ){(˘ν1 + ν˜ 1 )ξ1 uˆ 1 + (˘ν2 + ν˜ 2 )ξ2 uˆ 2 }. 2
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Under the assumption ν˘ 1 + ν˜ 1 = ν˘ 2 + ν˜ 2 , the above Eq. (32) reduces to an equation for ξ · u. ˆ Combining it with the first equation in (29) for ρˆ yields a linear system:
ρˆ ξ · uˆ
∂t
ρˆ = M(ξ ) . ξ · uˆ
(33)
Here, the elements of the matrix M(ξ ) are polynomials in ξ . Let us suppose that (ρ(0, ·), u(0, ·)) ∈ E , i.e. ρ(0, ˆ ξ ) = ξ · u(0, ˆ ξ ) = 0. The solution of (33) then clearly shows that ρ(t, ˆ ξ ) = ξ · u(t, ˆ ξ ) = 0. Hence, E is invariant.
4 L 2 -Stability Analysis The aim of this section is to present the results of an L 2 -stability analysis of the fully discrete scheme (23)–(25). For convenience, for U = (ρ, u) and U = (ρ , u ), let us introduce the new inner product: (U, U ) :=
a¯ 2
ρ, ρ + ρ u, ¯ u , ρε ¯ 2
(34)
where ·, · denotes the usual L 2 inner product. It is very easy to verify that the norm induced by this new inner product is equivalent to the usual norm in L 2 . With the help of (·, ·), we define the energy of U ∈ L 2 (Td )1+d as E(t) := (U (t), U (t)).
(35)
From the previous Theorem 4, we notice that the diffusion matrices D˘ and D˜ ˜ Henceforth, we set νρ = ν˘ ρ + ν˜ ρ and νu = ν˘ 1 + ν˜ 1 = always occur as a sum D˘ + D. 2 ν˘ 2 + ν˜ 2 . The L -stability of the scheme is as follows. Theorem 5. Under the CFL-like condition Δt ≤
min(νρ , νu ) min(Δx1 , Δx2 ), u ¯ 2
(36)
the scheme (23)–(25) is L 2 -stable. Proof. Note that dE 2a¯ 2 ¯ ∂t u . = 2(U, ∂t U ) = 2 ρ, ∂t ρ + 2ρ u, dt ρε ¯
(37)
Using the modified Eq. (27) in (37), we find that the first-order derivative terms on the left hand side of (27) do not contribute to the inner products due to
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periodic boundary conditions. Applying integration by parts to the remaining terms lead to
νρ a¯ 2 2 2 2 (Δx1 ∂x1 + Δx2 ∂ x2 )ρ Δ − (u¯ · ∇) ρ + (38) 2 ε2 νρ a¯ 2 Δt 2 2 2 2 ∇ · (ρ u) ¯ (Δx1 ∂x1 ρ + Δx2 ∂x2 ρ ) ∇ρ + − 2 2 ε2 a¯ 2 2 − νρ (Δx ∂ ρ2 + Δx ∂ ρ2 ) − Δt ∇ρ2 u 2 ∇ρ ¯ 1 x1 2 x2 2 2 ε2 2 2 ¯ ¯ 2 a¯ 2 − νρ Δx 1 − Δtu 2 + νρ Δx 2 − Δtu 2 ∇ρ ρ ρ ∂ ∂ x1 x2 2 2 ε2
Δt
ρ, ∂t ρ = ρ, 2 =−
Δt 2
≤−
Δt 2
=−
Δt 2
and
a¯ 2 Δt Δt νu (u¯ · ∇)2 u + (Δx1 ∂x21 + Δx2 ∂ x2 2 )u
u, ∂t u = u, ∇ ⊗ ∇u − 2 2 2ε2
(39)
a¯ 2 Δt Δt νu (u¯ · ∇)u2 − (Δx1 ∂x1 u2 + Δx2 ∂x2 u2 ) ∇ · u2 + 2 2 2ε2 2 2 νu Δx1 − Δtu ¯ ¯ 2 νu Δx2 − Δtu a¯ Δt 2 2 2 ≤ − 2 ∇ · u − ∂x1 u + ∂x2 u . 2 2 2ε =−
Hence, from (37)–(39), we infer that under the following CFL-like condition: Δt ≤
min(νρ , νu ) min(Δx1 , Δx2 ) u ¯ 2
(40)
the energy defined in (35) dissipates, leading to L 2 -stability of the scheme.
5 Numerical Test Problems In this section, we present the results of two numerical case studies to demonstrate the performance of the proposed scheme. In all the test problems considered, the implicit fluxes are discretised using simple central differences and the explicit fluxes are approximated using the Rusanov flux. Note that this choice of the flux discretisations is in accordance with the current framework.
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5.1 Order of Convergence and Accuracy This is a one-dimensional test problem, describing the advection of a pure long wave, and the aim of this problem is to show the uniform accuracy of the scheme with respect to ε. First, we consider the wave equation (1)–(2) in one dimension and transform the dependent variables into characteristic variables w1 and w2 , corresponding to the eigenvalues u¯ + a/ε ¯ and u¯ − a/ε. ¯ In order to derive the initial data, we set w2 = ¯ u(t, x1 ). Let us choose const = 1/2; which yields the condition ρ(t, x1 ) = 1 + ρε a¯ the well-prepared initial data: ρε ¯ 2 sin(2π x1 ), a¯ u(0, x1 ) = ε sin(2π x1 ).
ρ(0, x1 ) = 1 +
The linearisation parameters are initialised as (ρ, ¯ u, ¯ a) ¯ = (1, 1, 1.2). The computational domain is [0, 1] and we apply periodic boundary conditions. The computations are done up to the time taken by the pulse to complete three cycles. The following table gives the L 1 and L 2 error computed using the exact solution and the experimental order of convergence (EOC) obtained. When ε = 0.5, the time-step Δt is chosen as Δt = Δx1 /4, so that the advective CFL number is 0.25 and the acoustic CFL number is 0.85. When ε = 0.05, we set Δt = Δx1 /10 and hence, the advective CFL number is 0.1 and the acoustic CFL number is 2.5. The Tables 1 and 2 clearly shows that the scheme achieves first-order convergence for ε = 0.5 and ε = 0.05 respectively. We have also tested the EOC for a non well-prepared initial data: ρε ¯ sin(2π x1 ), a¯ u(0, x1 ) = sin(2π x1 )
ρ(0, x1 ) = 1 +
and our results show uniform first-order accuracy with respect to ε also for this case. Table 1 L 1 and L 2 erros in ρ and u along with EOC for Sect. 5.1 corresponding to ε = 0.5 N L 1 error in ρ L 1 error in u EOC L 2 error in ρ L 2 error in u EOC 20 40 80 160 320 640 1280
0.115539 0.082288 0.050634 0.028294 0.014983 0.007747 0.003996
0.273778 0.194993 0.119996 0.067056 0.035509 0.018361 0.009470
0.48962 0.70057 0.83961 0.91716 0.95161 0.95508
0.128122 0.091493 0.056264 0.031429 0.016642 0.008614 0.004436
0.303599 0.216807 0.133337 0.074485 0.039441 0.020414 0.010513
0.48578 0.70144 0.84011 0.91726 0.95007 0.95742
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Density at t = 0
Density at t = 1
2.5
1.0001
2
1.0001
1.5
rho
rho
1.0001
1 1
1 0.5 0 1
1 0.75 0.5 0.25 0 0
x2
0.25 x1
0.5
0.75
1
0.75 0.5 0.25
0.25 x1
0.75
1
x2 Velocity at t = 1
0.4
0.4
0.2
0.2
0
ux 2
1
x1 Velocity at t = 1
ux
0 0
x2
0.5
−0.2
0
−0.2
−0.4 1 0.75 0.5 0.25 0 0
x2
0.25
0.5
0.75
−0.4 1
1
0.75 0.5 0.25 x
x1
Divergence of Velocity at t = 0
0 0 2
0.25 x
0.5
0.75
1
1
Divergence of Velocity at t = 1
1
Div(ux1,ux2)
0.5
0
1
Div(ux ,ux2)
0.05
0
−0.5
−0.05 1 0.75 0.5 0.25 0 0 x2
0.25
0.5
0.75
1
x1
−1 1 0.75 0.5 0.25 x2
0 0
0.25
0.5
0.75
1
x1
Fig. 1 Density, velocity components, and divergence of the velocity computed at t = 1 with ε = 0.01
5.2 Asymptotic Preserving Property In this section, we present the results of a 2-D test problem and the aim of this numerical experiment is to demonstrate the ability of the scheme to capture the incompressible solution as ε → 0. The initial data are well-prepared which read ρ(0, x1 , x2 ) = 1 + ε2 sin2 (2π(x1 + x2 )), u 1 (0, x1 , x2 ) = sin(2π(x1 − x2 )) + ε sin(2π(x1 + x2 )), u 2 (0, x1 , x2 ) = sin(2π(x1 − x2 )) + ε cos(2π(x1 + x2 )).
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Table 2 L 1 and L 2 erros in ρ and u along with EOC for Sect. 5.1 corresponding to ε = 0.05 N L 1 error in ρ L 1 error in u EOC L 2 error in ρ L 2 error in u EOC 20 40 80 160 320 640 1280
0.001354 0.001307 0.001100 0.000771 0.000467 0.000258 0.000137
0.032099 0.030986 0.026068 0.018271 0.011065 0.006124 0.003235
0.0509 0.2487 0.5127 0.7233 0.8560 0.9131
0.001498 0.001451 0.001222 0.000856 0.000519 0.000287 0.000151
0.035507 0.034399 0.028958 0.020293 0.012291 0.006807 0.003590
0.0459 0.2478 0.5135 0.7218 0.8546 0.9265
The initial density and velocity are smooth and periodic and we set ε = 0.01. The linearisation parameters are taken as in the previous problem. The computational domain is [0, 1] × [0, 1] and the boundaries are periodic. We run the numerical simulations on a coarse mesh of 50 × 50 points with a time-step Δt = 0.009. Since the initial data are well prepared, we expect the convergence of the solution to the asymptotic solution ρ (0) (t, x1 , x2 ) = 1, u 1 (0) (t, x1 , x2 ) = u 2 (0) (t, x1 , x2 ) = sin(2π(x1 − x2 )) which is incompressible. In Fig. 1, we plot the density at t = 0 and t = 1, the velocity components at t = 1 and the divergence of the velocity at t = 0 and t = 1. The figure clearly shows that at time t = 1, the density has the constant value 1, both the velocities are identical, and the divergence of the velocity is 0. Hence, we conclude that the scheme is able to capture the incompressible limit solution as ε → 0.
6 Conclusions In this paper, we have derived and analysed a first-order accurate IMEX-RK finite volume scheme for the wave equation system in the low Mach number limit. The accuracy of the scheme is ensured by its ability to leave a well-prepared space of constant densities and divergence-free velocities invariant. The stability requirements of the scheme are shown to be independent of the Mach number. The results of numerical experiments validate the accuracy, stability, and robustness of the proposed scheme.
References 1. S. Dellacherie, Analysis of Godunov type schemes applied to the compressible Euler system at low mach number. J. Comput. Phys. 229(4), 978–1016 (2010) 2. H. Guillard, C. Viozat, On the behaviour of upwind schemes in the low mach number limit. Comput. Fluids 28(1), 63–86 (1999)
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3. S. Klainerman, A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math. 34(4), 481–524 (1981) 4. R. Klein, Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics I: one-dimensional flow. J. Comput. Phys. 121(2), 213–237 (1995) 5. S. Schochet, Fast singular limits of hyperbolic PDEs. J. Differ. Equ. 114(2), 476–512 (1994)
Bose–Einstein Condensation and Global Dynamics of Solutions to a Hyperbolic Kompaneets Equation Joshua Ballew
Abstract In this article, a simplified, hyperbolic model of the nonlinear, degenerate parabolic Kompaneets equation for the number density of photons is considered. It is shown that for non-negative, compactly supported initial data, weak solutions obeying a Kružkov entropy condition are unique. Other consequences for entropy solutions resulting from a contraction estimate are explored. Certain properties of entropy solutions are investigated and convergence in time of entropy solutions with compactly supported initial data to stationary solutions is shown. The development of a Bose–Einstein condensate for initial data under certain conditions is proven. It is also shown that the total number of photons not in a Bose–Einstein condensate is non-increasing in time, and that any such loss of photons is only to the condensate. Keywords Bose–Einstein Condensation · Hyperbolic · Kompaneets Equation
1 Introduction Compton scattering is the dominant process for energy transport in low-density or high-temperature plasmas. The seminal work of Kompaneets [6], which was published in 1957, derives an equation modeling the behavior of this scattering. Kompaneets’ work today has applications in several areas of astrophysics including the interaction between matter and radiation early in the history of the universe and black holes [2, 9, 10]. In his work, Kompaneets considers the regime of a nonrelativistic, spatially uniform, and isotropic plasma at a constant temperature and derives a Fokker–Planck approximation for the Boltzmann–Compton scattering. The photons’ heat capacities are taken to be negligible. The equation that governs the evolution of the photon density f is
J. Ballew (B) Carnegie Mellon University, 5000 Forbes Ave., Pittsburgh, PA 15213, USA e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_9
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∂t f =
1 4 ∂x x ∂x f + f + f 2 . 2 x
(1)
The photon density f is a function of the non-dimensionalized energy x ∈ (0, ∞) and time t ∈ [0, ∞). The total number of photons is given by
∞
N (t) :=
x 2 f (x, t) dx
(2)
0
and the total energy of the photons is given by E(t) :=
∞
x 3 f (x, t) dx.
(3)
0
It is also known that (1) possesses an entropy structure. More specifically, the quantum entropy ∞ x 2 h(x, f (x, t)) dx (4) H (t) := 0
where h(x, y) := y ln y − (1 + y) ln(1 + y) + x y
(5)
formally dissipates in time (see [4, 8]). This suggests that the solutions to (1) converge to some equilibrium solution as t → ∞. Such non-negative solutions are given by f μ (x) :=
1 e x+μ − 1
(6)
for μ ≥ 0. Taking the total photon number for each of these equilibrium solutions by using (2) yields an upper bound for the total number of photons at equilibrium of
∞
sup μ≥0
0
x2 dx = 2ζ (3) < ∞ e x+μ − 1
(see, for example, [1]). Thus, if the initial photon number is greater than this quantity, there must be some loss of photons as t → ∞. However, the Kompaneets equation for photon number density ∂t n = ∂x
x 2 − 2x)n + n 2 + x 2 ∂x n
(7)
shows that formally, the total photon number must be conserved, which is not possible if the initial photon number is large. Previous work on the Kompaneets equation suggests that an out-flux of photons at x = 0 can occur due to a concentration of low-energy photons. In the literature, this is interpreted as a Bose–Einstein condensate. It is noted, however, that physically, there may be other effects at play, such as Bremsstrahlung radiation which would
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tend to suppress such an out-flux at x = 0. It is still worthwhile to investigate (1) mathematically to increase the understanding of how a photon flux at x = 0 can develop due to Compton scattering. Following conventions in previous work, the x = 0 out-flux is referred to as a Bose–Einstein condensate. In order to investigate the Kompaneets equation, this workconsiders a hyperbolic model obtained from (7) by neglecting the diffusive term ∂x x 2 ∂x n : ∂t n + ∂ x
2x − x 2 n − n 2 = 0.
(8)
The work in [8] found the omitted diffusion term to have a negligible contribution to the flux at x = 0 in the limit of small x. In addition, the model (8) has infinitely many stationary solutions. The largest of these agrees with the classical Bose–Einstein distribution near x = 0. The model (8) is considered on the domain x > 0, t > 0. It is also assumed that as x → ∞, F(x, n) := x 2 − 2x n − n 2 → 0 based on physical considerations. There is no boundary condition imposed at zero. Even so, there is a uniqueness result for this model. For convenience, the initial data will have compact support; this property is propagated to all times t > 0. The rest of this article is a summary of the results for the dynamics of (8) from [1]. In Sect. 2, the notion of solution to be considered is defined and the main results stated. In Sect. 3, results related to the L 1 -contraction for (8) are discussed. In Sect. 4, the regularity and compactness of solutions are investigated. Section 5 explains how the lemmas and propositions lead to the results in the main theorem and also proves some corollaries the main result. Finally, in Sect. 6, future research plans are discussed.
2 Definitions and Main Results In this article, the concern is with entropy solutions (8). These are defined with the following definition. Definition 1. Let T > 0. The function n : [0, T ] × [0, ∞) is a weak solution to (8) if n ∈ L 1 ([0, T ] × [0, ∞)) ∩ L 1 [0, T ] ; L 2 [0, ∞) and for each test function φ ∈ Cc∞ ((0, T ) × (0, ∞)), 0
T
∞
n(x, t)∂t φ(x, t) + F(x, n(x, t))∂x φ(x, t) dx dt = 0.
(9)
0
In addition, n is called an entropy solution to (8) if for each non-negative test function φ ∈ Cc∞ ((0, T ) × (0, ∞)),
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0
T
∞
|n − k| ∂t φ + sgn(n − k) [F(x, n) − F(x, k)] ∂x φ
0
−sgn(n − k)Fx (x, k)φ dx dt ≥ 0
(10)
for any k ∈ R. The formulation (10) is called the Kružkov entropy (see [7]). The existence of entropy solutions is proved using a vanishing viscosity technique along the lines of [7, Sects. 4 and 5]. The proof of the following is found in [1, Sect. 3.3] and uses standard extension and vanishing viscosity techniques (see [3, 5] for example), and so is omitted here. Proposition 1. Let n 0 ∈ L 1 [0, ∞) be non-negative with support on [0, R] for some R > 2. Then there exists a non-negative entropy solution to (8) in the sense of Definition 1. The main result of the analysis of the hyperbolic Kompaneets equation considered here and proven in the ensuing sections is as follows. Theorem 1. Let n 0 ∈ L 1 [0, ∞) be non-negative and compactly supported on [0, R] for some R > 2. Then there exists a unique, non-negative, global-in-time entropy solution n to (8) such that n ∈ L ∞ [0, ∞), L 1 [0, ∞) 1 − e−t n(·, t) ∈ L ∞ ([0, ∞), BV[0, ∞))
(11)
where the boundary condition F(x, n) → 0 as x → ∞ is satisfied in the L 1 sense. In addition, the solution satisfies the following. 1. There exists a unique α ∈ [0, 2] such that lim
t→∞ 0
∞
|n(x, t) − n α (x)| dx = 0.
(12)
Here, the n α ’s are the equilibrium entropy solutions defined by n α (x) :=
0, x∈ / (α, 2) 2x − x 2 , x ∈ (α, 2).
(13)
2. The total photon number N (t) is non-increasing in time. Indeed, the total photon number obeys the loss formula
T
N (T ) + 0
for T > 0.
n 2 (0, t) dt = N(0)
(14)
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3 Contraction The key result of this section is to use the structure of entropy solutions in the sense of Definition 1 to prove the following L 1 contraction property. In light of this, it is clear that the L 1 distance between any bounded, non-negative entropy solution and any stationary solution n α defined above is non-increasing in time. Proposition 2 (Contraction Principle). Let n and m be two non-negative, bounded entropy solutions of (8) in the sense of Definition 1 with L 1 initial data n(·, 0) and m(·, 0), respectively. Then for R > 2, T n(0, t)2 − m(0, t)2 dt |n(x, T ) − m(x, T )| dx + 0 0 R T |n(x, 0) − m(x, 0)| dx + |F(R, n(R, t)) − F(R, m(R, T))| dt. (15) ≤
R
0
0
It is noted that Proposition 2 is similar to the L 1 contraction result in [7]. However, Proposition 2 is different because the flux in (8) does not have the same Lipschitz property as used in [7]. This problem also has a boundary unlike the Cauchy problem. Proposition 2 follows immediately from the following two lemmas by using a = 1 and b = 0 in the definition of below. Lemma 1. Let n and m be entropy solutions to (8) in the sense of Definition 1. Then for (s) := a|s| + bs where a ≥ 0 and b ∈ R, for any non-negative test function φ, 0
T
∞
(n(x, t) − m(x, t)) [F(x, n) − F(x, m)] ∂x φ
0
+ (n(x, t) − m(x, t))∂t φ dx dt ≥ 0.
(16)
Proof. This lemma is proven by using a family of test functions gh (x, t, y, s) := φ
x +y t +s , 2 2
where ηh (x) :=
ηh
t −s 2
ηh
x−y 2
1 x η h h
for η ∈ Cc∞ (R) such that η(x) ≥ 0, η(x) = 0 for |x| ≥ 1 and R
η(x) dx = 1.
Using m(y, s) as k in the entropy condition for n(x, t) with gh as the test function and integrating over y and s, and doing a similar procedure for the entropy formulation
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of m(y, s), and for the weak formulations for n and m, some elementary calculations and taking h → 0 yields (16). The reader is referred to the proof of Lemma 3.4 in [1] for the details. Using the previous result, the following lemma immediately yields the result of Proposition 2. Lemma 2. Let n and m be bounded entropy solutions to (8) with initial data n(·, 0) and m(·, 0), respectively. Then −
R
(n(x, T ) − m(x, T )) dx ≤
0 T
R
(n(x, 0) − m(x, 0)) dx
0
(n(R, t) − m(R, T ))[F(R, n(R, t)) − F(R, m(R, t))] dt 0 T (n(0, t) − m(0, t))[F(0, n(0, t)) − F(0, m(0, t))] dt. +
(17)
0
Proof. This lemma is proven by using the test function φ(x, t) = [αh (t − ε) − αh (t − T + ε)] [αh (x − ε) − αh (x − R + ε)]
where αh (x) :=
x
−∞
(18)
ηh (s) ds
in (16). This φ is an approximation for the characteristic function on the space-time domain [0, T ] × [0, R]. Taking h → 0 and ε → 0 completes the proof. The reader is referred to the proof of Lemma 3.5 in [1] for the details. The next lemma shows that for entropy solutions, if n is initially compactly supported, it remains so for all time. Lemma 3. Let n be a non-negative entropy solution of (8). Assume that n(·, 0) is compactly supported on [0, R] for some R > 2. Then n(·, T ) is compactly supported on [0, R] for all T > 0. Proof. Using m ≡ 0 and following a similar technique for proving (17), the following holds: T ∞ ∞ |n(R, t)| 2R − R2 − n(R, t) dt ≤ |n(x, T )| dx − |n0 | dx. (19) R
0
R
Since R is larger than 2 and n is non-negative, the result follows immediately from (19). Remark 1. It can be shown that in fact, the support of n contracts to [0, 2] as T → ∞, which agrees with formal calculation of the characteristics of (8). See [1] for the details.
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It is also noted that Lemma 2 also leads to the following comparison principle. Proposition 3 (Comparison Principle). Let n and m be non-negative entropy solutions to (8) with compactly supported initial data n(·, 0) and m(·, 0), respectively. If n(x, 0) ≤ m(x, 0) on (0, ∞), then for all T > 0, n(x, T ) ≤ m(x, T ) almost everywhere on (0, ∞). Proof Letting a = b = 21 in the definition of in (16), becomes the positive part of s. Letting R be an upper bound for the supports of n(·, 0) and m(·, 0) and using the definitions of F and the fact that n and m are non-negative,
R
0
[n(x, T ) − m(x, T )]+ dx ≤
The result follows immediately.
R
0
[n(x, 0) − m(x, 0)]+ dx.
(20)
In light of Proposition 2 and Lemma 3, it becomes clear that non-negative entropy solutions are unique. Proposition 4 (Uniqueness of Entropy Solutions). Let n(·, 0) ∈ L 1 (0, ∞) be nonnegative and compactly supported on [0, R]. Then there is at most one non-negative entropy solution to (8) with initial data n(·, 0). Proof Let n and m be non-negative entropy solutions to (8) with initial data n(·, 0). Using Proposition 2 along with the fact that the support for n and m is in [0, R] for all positive times from Lemma 3, it is clear that
R
T
|n(x, T ) − m(x, T )| dx ≤ −
0
n(0, t)2 − m(0, t)2 dt.
(21)
0
This is only possible if n = m.
4 Regularity and Compactness In this section, the regularity and compactness of the entropy solutions are investigated. Particularly, the interest is in BV bounds for the entropy solutions as t → ∞. With these bounds, the convergence to stationary solutions can be shown. All entropy solutions here are taken to be non-negative with non-negative, compactly supported initial data. To begin with, a bound on the entropy solutions as t → ∞ by stationary solutions is shown. Lemma 4. Let n be an entropy solution to (8). Then lim sup n(x, t) ≤ n 0 = 2x − x 2 + . t→∞
(22)
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Formally, the main idea for the proof of Lemma 4 is to find a hyperbolic counterpart to the idea of super-solution for parabolic equations, that is, to find some function n such that (23) ∂t n + ∂x F ≥ 0 where F := F(x, n). Formally, n is chosen such that 2x − x 2 n − n 2 = −K (t)G(x), or n(x, t) =
1 g + g 2 + 4K G 2
(24)
(25)
where g(x) := 2x − x 2 . The functions K and G are chosen such that (23) holds. Some straight-forward calculations and the choice that K satisfies ∂t K ≤ 0 lead to the realization that 1 (26) K (t) = (βt + c1 )2 and G(x) =
β 2 (R + c2 − x)2 4
(27)
satisfy (23) for non-negative constants β, c1 and c2 . If the hyperbolic equation (8) has a notion of entropy super-solutions and a comparison principle (note the comparison result proved in Proposition 3 only compares entropy solutions, not super-solutions), then noting that choosing c1 = 0 and c2 > 0 in the formulas for K and G formally would yield a super-solutions with initial values n(·, 0) = ∞. Then, it could easily be shown that (28) n(x, t) ≤ n(x, t) → g(x)+ = n 0 (x) with the limit being taken as t → ∞, which would prove Lemma 4. This argument is made rigorous by considering the viscous limit of ˆ n ε ) = ε∂x2 n ε ∂t n ε + ∂x F(x,
(29)
where Fˆ is an appropriate extension of F over the entire real line. The details of this analysis are rather technical and the reader is referred to Sect. 4.1 in [1]. The next key step in gathering BV bounds is to prove the following one-sided spatial Lipschitz bound for entropy solutions. Lemma 5. Let n be an entropy solution to (8) with non-negative L 1 initial data supported on [0, R] for some R > 0. Then for any t > 0, there exists some negative function m(R, t) increasing in t such that for all 0 ≤ x ≤ y ≤ R,
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n(y, t) − n(x, t) ≥ m(R, t)(y − x).
(30)
Proof. As in the proof of Lemma 4, the fact that n can be written as a vanishing viscosity limit is exploited. Letting m ε = ∂x n ε , differentiating (29) with respect to x yields (31) ∂t m ε − 2m 2ε + 2g m ε + g n ε + (g − 2n ε )∂x m ε − ε∂x2 m ε = 0. Defining m ε (t) := −
where
C1 − 4
2 1 + exp −t 8Cε + C1 8Cε +
4 1 − exp −t 8Cε + C12 C12
C1 := 2 g ∞ and Cε := sup g n ε ,
(32)
(33)
straight-forward calculations show that m ε is a sub-solution of (31) such that m ε (0) = −∞. Thus, taking the limit as ε → 0, the function m(R, t) can be defined as
C1 m(R, t) := − − 4
1 + C12
2 1 − exp −t 1 + C12
(34)
from which it can be shown that for any 0 ≤ x ≤ y ≤ R and t > 0, n(y, t) − n(x, t) = lim n ε (y, t) − n ε (x, t) ≥ m(R, t)(y − x), ε→0
(35)
completing the proof. The omitted details justifying the taking of the limit ε → 0 are in Sect. 4.2 of [1]. This section is concluded with a lemma specifying the compactness of the trajectory {n(·, t)}t≥0 . The lemma is proven by using the L 1 contraction result of Proposition 2 and control of the total variation using the one-sided Lipschitz bound of Lemma 5 (see [1, Sect. 4.3]) for the details). Lemma 6. Let n be a non-negative entropy solution of (8) with initial data n(·, 0) supported on [0, R] for some R > 0. Then (11) holds and the trajectory {n(·, t)}t≥0 is relatively compact in L 1 .
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5 Proof of the Main Theorem The focus now turns to the proof of Theorem 1. The solution space for n is proven by Lemma 6. Also from Lemma 6, the trajectory {n(·, t)}t≥0 is relatively compact in L 1 . Thus, it suffices to show that subsequential limits are unique in order to show that n(·, t) converges in L 1 as t → ∞. Let (tk ) be a sequence of times such that tk → ∞ and n ∞ ∈ L 1 such that n(·, tk ) → n ∞ as k → ∞. Defining for β ∈ [0, 2]
∞
Cβ (t) :=
|n(x, t) − n β (x)| dx
(36)
0
and letting r < t, the results of Sect. 3 lead to Cβ (t) ≤ Cβ (r ) −
t
n(0, s)2 ds ≤ Cβ (r).
(37)
r
Thus, Cβ (t) is monotone and converges to some C β as t → ∞ which is independent of the sequence (tk ). Additionally, Cβ =
∞
n ∞ − n β dx.
(38)
0
Since n ∞ (x) ≤ n 0 (x) by Lemma 4,
β
Cβ =
n ∞ dx +
0
β
2
n 0 − n ∞ dx,
(39)
thus, 2n ∞ (β) = n 0 (β) + ∂β C β
(40)
and n ∞ is determined by C β , proving the limit does not depend on the sequence (tk ). Using standard arguments, it is clear that n ∞ is a stationary entropy solution of (8). This completes proof of (12). The loss formula (14) follows from using the test function φ(x, t) = [αh (t − ε) − αh (t − T + ε)] [αh (x − ε) − αh (x − R + ε)]
(41)
in the weak formulation (9). Using similar techniques as used to prove Proposition 2, the loss formula (14) is obtained after noting that lim
R→∞ 0
T
F(R, n(R, t)) dt = 0
(42)
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from the compact support of n. Thus, Theorem 1 is proven. From the main result, some corollaries arise. Corollary 1. Let n be a non-negative entropy solution to (8) with compactly supported initial data n(·, 0) ∈ L 1 . If
∞
2
n(x, 0) dx >
0
2x − x 2 dx,
(43)
0
then there exists T > 0 such that ∞ n(x, T ) dx < 0
∞
n(x, 0) dx
(44)
0
and therefore a Bose–Einstein condensate forms in finite time. Proof. From (13), it is clear that lim N (t) =
t→∞
α
2
2x − x 2 dx ≤
2
2x − x 2 dx.
(45)
0
Thus, if
2
N (0) >
2x − x 2 dx,
(46)
0
it must be true that for some T > 0, 1 N (T ) < 2 implying the result.
2
N (0) +
2x − x dx < N (0), 2
(47)
0
The next corollary states that even though it cannot be determined which stationary solution an entropy solution approaches as t → ∞, a nonzero lower bound on the total photon number in the equilibrium state can be found. Corollary 2. Let n be a non-negative entropy solution to (8) with compactly supported, L 1 initial data n(·, 0). Let n α be the limiting stationary solution as t → ∞. Then 2 2 2x − x 2 dx ≥ sup min{n(x, t), 2x − x 2 } dx. (48) α
t≥0
0
Further, if n(·, 0) is not identically zero, then α < 2.
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Proof. Let t0 > 0 and let n be the solution of (8)with initial data n(·, 0) = min n(x, t0 ), (2x − x 2 )+ . Then n(x, t) ≤ min n(x, t0 + t), (2x − x 2 )+ and α
2
2
2x − x 2 dx ≥
min n(x, t0 ), 2x − x2 dx
(49)
0
by the loss formula (14) and the fact that n(0, t) = 0. This proves (48). The fact that α < 2 follows from the fact that entropy solutions can only have upward jumps by virtue of Lemma 5. For the details, see [1].
6 Future Directions The current and future work of the authors of [1] is to find an analogous result for the full Kompaneets equation (7) to the results for (8) presented here and in [1]. Considering the full Kompaneets equation (7), it can be shown that there are stationary super-solutions of the form n(x) = m(x) +
x2 e2 − 1
(50)
where m solves the ordinary differential equation x2
dm + g(x)m = 0 dx
where g(x) := x 2 − 2x +
2x 2 . ex − 1
(51)
(52)
Preliminary analysis shows that these super-solutions are well-behaved at x = 0. Thus, the plan is to use the techniques outlined in [1] and in [8] in application to the full Kompaneets equation and determine if similar results, such as showing convergence of solutions to a stationary solution and finding conditions for which it is known a Bose–Einstein condensate can form in finite time. It is anticipated that the physical entropy H (t) defined in (4) will be needed to perform some of the estimates. Acknowledgements J. B. acknowledges support from the National Science Foundation under the grant DMS-1401732 and from the Center of Nonlinear Analysis at Carnegie Mellon University.
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References 1. J. Ballew, G. Iyer, R.L. Pego, Bose–Einstein Condensation in a hyperbolic model for the Kompaneets equation. SIAM J. Math. Anal. 48(6), 3840–3859 (2016). https://doi.org/10.1137/ 15M1054730 2. M. Birkinshaw, The Sunyaev–Zel’dovich effect. Phys. Rep. 310(2), 97–195 (1999) 3. A. Bressan, Hyperbolic systems of conservation laws, in Oxford Lecture Series in Mathematics and its Applications, vol. 20. (Oxford University Press, Oxford, 2000) (The one-dimensional Cauchy problem) 4. R.E. Caflisch, C.D. Levermore, Equilibrium for radiation in a homogeneous plasma. Phys. Fluids 29(3), 748–752 (1986). https://doi.org/10.1063/1.865928 5. C.M. Dafermos, Hyperbolic conservation laws in continuum physics, in Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, 3rd edn. (Springer, Berlin, 2010). https://doi.org/10.1007/978-3-642-04048-1 6. A. Kompaneets, The establishment of thermal equilibrium between quanta and electrons. Sov. Phys. JETP-USSR 4(5), 730–737 (1957) 7. S.N. Kružkov, First order quasilinear equations with several independent variables. Math. Sb. (N.S.) 81(123), 228–255 (1970). Translated by N. Koblitz 8. C.D. Levermore, H. Liu, R.L. Pego, Global dynamics of Bose–Einstein condensation for a model of the Kompaneets equation. SIAM J. Math. Anal. 48(4), 2454–2494 (2016). https:// doi.org/10.1137/15M1054377 9. N. Shakura, R. Sunyaev, Black holes in binary systems. observational appearance, in Accretion: A Collection of Influential Papers (1989), p. 130 10. R. Sunyaev, Y.B. Zeldovich, The interaction of matter and radiation in the hot model of the universe II. Astrophys. Space Sci. 7(1), 20–30 (1970)
Finite Volume Methods for Hyperbolic Partial Differential Equations with Spatial Noise Andrea Barth and Ilja Kröker
Abstract Various real-world applications require the consideration of the influence of uncertain parameters on the solution to some nonlinear hyperbolic problem. The topic of this paper is to study the solution to a nonlinear hyperbolic PDE perturbed by a spatial noise term. In the first part of this paper, a definition of the corresponding stochastic entropy solution is defined and the required properties for the existence and uniqueness of the defined solution are discussed. The second part is devoted to numerical simulation. An approximation of the spatial noise is introduced. Further, the influence of the noise on the solution to the nonlinear hyperbolic problems is investigated. Several nonlinear numerical examples illustrate the discussion. Keywords Nonlinear hyperbolic PDE · Spatial noise · Random field Uncertainty quantification · Stochastic entropy solution · Finite volume method MSC2010 35R60 · 65M08 · 68U20
1 Model Problem Multiple technical applications, e.g. porous media flow require to deal with a flux function perturbed by spatial noise. In this paper, we consider a nonlinear hyperbolic partial differential equation perturbed by spatial noise (for D ⊂ R and T ∈ (0, +∞)): u t + (h(u)β(x, ω))x = 0, u(·, 0) = u 0 (·),
on × D × (0, T ), on D.
(1) (2)
A. Barth · I. Kröker (B) IANS, Universtität Stuttgart, Allmandring 5b, 70569 Stuttgart, Germany e-mail: [email protected] I. Kröker e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_10
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Here the flux function h : R → R is assumed to be nonlinear, in particular we consider the Burgers’ and Buckley–Leverett flux. The log-normally distributed random field β is given by the decomposition ⎛ β(x, ω) := exp ⎝
∞
⎞ λ j e j (x)β j ⎠ , x ∈ D,
(3)
j=0
where the random variables β j are normally distributed, i.e. β j ∼ N (0, σ 2 ), for all j ∈ N0 . This expansion stems from the spectral representation (with respect to the covariance operator) a Gaussian measure in an Hilbert space naturally admits. To model uncertain material properties by a Gaussian measure is common in a porous media framework. Different aspects of nonlinear hyperbolic problems with time-dependent stochastic source terms were initially considered by Holden and Risebro in [6]. This concept was extensively discussed in the literature during the last decade. The concept of stochastic weak and entropy solutions was introduced by Holden and Risebro in [6] resp. Kim in [8]. Feng and Nualart, as well as Biswas et al. extended it to conservation laws with multiplicative Gaussian and Lévy noise in [5] and [2]. The convergence of a numerical schemes was discussed, e.g. by Kröker and Rohde in [9] and by Bauzet et al. in [1]. Another way to introduce uncertainty into a system is by considering random parameters. This may be some random initial data as considered by Mishra and Schwab in [15], or by some randomly perturbed flux function as considered by Després et al. [16], were the hyperbolicity of the stochastic Galerkin representation of an hyperbolic problem was shown. Tryoen et al. [17] and Bürger et al. [3] extended the stochastic Galerkin representation by use of Multi-Resolution and Mulit-Wavelet methods. In this paper, we introduce an appropriate definition of stochastic entropy solution in Sect. 2 and show some brief sketches of the proofs of uniqueness and existence of the defined solutions. After a short introduction into solution concepts for spatially hyperbolic PDEs, we proceed with numerical simulations. In Sect. 3, we discuss the representation and an approximation of the random field β and the finite volume discretization of the problem (1)–(3). We discuss, furthermore, the influence of the spatial noise on the solution in several numerical examples.
2 Stochastic Entropy Solution In this section, we give an existence and uniqueness result and mention properties of the solution to the problem (1)–(2). For this purpose, we rewrite problem (1)–(2) into the following form: Given a probability space (, A , P), find u : R × [0, T ] × → R such that for P-a.e. ω ∈
Finite Volume Methods for Hyperbolic Partial Differential Equations …
u t (x, t, ω) + ∂x f (ω, x, u(x, t, ω)) = 0, u(·, 0, ω) = u 0 (·),
127
on R × (0, T ),
(4)
on R.
(5)
Throughout this section, we make the following assumptions on the flux function f : D × R × → R. (A.1) For almost all ω ∈ , the flux function f (·, ·, ω) ∈ C 2,2 (D × R) satisfies the following conditions. | f (x, u, ω)|, | f x (x, u, ω)|, | f u (x, u, ω)|, | f x x (x, u, ω)|, | f uu (x, u, ω)|, | f xu (x, u, ω)| ≤ C(|u| N + 1) for all (x, u) ∈ D × R. (6) for some positive integer N ≥ 2 and positive constant C > 0. (A.2) The flux function f (x, u, ω) has bounded support in x for almost all ω ∈ . (A.3) meas{ f uu (x, u, ω) = 0} = 0. (A.4) For almost all ω ∈ and x ∈ R the flux function f satisfies (for L finite P-a.s.) | f x (x, u, ω) − f x (x, v, ω)| ≤ L(ω) |u − v| . With this assumption in hand, we define the stochastic entropy solution of the problem (4)–(5) as follows. Definition 1 (Stochastic entropy solution). Let T > 0 be given. We call an L p (R × [0, T ])-valued random variable u stochastic entropy solution of the problem (4)–(5) if for almost all ω ∈ • u ∈ L p (R × [0, T ]) for all 1 ≤ p < ∞; • u is L 2 (R)-weakly continuous on [0, T ]; • for all ϕ ∈ Cc∞ (R × (−T, T )), with ϕ ≥ 0, and k ∈ R holds
T
R
0
(|u(x, t) − k| − |u 0 − k|) ϕt (x, t) + sign(u(x, t) − k) [ f (x, u(x, t)) − f (x, k)] ϕx (x, t) − sign(u(x, t) − k) ( f x (x, k)) ϕ(x, t) dx dt ≥ 0. (7)
• Furthermore, for all ϕ ∈ Cc∞ (R × (0, T )), with ϕ ≥ 0 holds T 0
R
|u(x, t) − k| ϕt (x, t) + sign(u(x, t) − k) [ f (x, u(x, t)) − f (x, k)] ϕx (x, t) −sign(u(x, t) − k) ( f x (x, k)) ϕ(x, t) dx dt ≥ 0. (8)
Theorem 1 (Uniqueness). Let Assumptions A.1–A.3 be satisfied, and set T > 0. Further, let u and uˆ be stochastic Kružkov entropy solutions with initial values
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u 0 , uˆ 0 ∈ L p (R) for all 1 ≤ p < ∞. Then, for u 0 = uˆ 0 , we have u(ω) = u(ω), ˆ for almost all ω ∈ . Proof (sketch). The main argument used in the proof of uniqueness of the solution as in Definition 1 is the Kružkov integral doubling argument, which was introduced by Kružkov in [11] end extended by Kim in [8] to the stochastic setting. The proof can be performed in three steps. In the first step we fix an ω ∈ such that Assumptions A.1–A.4 on the flux function f are satisfied. Further, let u and uˆ be stochastic entropy solution with corresponding initial values u 0 and uˆ 0 , and consider ϕ ∈ Cc∞ (R × (0, T )), with ϕ ≥ 0. By means of integral doubling we show that T u(x, t) − u(x, ˆ t) ϕt (x, t) dx dt R 0
+
T R 0
sign(u(x, t) − u(x, ˆ t)) f (x, u(x, t)) − f (x, u(x, ˆ t)) ×ϕx (x, t) dx dt ≥ 0
In the second step we show, also by integral doubling, that for ϕ ∈ Cc∞ (R × (−T, T )), with ϕ ≥ 0, for all ε > 0 and an appropriate constant C > 0 the inequality
T u(x, t) − u ε (x) ϕt (x, t) dx dt − u 0 (x) − u ε (x) ϕt (x, t) dx dt 0 0 R R 0 0 T
sign(u(x, t) − u ε0 (x)) f (x, u(x, t)) − f (x, u ε0 (x)) ϕx (x, t) dx dt + 0 R T T f x (x, u ε (x))ϕ(x, t) dx dt + C ϕ(·, t) L ∞ (R) dt ≥ 0 + 0 ε 0 0 R T
holds. Here, for a mollifier ρε , the mollified initial distribution is given by u ε0 (x) =
R
u 0 (y)ρε (x − y) dy for x ∈ R.
These steps allow us to show that the contraction property R
u(x, t) − u(x, ˆ t) dx ≤
R
u 0 (x) − uˆ 0 (x) dx
holds for almost all t ∈ [0, T ] and ω ∈ . This, in turn, implies the pathwise uniqueness of the stochastic entropy solution.
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Theorem 2 (Existence of the stochastic entropy solution). Let Assumptions A.1– A.4 be satisfied for almost all ω ∈ and let u 0 ∈ L p (R) for all 1 ≤ p ≤ ∞. Then for T > 0 and almost all ω ∈ there exists a stochastic entropy solution to problem (4)–(5). Proof (sketch). We start our proof with the viscous problem u εt + f (x, u ε , ω)x = ε u ε u ε (·, 0, ω) = u 0 (·)
in R × (0, T ) × , in R.
(9)
Using of classical existence results requires to restrict f and u 0 to f k (x, u, ω) := f (x, u, ω)σk (u), and u ε,k (x) :=
1 u 0 (y) ρ ε |y|≤k
x−y ε
dy.
Here σk ∈ Cc∞ (R) is a cut-off function given by σk (y) :=
1, |y| < k, 0, |y| ≥ k,
0 ≤ σk (y) ≤ 1, σk (y) ≤ 2,
y ∈ R.
This yields the corresponding restricted problem k ε,k ε,k u ε,k t + f (x, u , ω)x = ε u
in R × (0, T ) × ,
u ε,k 0 (·)
in R.
u
ε,k
(·, 0, ω) =
(10)
Now, we fix again an ω ∈ such that Assumptions A.1–A.4 are satisfied. By the application of the parabolic framework (cf. par example [7]) we can obtain the solution of the restricted problem (10). For k → ∞ we obtain the solution u ε of the viscous problem (10), such that (i) The functions u ε and u εx x are continuous for almost all ω ∈ . (ii) For all 1 ≤ p < ∞ and almost all ω ∈ u ε (ω) ∈ C([0, T ]; L 2 (R)) ∩ L ∞ (0, T ; L 1 (R) ∩ L p (R)) ∩ L 2 (0, T ; H 1 (R)). (iii) For all 2 ≤ p < ∞, there is a positive constant C p which is independent of p, and ε such that p 2 E u ε L ∞ (0,T ;L 1 (R)) + E u ε L ∞ (0,T ;L p (R)) + εE u ε L 2 (0,T ;H 1 (R)) ≤ C p .
In the next step we let the viscosity coefficient ε tend to zero. For this purpose we use the compensated compactness framework [4, 13, 14]. We fix an ω ∈ such that u ε (ω) exists and choose η1 ∈ C 2 (R), η2 ∈ C 2,2 (R × R) to define
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u
q 1 (x, u) :=
η1 (s) f u (x, s) ds, q 2 (x, u) :=
0
0
u
ηu2 (x, s) f u (x, s) ds.
Murats’ Lemma (cf. [4, 14]) then yields ∂ 1 ε ∂ 1 η (u (x, t)) + q (x, u ε (x, t)) ⊂ A, ∂t ∂x ∂ 2 ∂ 2 η (x, u ε (x, t)) + q (x, u ε (x, t)) ⊂ B, ∂t ∂x −1 (R × (0, T )). Now we can apply the (modwhere A, B are compact subsets of Hloc ified) results of Liu [13] to obtain for almost all ω ∈ that
u ε → u strongly in L loc (R × [0, T ]) for all 1 ≥ p < ∞. p
The Dominated Convergence Theorem and Hölder’s inequality provide then the convergence to the stochastic entropy solution.
3 Numerical Experiments In this section, we approximate the random field β by a truncated field β¯ and present a discretization of the problem (1)–(3) by a Upwind finite volume method. We then apply these methods on some spatially perturbed Burgers’ equation, i.e. where the 2 flux function in (1) is given by h(u) := u2 , and a spatially perturbed Buckley–Leverett u2 problem, i.e. for h(u) := u 2 +(1−u) 2 , and discuss the obtained results of the presented numerical experiments.
3.1 Noise Approximation For the numerical experiments, we replace the log-normal random field given in (3) by the corresponding truncated decomposition ⎛
⎞ M ¯ ω) := exp ⎝ β(x, λ j e j (x)β j (ω)⎠ , x ∈ D, M ∈ N, ω ∈ . j=1
Here the sequences (e j , j = 1, . . . , M) and (λ j , j = 1, . . . , M) are given by λ j := j
−α
, α ∈ N, and e j (x) :=
2 x−D sin 2π j |D| |D|
Finite Volume Methods for Hyperbolic Partial Differential Equations … Fig. 1 Five realizations of β¯
131
5
4
3
2
1
0
0
0.2
0.4
0.6
0.8
1
for all j ∈ N. The length of the interval D is denoted by |D|, and D denotes the minimum of the interval D. The sequence (β j , j = 1, . . . , M) consists of normally distributed random variables β j ∼ N (0, σ 2 ) for all j ∈ N0 . Figure 1 shows five realizations of β¯ computed with α = 4, σ = 0.25 and truncated at M = 10. Further, Fig. 2 shows the expectation and the variance of β¯ computed with 10,000 Monte Carlo samples. The choice of the parameters M, α and σ determines the amplitude of a single sample and the magnitude of the variance. We use the presented setting in the numerical experiments in Sect. 3.3. Convergence of the truncated field β¯ to the corresponding Gaussian random field β can be proven in mean square sense (and, in fact, L p -sense) and almost surely.
3.2 Finite Volume Discretization For the pathwise spatial discretization, we use a classical finite volume discretization in conservation form given by
t g j+ 21 − g j− 21 , :=u nj − u n+1 j
x x j+1/2 1 u 0j := u 0 (x) dx,
x x j−1/2 for j ∈ Z. For the presented numerical experiments, the Upwind flux is used, which is defined by ¯ g j+ 21 := β(ω, x j )h(u j ), for j ∈ Z, ω ∈ .
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(a)
(b) 0.15
1.06
0.1
1.04
0.05
1.02
1
0
0.2
0.4
0.6
0.8
1
0
0
0.2
0.4
0.6
0.8
1
¯ truncated on M = 10, computed with Fig. 2 Monte Carlo expectation (a) and variance (b) of β, 10,000 samples
(a)
(b)
1
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0.8
0.8
0.6
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0.4
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0.2
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0
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1
0
0
0.1 0.2
0.3 0.4
0.5 0.6 0.7
0.8 0.9
1
Fig. 3 Initial distribution: a Discontinuous case; b continuous case
For a comprehensive discussion of finite volume methods for conservation laws, we refer to the monographs of LeVeque and Kröner [10, 12]. The time step t for each realization ω ∈ is given by the (sample-specific) CFL condition
t
x
max
x∈D,u∈[0,1]
¯ β(ω, x)h (u) < 1.
In particular, this implies that each realization is calculated with an appropriate t, ¯ and that the number of time steps strongly depends on the magnitude of β(ω).
3.3 Numerical Experiments In this section, we consider the application of the finite volume method defined in Sect. 3.2 on the problem (1)–(2) with Burgers’ and Buckley–Leverett flux functions and truncated spatial noise, as defined in Sect. 3.1. The numerical experiments were
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shows a strong influence on the shock speed and a minor influence on the shape of the rarefaction wave. According to these results, the expectation shows the diffusionlike behaviour near the shock position. Further, the variance has its highest magnitude near the shock. Acknowledgements The research leading to these results has received funding from the German Research Foundation (DFG) as part of the Cluster of Excellence in Simulation Technology (EXC 310/2) at the University of Stuttgart, and it is gratefully acknowledged.
References 1. C. Bauzet, J. Charrier, T. Gallouët, Convergence of monotone finite volume schemes for hyperbolic scalar conservation laws with multiplicative noise. Stoch. Partial. Differ. Equ. Anal. Comput. 4(1), 150–223 (2016). https://doi.org/10.1007/s40072-015-0052-z
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2. I.H. Biswas, K.H. Karlsen, A.K. Majee, Conservation laws driven by Lévy white noise. J. Hyperbolic Differ. Equ. 12(3), 581–654 (2015). https://doi.org/10.1142/S0219891615500174 3. R. Bürger, I. Kröker, C. Rohde, A hybrid stochastic Galerkin method for uncertainty quantification applied to a conservation law modelling a clarifier-thickener unit. ZAMM Z. Angew. Math. Mech. 94(10), 793–817 (2014). https://doi.org/10.1002/zamm.201200174 4. C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, vol. 325, 3rd edn., Grundlehren der Mathematischen Wissenschaften (Springer, Berlin, 2010). https://doi.org/10. 1007/978-3-642-04048-1 5. J. Feng, D. Nualart, Stochastic scalar conservation laws. J. Funct. Anal. 255(2), 313–373 (2008). https://doi.org/10.1016/j.jfa.2008.02.004 6. H. Holden, N.H. Risebro, Conservation laws with a random source. Appl. Math. Optim. 36(2), 229–241 (1997) 7. L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations (Springer, Berlin, 1997) 8. J.U. Kim, On a stochastic scalar conservation law. Indiana Univ. Math. J. 52(1), 227–256 (2003) 9. I. Kröker, C. Rohde, Finite volume schemes for hyperbolic balance laws with multiplicative noise. Appl. Numer. Math. 62(4), 441–456 (2012). https://doi.org/10.1016/j.apnum.2011.01. 011 10. D. Kröner, Numerical Schemes for Conservation Laws, Wiley-Teubner Series Advances in Numerical Mathematics (Wiley, Chichester, 1997) 11. S.N. Kružkov, First order quasilinear equations with several independent variables. Mat.USSR Sb. 10, 217–243 (1970) 12. R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics (Cambridge University Press, Cambridge, 2002) 13. Y. Lu, Hyperbolic Conservation Laws and the Compensated Compactness Method, vol. 128, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics (Chapman & Hall/CRC, Boca Raton, 2003) 14. J. Málek, J. Neˇcas, M. Rokyta, M. Ruzicka, Weak and Measure-Valued Solutions to Evolutionary PDEs, vol. 13, Applied Mathematics and Mathematical Computation (Chapman & Hall, London, 1996). https://doi.org/10.1007/978-1-4899-6824-1 15. S. Mishra, C. Schwab, Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random initial data. Math. Comput. 81(280), 1979–2018 (2012). https://doi.org/10.1090/S0025-5718-2012-02574-9 16. G. Poëtte, B. Després, D. Lucor, Uncertainty quantification for systems of conservation laws. J. Comput. Phys. 228(7), 2443–2467 (2009). https://doi.org/10.1016/j.jcp.2008.12.018 17. J. Tryoen, O.L. Maître, A. Ern, Adaptive anisotropic spectral stochastic methods for uncertain scalar conservation laws. SIAM J. Sci. Comput. 34(5), A2459–A2481 (2012). https://doi.org/ 10.1137/120863927
A Hyperbolic Approach for Dissipative Magnetohydrodynamics Hubert Baty and Hiroaki Nishikawa
Abstract The hyperbolic method initially introduced by Nishikawa (J Comput Phys 227:315–352, 2007, [1]) for solving the diffusion equation is extended to a twodimensional magnetohydrodynamic (MHD) model. The approach is based on reformulation of the dissipative terms in order to solve an equivalent first-order hyperbolic system. This enables the use of approximate Riemann solvers for handling dissipative and advective fluxes in the same way. The advantages of our method compared to traditional ones are illustrated by finding steady-state solutions for magnetic reconnection process. In particular, the numerical solution is obtained with the same order of accuracy for the main and gradient variables over a wide range of magnetic Reynolds numbers, without any deterioration characteristic of more conventional schemes. Second, the convergence towards the steady-state scales only linearly with the cell width h, giving thus a O(1/ h) acceleration factor. The improvement of the hyperbolic method and its extension to time-dependent MHD problems are presented. Finally, we discuss the importance of developing such new numerical methods for the sake of understanding the physical mechanisms underlying flares phenomena in plasmas. Keywords Hyperbolic method · Finite volume method · Riemann solver Magnetohydrodynamics
1 Introduction Most conventional magnetohydrodynamic (MHD) codes are based on methods using different Riemann-type solvers in order to handle discontinuities and shocks, as for example in the finite-volume based code AMRVAC [2]. Such methods are particularly adapted and efficient for purely hyperbolic equations. This is however not H. Baty (B) Observatoire Astronomique, CNRS UMR 7550, Université de Strasbourg, Strasbourg, France e-mail: [email protected] H. Nishikawa National Institute of Aerospace, Hampton, VA 23666, USA e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_11
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the case of the dissipative MHD model, where dissipative (resistive and/or viscous) terms play an important role. The dissipation is generally computed by a separated scheme (using a finite-difference source term approximation for example), making the comparison between the evaluation of the advective flux and the dissipative one not fully consistent. The reason for this separated treatment is due to different orders of spatial derivative of the main variable defining the fluxes, e.g., first and second order for the advective and dissipative flux respectively. An example of inconsistent behavior due to inconsistent construction of diffusive flux has been pointed out in the context of two-dimensional (2D) advection-diffusion problem. Indeed, a degradation of the order of accuracy has been observed for Reynolds numbers of order unity, when compared to pure advective/diffusive limits [3]. In MHD context, magnetic reconnection process involves large regions where the advection is dominant (i.e., where the local magnetic Reynolds number is much greater than one) and much smaller regions where the diffusion is dominant (i.e., where the local magnetic Reynolds number is much lower than one). As a consequence, a local deterioration of the spatial accuracy could propagate towards other regions and affect the solution in the whole domain. The solutions are also very sensitive to the detailed spatial profile of the resistivity parameter [4]. Thus, dissipative MHD requires a careful treatment of the dissipation terms and is clearly appealing for improved numerical schemes. We propose to cure the inconsistency by using the hyperbolic approach initially introduced for the diffusion equation [1], and later extended to other systems like Navier–Stokes one [5]. The basic idea is to transform the diffusive terms into hyperbolic ones by introducing diffusive fluxes as additional variables. These new variables that are gradients of the main variables, and are also solutions of additional hyperbolic equations. Despite the common idea of using flux variables, this approach is different from the so-called mixed methods commonly employed in fluid mechanics. The plan of the paper is as follows. Section 2 presents the principles of hyperbolization and associated numerical procedure to find stationary solutions of a model equation. Section 3 is devoted to the extension of the hyperbolic method four our 2D MHD model. The advantages and performances of the method are illustrated in Sect. 4, with a comparison of 2D stationary magnetic reconnection solutions obtained using our method with exact analytical solutions. Finally, we present ameliorations of our scheme and extensions (actually under development) to time-dependent problems in the last section before concluding.
2 The Hyperbolic Method We consider a simple one-dimensional (1D) advection-diffusion equation for the real variable ψ(x, t) (the real variable x denoting the space and the real positive one t is for time): ∂(V ψ) ∂ 2ψ ∂ψ + = η 2 + E(x), (1) ∂t ∂x ∂x
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where the real variables V (x) and E(x) are given velocity and source terms, respectively. The dissipation is represented by the real coefficient η that is assumed to be uniform for simplicity. Following previously related works on the hyperbolic method, we consider the equivalent hyperbolic system: ∂(V ψ) ∂(ηp) ∂ψ + − = E(x), ∂τ ∂x ∂x
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where p is an additional variable, and τ is a pseudo-time. The crucial remark is that, solving these two equations in the pseudo-steady state (i.e., for vanishing pseudo-time derivative terms) is equivalent of solving the original equation in the true steady state, as p is the x derivative of ψ in this limit. Tr is a relaxation time parameter defined as, Tr = L r2 /η, where L r is a length scale parameter. The hyperbolic method is different from classical relaxation methods, which results in stiff relaxation systems because of the requirement of a vanishing equivalent relaxation time [6]. In our model, Tr is not required to vanish because our hyperbolic model reduces to the advection-diffusion equation exactly in the steady state for any nonzero Tr . The choice of the L r value, which also determines the Tr value, is mainly based on fast steady convergence requirement. For example, an optimal value of L r = 1/(2π ) has been proposed in diffusion problems [1]. Different optimal formulae for L r depending on the typical problems addressed can be deduced, showing that L r must be reduced as much as the Reynolds number is high [7].
2.1 Steady State: Discretization and Implementation We rewrite our system as: ∂U ∂F + = S, (4) ∂τ ∂x E ψ V ψ − ηp , and S = . The flux F can be split where U = ,F= − p/Tr p −ψ/Tr Vψ a d a into two terms, an advective one F and a dissipative one F , with F = , and 0 −ηp . A finite-volume method is used with a spatial discretization of the Fd = −ψ/Tr solution U, namely Uj , where Uj is defined as the cell average of the solution of the jth spatial grid cell centered at x j (see Appendix). Steady-state solutions can be thus obtained by using a pseudo-time explicit iteration, Ujk+1 = Ujk − τ Resjk , where k is the iteration counter, Δτ is the pseudo-time step, and −Resjk is the residual which is required to vanish (up to a predefined given accuracy) when a steady-state solution is reached.
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The numerical fluxes are computed using upwind-type formula, and separating ∂F into two parts, |A| ≈ |Aa | + Ad . We can thus take the Jacobian matrix, A = ∂U the maximum wave speeds to evaluate the Jacobians. The advective Jacobian |Aa | has a nonzero V , while the dissipative eigenvalue that is equal to the local velocity √ Jacobian Ad has the following eigenvalues λ = ± η/Tr . There is an alternative way that consists in computing the eigenvalues of the full Jacobian in order to get an exactly unified scheme, but leading to more complicated implementation (especially for MHD equations). The explicit pseudo-time iteration to march towards the steady state is limited by a criterion, h , Δτ = C F L × Min √ η/Tr + an
(5)
where CFL is the Courant–Friedrichs–Lewy number less than or equal to one, and an is the local absolute velocity. One can also refer to the Appendix for more details. The ability of the above numerical scheme (using a second-order finite-volume discretization with an upwind flux) to obtain steady-state solutions is demonstrated, even when a quasi-singular layer tends to form [7]. It is also shown that the same order of accuracy for the main solution and gradient is attained for a wide range of magnetic Reynolds numbers, without any deterioration characteristic of more conventional schemes. Second, the convergence towards the steady-state scales only linearly with the cell width h, giving thus a O(1/ h) acceleration factor. This is an expected advantage from using our CFL criterion over explicit a 2conventional scheme, where the CFL criterion is Δt = C F L × Min ahn , hη .
3 Hyperbolic Method for the 2D MHD Equations We consider the 2D incompressible set of dissipative (viscous and resistive) MHD equations written in flux-vorticity scalar variables as follows: ∂ψ + V.∇ψ = η∇ 2 ψ + E(x, y), ∂t
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magnetic permeability constant equal to one in our units. The dissipative terms are the second derivatives terms involving the viscosity coefficient ν in Ω-equation, and the resistivity coefficient η in ψ-equation. For simplicity, these dissipative coefficients are assumed constant and uniform in the present work. The source term E represents a given electric field component perpendicular to the plane. Compared to a more conventional formulation using magnetic field and fluid velocity vectors as main variables, this set of equations has the advantage of ensuring the divergence free conditions on B and V. Two extra equations must however be introduced in order to deduce the fluid velocity components from the two main variables, that can be shown to be (using the incompressibility assumption ∇V = 0), namely ∇ 2 Vx = − ∇ 2 Vy =
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3.1 Steady State: Discretization and Implementation The hyperbolization of the ψ-equation follows: ∂ψ ∂(Vx ψ) ∂(Vy ψ) ∂(ηpψ ) ∂(ηqψ ) + + − − = E ψ (x, y), ∂τ ∂x ∂y ∂x ∂y
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where we use the notation pψ and qψ for the x and y derivatives of ψ respectively in the true steady state, and where E ψ is the source term. We also introduce an associated relaxation time parameter, Tψ = L r2 /η. The equation for the vorticity function Ω can be also hyperbolized exactly in the same way (not shown here), where two associated gradient variables are now introduced, pΩ and qΩ . A second corresponding relaxation time is also introduced, TΩ = L r2 /ν, as a second dissipative coefficient ν (viscosity) is present for viscous plasmas. The velocity components (Vx , Vy ) are computed using the vorticity trough, , and ∇ 2 Vy = ∂Ω . The first Poisson equations can also be hyper∇ 2 Vx = − ∂Ω ∂y ∂x bolized as ∂ Vx ∂ px ∂qx ∂Ω − − = , (13) ∂τ ∂x ∂y ∂y
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4 Results on Steady-State Magnetic Reconnection The hyperbolic method is tested for the MHD equations, in order to retrieve a particular solution for 2D steady-state magnetic reconnection process, namely the reconnective diffusion solution in an inviscid medium (i.e., for ν = 0) [8]. The latter solution corresponds to the velocity and magnetic field profiles of the form: V = (−αx, αy − and
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x respectively. Daw(x) is the Dawson function given by, Daw(x) = 0 exp (t 2 − x 2 )dt. E d is the magnitude of a uniform electric field perpendicular to the 2 −β 2 with (x, y) plane, and the three real constants (α, β, μ) are related via μ2 = α 2ηα β < α. Typically, we use E d = 0.1, α = 1, and β is varied between 0 and unity. Exact Dirichlet boundary conditions are imposed. The numerical results using the hyperbolic method are computed following the procedure described in detail elsewhere [7], and compared to the expected analytical solution. A second-order finite-volume discretization on rectangular grids with upwind flux is employed. First, the optimal L r value for fastest convergence is shown to be dependent on the resistivity with L r scaling approximately as η1/2 , as one can see in Fig. 1 for β = 0 case. As an illustration, the typically obtained solutions for main and gradient variables are plotted in Fig. 2, and the L 1 errors on main and gradient variables are displayed in Fig. 3 for a few cases. As expected, the same second order of convergence is obtained for the main and gradient variables for a wide range of Lundquist numbers (i.e., resistivity values). We have also checked that the convergence towards the steady-state scales linearly with the cell width [7].
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Fig. 1 Optimal 1/L r value (in units of π ) required for fastest numerical convergence towards steady state, for 2D MHD reconnection with β = 0 at different η values. The η−1 and η−1/2 dependences are indicated with plain line (a) ψ and magnetic field lines
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5 Extension to Time-Dependent Problems The time-dependent version of the hyperbolic method for our 1D model equation (Eq. 1) can be written as, ∂ψ ∂(V ψ) ∂(ηp) ∂ψ + − = E(x) − ∂τ ∂x ∂x ∂t
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These schemes have been initially introduced and investigated for time-dependent advection-diffusion equation [9]. The extension of our previous scheme (developed for steady-state solutions) results now in a point-implicit pseudo-time iteration (with k index),
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Fig. 6 Solution obtained for a time-dependent Burger equation using two different resistivity values, η = 10−2 and 10−4 . The final integration time is t = 3.5 for a physical time step Δt = 10−4 . The number of grid cells is 180. Left panel corresponds to a run without slope limiter, and right panel is for MC limiter. The initial ψ profile (at t = 0) is plotted using plain line
the second order expected for spatial and also time discretizations are effectively obtained. Secondly, the implementation of the method to solve a viscous Burgers equation (see Fig. 6) clearly demonstrates the necessity to implement a slope limiter in order to evaluate nonphysical oscillations when a shock is forming (see case at η = 10−4 ).
6 Conclusion Our hyperbolic scheme is shown to reproduce stationary solutions for 2D MHD reconnection problems, with the expected same convergence order for main and gradients variables. The extension for accurate time-dependent reconnection is
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actually under development. Preliminary results obtained using the BDF2 scheme are promising. In order to speed up the convergence procedure iteration at each time step, the scheme presented in this work would probably require some improvements like higher order spatial reconstruction and/or an implicit pseudo-time solver [10]. There is actually a great deal of interest on the subject, as the reconnection mechanism is considered to give an efficient energy mechanism on a fast timescale that is nearly independent of the resistivity. This is of considerable importance to explain flares occurring in many astrophysical plasmas. For example, the numerical investigation of the role of plasmoids in magnetic reconnection is complicated, as conventional codes lack some convergence properties due to the stochastic feature of the associated reconnection mechanism in such very high Lundquist plasmas [11]. The existence of fast regime of reconnection is also very dependent on the spatial variation of the resistivity [4]. We have checked that the hyperbolic method also works when a spatially localized profile is prescribed for the resistivity. The use of our hyperbolic scheme would be thus particularly promising in this context. Finally, we mention that another interesting hyperbolic approach for Navier–Stokes equations has been proposed very recently [12]. Acknowledgements H. Baty acknowledges support by French National Research Agency (ANR) through Grant ANR-13-JS05-0003-01 (Project EMPERE). We also acknowledge computational facilities available at Equip@Meso of the Université de Strasbourg.
Appendix We consider a Riemann problem for the jth cell where the fluxes are evaluated at the two cell interfaces x j−1/2 and x j+1/2 respectively. The semi-discrete discretization of the system (Eq. 5) with our cell-centered scheme over the dual volume I j = [x j−1/2 , x j+1/2 ] is thus ∂Uj 1 1 = − [Fj+1/2 − Fj−1/2 ] + ∂τ h h
Sd x,
(23)
Ij
where h is the grid cell volume (not necessarily uniform). The numerical flux is computed using the upwind formula: Fj+1/2 =
1 1 (FL + FR ) − |A| (UR − UL ), 2 2
(24)
where the subscripts L and R stand for the left and right sides of the cell interface situated at x j+1/2 respectively. The first term is computed from an average value of the two fluxes FL = F(UL ) and FR = F(UR ). We can therefore construct a numerical scheme by using advective/dissipative numerical fluxes based on upwind-type formulation, and taking the maximum wave speeds to evaluate the Jacobians,
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d Fj+1/2
=
1 2
[(V ψ) L + (V ψ) R ] − 0
1 [(−ηp) L 2 1 [(−ψ/T r )L 2
an [ψ R 2
− ψL ]
,
√ + (−ηp) R ] − 21 η/T r [ψ R − ψ L ] √ , + (−ψ/Tr ) R ] − 21 η/Tr [ p R − p L ]
(25)
(26)
where an = |V |. Finally, the source term is computed using a simple point integration approximation,
hEj . (27) Sd x ≈ Sj h = −hp j /Tr Ij We use a second-order accuracy method, where the nodal gradients are evaluated by a linear least-squares (LSQ) method. The left and right states are thus evaluated by a linear extrapolation from the cell centers. More explicitly we use, 1 1 ψ L = ψ j + h∇ψ j , ψ R = ψk − h∇ψk , 2 2
(28)
1 1 p L = p j + h∇ p j , p R = pk − h∇ pk , 2 2
(29)
where ∇ψ j is the gradient of ψ computed by the LSQ method at node j, and similarly for ∇ p j (we have k = j + 1).
References 1. H. Nishikawa, A first-order system approach for diffusion equation. I: second-order residual distribution schemes. J. Comput. Phys. 227, 315–352 (2007) 2. O. Porth, C. Xia, T. Hendrix, S.P. Moschou, R. Keppens, MPI-AMRVAC for solar and astrophysics. Astrophys. J. Suppl. Ser. 214, 4–26 (2014) 3. H. Nishikawa, P.L. Roe, On high-order fluctuation-splitting schemes for Navier–Stokes equations, in Computational Fluid Dynamics (Springer, 2004), pp. 799–804 4. H. Baty, T.G. Forbes, E.R. Priest, The formation and stability of Petschek reconnection. Phys. Plasmas 21, 11211 (2014) 5. H. Nishikawa, Two ways to extend diffusion schemes to Navier–Stokes schemes: gradient formula or upwind flux. in Proceedings of 20th AIAA Computational Fluid Dynamics Conference, AIAA Paper, Honolulu, Hawaii (2011), pp. 2011–3044 6. R.B. Lowrie, J.E. Morel, Methods for hyperbolic systems with stiff relaxation. Int. J. Numer. Methods Fluids 40, 413–423 (2002) 7. H. Baty, H. Nishikawa, Hyperbolic method for magnetic reconnection process in steady state magnetohydrodynamics. MNRAS 459, 624–637 (2016) 8. I.J.D. Craig, S.M. Henton, Exact solutions for steady state incompressible magnetic reconnection. Astrophys. J. 450, 280–288 (1995) 9. A. Mazaheri, H. Nishikawa, Very efficient high-order hyperbolic schemes for time-dependent advection-diffusion problems: third-, fourth-, and sixth-order. Comput. Fluids 102, 131–147 (2014)
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10. Y. Nakashima, N. Watanabe, H. Nishikawa, Hyperbolic Navier–Stokes solver for threedimensional flows. in Proceedings of 54th AIAA Aerospace Sciences Meeting, AIAA Paper 2016-1001, San Diego, California (2016) 11. R. Keppens, O. Porth, K. Galsgaard, J.T. Frederiksen, A.L. Restante, G. Lapenta, C. Parnell, Resistive magnetohydrodynamic reconnection: resolving long-term, chaotic dynamics. Phys. Plasmas 20, 092109 (2013) 12. I. Peskkov, E. Romenski, A hyperbolic model for viscous Newtonian flows. Contin. Mech. Thermodyn. 28, 85–104 (2016)
A General Well-Balanced Finite Volume Scheme for Euler Equations with Gravity Jonas P. Berberich, Praveen Chandrashekar and Christian Klingenberg
Abstract We present a second-order well-balanced Godunov-type finite volume scheme for compressible Euler equations with a gravitational source term. The scheme is designed to work for any hydrostatic equilibrium, which must be known á priori. It can be combined with any numerical flux function, time-stepping method, and grid topology. The scheme is based on the reconstruction of a special set of variables and a special source term discretization. We show the well-balanced property numerically for isothermal and polytropic equilibria in one and two dimensions using the Roe flux function and an explicit three-stage Runge–Kutta scheme. We demonstrate the superior resolution of small pressure perturbations of hydrostatic equilibria, down to an order 10−10 and below compared to the hydrostatic background. Keywords Euler equations with gravity · Finite volume method · Well-balancing
1 Introduction Astrophysical processes involve many orders of magnitude in space, time, and thermodynamical quantities such as density and pressure. The overall evolution of our sun, for example, can be measured in a scale of 1010 years, but from a hydrodynamical point of view, the sun can be seen as being in a hydrostatic equilibrium locally in time. Many hydrodynamical processes are small perturbations to that equilibrium. It is common to model these dynamics using compressible Euler equations with gravity. The astrophysical equation of state is very complex since they take into account a J. P. Berberich (B) · C. Klingenberg Department of Mathematics, University of Würzburg, Würzburg, Germany e-mail: [email protected] C. Klingenberg e-mail: [email protected] P. Chandrashekar TIFR Center for Applicable Mathematics, Bangalore, India e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_12
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whole bunch of physical effects including relativistic and quantum effects. Typically, there is no closed formulation for these equations of state. Hence, one cannot hope to calculate the hydrostatic equilibria analytically. Also, many processes occur at low Mach numbers that require special schemes. It is also necessary to use general grids in order to be able to compute flows inside star-like geometry, where Cartesian grids may not be suitable. In this paper, such a general well-balanced finite volume scheme will be presented which can preserve hydrostatic solutions exactly on even coarse meshes. It is designed to balance any arbitrary hydrostatic equilibrium, even if it is only given numerically on the grid. Thus, it can cope with arbitrary equations of state. It can be combined with any numerical flux function including low Mach number fluxes. Also, any time-stepping method can be used including implicit methods. We will present the scheme for one spatial dimension, but it can be easily extended to two or three spatial dimensions. In numerical simulations it can be practical to use non-Cartesian grids, which can be mapped to a Cartesian grid, to reflect the systems symmetries. Our scheme can be used on this kind of grid also. Numerical results are presented to show the well-balanced property and also the ability of the scheme to accurately resolve small perturbations around the hydrostatic solution.
2 One-Dimensional Euler Equations with Gravity Consider the system of compressible Euler equations in one dimension, which models conservation of mass, momentum, and energy and is given by ∂ρ ∂ + (ρu) = 0, ∂t ∂x ∂ dφ ∂ (ρu) + ( p + ρu 2 ) = −ρ , ∂t ∂x dx ∂E ∂ + (Eu + pu) = 0. ∂t ∂x Here, ρ is the density, u is the velocity, p is the pressure, E is the energy per unit volume including the gravitational energy, and φ is the gravitational potential. The total energy E is given by E = ρε + 21 ρu 2 + ρφ where ε is the internal energy per unit mass. We can write the above set of coupled equations in a compact notation as ⎡ ⎤ 0 ∂f ∂q + = ⎣s ⎦ , ∂t ∂x 0
s = −ρ
dφ dx
where q is the set of conserved variables and f is the corresponding flux vector.
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2.1 Hydrostatic States Consider the hydrostatic stationary solution, i.e., for which the velocity is u¯ = 0. In this case, the mass and energy conservation equations are automatically satisfied. The momentum equation becomes an ordinary differential equation given by dφ d p¯ = −ρ¯ . dx dx
(1)
We will write the hydrostatic solution as ρ¯ = ρ0 α(x),
p¯ = p0 β(x),
where p0 , ρ0 are reference values at some location x = x0 and α(x), β(x) are nondimensional functions. These functions must satisfy the hydrostatic equation (1) p0 β (x) = −ρ0 α(y)φ (x),
i.e.,
φ (x) = −
p0 β (x) , ρ0 α(x)
(2)
Note that since the pressure and density are strictly positive, we have α(x) > 0, β(x) > 0.
3 One-Dimensional Finite Volume Scheme Let us divide the domain into N finite volumes each of size Δx. The ith cell is given by the interval (xi− 21 , xi+ 21 ). Consider the semi-discrete finite volume scheme for the ith cell ⎡ ⎤ 0 ˆf i+ 1 − ˆf i− 1 dq i 2 2 + = ⎣si ⎦ , (3) dt Δx 0 L R ) is a consistent numerical flux. For the source term, we where ˆf i+ 21 = ˆf (q i+ 1,q i+ 21 2 make use of the representation of the gravitational potential in terms of the hydrostatic functions α, β given by (2), and we will write the source term in the general case as
s(x, t) =
p0 β (x) ρ(x, t). ρ0 α(x)
Applying a central difference discretization to the source term, we obtain si =
p0 βi+ 21 − βi− 21 ρi , ρ0 Δx αi
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which is a second-order accurate approximation. Note that even though we could calculate the source term exactly, we use an approximation since this helps to construct the well-balanced scheme. L R at the cell boundary which are required to calculate To obtain the states q i+ 1,q i+ 1 2
2
the numerical flux ˆf i+ 21 , we will reconstruct the following set of variables w = [ρ/α, u, p/β] . L Once wi+ 1 , etc., are computed, the primitive variables are obtained as 2
L L ρi+ , 1 = αi+ 1 (w1 ) i+ 1 2 2
2
L L u i+ , 1 = (w2 ) i+ 1 2
2
L L pi+ , etc., 1 = βi+ 1 (w3 ) i+ 1 2 2
2
where αi+ 21 = α(xi+ 21 ) and βi+ 21 = β(xi+ 21 ).
3.1 Well-Balanced Property We now state the basic result on the well-balanced property. Theorem 1. The finite volume scheme (3) together with any consistent numerical flux and reconstruction of w variables is well balanced in the sense that all initial conditions satisfying u i = 0,
ρi /αi = const,
pi /βi = const,
∀i
(4)
are preserved by the numerical scheme. Proof. Let us start the computations with an initial condition that satisfies (4). Since we reconstruct the variables w which are constant for a hydrostatic solution, at any interface i + 21 we have L R = 0, u i+ 1 = u i+ 1 2
2
L R ρi+ =: ρi+ 21 , 1 = ρ i+ 1 2
2
L R pi+ =: pi+ 21 . 1 = p i+ 1 2
2
Since the numerical flux is consistent, we get ˆf i− 1 = [0, pi− 1 , 0] , 2 2
ˆf i+ 1 = [0, pi+ 1 , 0] . 2 2
The flux in mass and energy equations are zero and the gravitational source term in the energy equation is also zero. Hence, the mass and energy equations are already well dq (1)
dq (3)
balanced, i.e., dti = 0 and dti = 0. It remains to check the momentum equation. On the left, we have ˆf (2) 1 − ˆf (2) 1 pi+ 21 − pi− 21 i+ 2 i− 2 = , Δx Δx
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while on the right si =
p0 βi+ 1 − p0 βi− 1 pi+ 1 − pi− 1 p0 βi+ 21 − βi− 21 ρi p0 βi+ 21 − βi− 21 2 2 2 2 = ρ0 = = ρ0 Δx αi ρ0 Δx Δx Δx dq (2)
and hence dti = 0. This proves that the initial condition is preserved under any time integration scheme. Remark 1. This is a general result in the sense that we have not made any assumptions on the equation of state (EOS) or on the type of gravitational field. It holds for any consistent numerical flux and reconstruction scheme to obtain the cell face values. Remark 2. It is straightforward to generalize this scheme to two or three spatial dimensions using the method of lines. It is also possible to use coordinates, which can be mapped smoothly on Cartesian coordinates. We will refer to this kind of coordinates as curvilinear coordinates. We will use these cases later in this article for numerical tests.
4 Numerical Tests The scheme has been implemented in the Seven-League Hydro Code [3, 7]. We use the standard Roe flux [8] for all numerical experiments. For time stepping, we use an explicit three-stage Runge–Kutta scheme. This scheme has been proposed in [9]. It is third order in time and has the total variation diminishing property. An ideal gas with a gas constant R = 1 and a specific heats ratio of γ = 1.4 is assumed.
4.1 One-Dimensional Numerical Results In the one-dimensional tests, the domain is the interval [0, 1]. If not stated explicitly, we use Dirichlet boundaries, which are consistent with the initial condition. This is achieved by using ghost cells and keeping them constant over time.
4.1.1
Isothermal Atmosphere
A solution of the hydrostatic equation (1) for constant temperature T = 1 is given by ρ(x) ¯ = p(x) ¯ = exp(−φ(x)). (5) ¯ β = p. ¯ We choose α0 = β0 = 1 and hence α = ρ, We consider the three different gravitational potentials φ(x) = x, φ(x) = 21 x 2 , and φ(x) = sin(2π x). For the latter, we use periodic boundary conditions. The errors
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Table 1 L 1 errors for the one-dimensional isothermal atmosphere of Sect. 4.1.1 φ N v p ρ x 1 2 2x
sin(2π x)
100 1000 100 1000 100 1000
1.62316e-15 1.83243e-14 3.21800e-16 1.27564e-14 1.15182e-15 3.55886e-15
1.12521e-15 1.28197e-14 1.72085e-16 6.03995e-15 5.72931e-15 3.37666e-14
1.20681e-15 1.26239e-14 8.52651e-16 1.31585e-14 3.43836e-15 1.61612e-14
Table 2 L 1 errors for the one-dimensional isentropic atmosphere of Sect. 4.1.2 φ N v p ρ x 1 2 2x
sin(2π x)
100 1000 100 1000 100 1000
1.41193e-15 1.61820e-14 1.37162e-15 1.39921e-14 5.37656e-15 2.35665e-14
9.19265e-16 1.16552e-14 5.20695e-16 7.03848e-15 1.27287e-15 2.25356e-14
1.56597e-15 1.66951e-14 2.13718e-15 1.84036e-14 1.51712e-15 1.78504e-14
after a time t = 2.0 in L 1 norm are given in Table 1. All errors are close to machine precision.
4.1.2
Isentropic Atmosphere
Isentropic solutions of Eq. (1) are of the form ν−1 1 φ(x), ρ¯ = T¯ ν−1 , T¯ (x) = 1 − ν
p¯ = ρ¯ ν
(6)
with ν = γ . Again, we choose α0 = β0 = 1 and hence α = ρ, ¯ β = p. ¯ The same tests as in Sect. 4.1.1 are conducted and the L 1 norms of the errors are shown in Table 2. All errors are close to machine precision.
4.1.3
Polytropic Atmosphere
Polytropic solutions of Eq. (1) are more general than isentropic solutions. We can write them in the form of Eq. (6), but with ν = γ possible. Let us use ν = 1.2 following [1]. We choose α0 = β0 = 1 and hence α = ρ, ¯ β = p. ¯ The same tests as in Sect. 4.1.1 are conducted and the L 1 norms of the errors are shown in Table 3. All errors are close to machine precision.
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Table 3 L 1 errors for the one-dimensional polytropic atmosphere of Sect. 4.1.3 φ N v p ρ x 1 2 2x
sin(2π x)
4.1.4
100 1000 100 1000 100 1000
1.21736e-15 1.75455e-14 1.39958e-15 1.31861e-14 3.82965e-15 1.49882e-14
9.60343e-16 1.24413e-14 3.96350e-16 6.04949e-15 1.90958e-15 1.79979e-14
1.44995e-15 1.60103e-14 1.20570e-15 1.45129e-14 1.68809e-15 8.44363e-15
Evolution of a Small Perturbation
To study the propagation of a small perturbation from the isothermal hydrostatic equilibrium (5), we use the following initial condition: 1 2 1 2 φ(x) = x , u = 0, ρ = exp(−φ(x)), p(x) = exp(−φ(x)) + ε exp −100 x − . 2 2
This setup corresponds to comparable tests in [1, 5]. We present this test for different ε and resolutions in Fig. 1. We conducted the tests using our well-balanced scheme and another scheme, which is not well-balanced. The non-well-balanced scheme is described in [7] and discretizes the source term pointwise via si = ρi g, where g = −φ . We set this derivative explicitly. The test results are illustrated in Fig. 1. In Fig. 1a, we see that for a large pressure perturbation with ε = 10−3 the well-balanced and the non-well-balanced scheme give comparable results for a short time even on a coarse grid of N = 100 cells. When the perturbation is decreased to ε = 10−5 , the non-well-balanced scheme with the same parameters shows significant spurious effects, as we can see in Fig. 1b. The well-balanced scheme still shows a good result. One has to increase the number of grid cells to get a good result with the non-well-balanced scheme for the small perturbation. We can see this in Fig. 1c for N = 500 cells. In Fig. 1d we show that the result with the well-balanced scheme does not improve significantly if we increase the number of grid cells to more than N = 100 even for the small perturbation. Hence, using this scheme can save computational effort by reducing the necessary resolution.
4.1.5
Nonideal Equation of State
In this section, we aim to show the ability of the scheme to preserve hydrostatic equilibria even if an other EOS than the ideal gas EOS is used. For that, let us use the EOS for an ideal gas with radiative pressure [2] p = ρT + T 4 .
(7)
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Fig. 1 Evolution of a small pressure perturbation on an one-dimensional isothermal equilibrium. The test setup is described in Sect. 4.1.4 Table 4 L 1 errors for the one-dimensional isothermal atmosphere with an nonideal EOS of Sect. 4.1.5 φ N v p ρ x 1 2 2x
sin(2π x)
100 1000 100 1000 100 1000
1.05583e-15 1.53555e-14 2.09505e-15 7.01467e-15 7.01467e-15 7.01467e-15
3.74811e-15 3.19185e-14 6.87894e-15 1.44657e-15 1.44657e-15 1.44657e-15
1.80023e-15 1.31348e-14 3.80584e-15 1.39411e-15 1.39411e-15 1.39411e-15
Isothermal Atmosphere. With the choice of T ≡ 1, Eqs. (1) and (7), we find an isothermal hydrostatic equilibrium given by ρ(x) ¯ = exp (−φ(x)) ,
p¯ = exp (−φ(x)) + 1.
(8)
With this modified EOS and equilibrium, we conduct the same tests as in Sect. 4.1.1. The corresponding errors in L 1 norm can be seen in Table 4. All errors are close to machine precision.
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Table 5 L 1 errors for the one-dimensional polytropic atmosphere with an nonideal EOS of Sect. 4.1.5 φ N v p ρ x 1 2 2x
sin(2π x)
100 1000 100 1000 100 1000
8.36066e-16 1.61427e-14 4.37774e-16 2.71318e-15 7.59499e-16 7.01467e-15
3.95795e-16 6.89920e-15 1.26565e-16 8.31668e-16 1.47105e-16 1.44657e-15
2.84772e-16 4.68958e-15 1.07692e-16 5.95635e-16 1.11022e-16 1.39411e-15
Polytropic Atmosphere. To derive the polytropic equilibrium shown in Eq. (6), the EOS is not used. Hence, Eq. (6) describes an equilibrium for every EOS. We conduct the same test as in Sect. 4.1.3 with the EOS in Eq. (7) instead of an ideal gas law. The results can be seen in Table 5. As in the isothermal setup, all errors stay close to machine precision.
4.2 Two-Dimensional Numerical Results The two-dimensional tests will be conducted on the unit square [0, 1] × [0, 1]. We use a Cartesian grid and the Dirichlet boundary condition, if not stated explicitly.
4.2.1
Isothermal Atmosphere
Consider the hydrostatic solution given by ρ(x, ¯ y) = ρ0 exp(−ρ0 φ(x, y)/ p0 ),
p(x, ¯ y) = p0 exp(−ρ0 φ(x, y)/ p0 ),
(9)
with φ(x, y) = x + y, ρ0 = 1.21, and p0 = 1. We apply our well-balanced scheme as well as the non-well-balanced scheme to evolve this initial condition to a time t = 1.0. The errors in L 1 norm with respect to the initial condition are shown in Table 6. For the well-balanced scheme, all errors are close to machine precision. The nonwell-balanced scheme on the other hand introduces significant errors, which decrease for the higher resolution. 4.2.2
Isentropic Atmosphere
Consider the hydrostatic solution (6) on the unit square with the gravitational potential φ(x, y) = x + y. We run the simulation on different Cartesian grids to the final time t = 1.0. The results are shown in Table 7 using the L 1 norm. Like in Sect. 4.2.1 the
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Table 6 L 1 errors for the two-dimensional isothermal atmosphere of Sect. 4.2.1 Cells Scheme vx vy p ρ 50 × 50
200 × 200
Well-balanced Non-wellbalanced Well-balanced Non-wellbalanced
1.00232e-15 2.55432e-03
9.33272e-16 2.55432e-03
4.35663e-16 1.71889e-03
5.76417e-16 3.82551e-03
2.53379e-15 6.38199e-04
2.54463e-15 6.38199e-04
1.74903e-15 4.15871e-04
1.85208e-15 9.42026e-04
Table 7 L 1 errors for the two-dimensional isentropic atmosphere of Sect. 4.2.2 Cells Scheme vx vy p 50 × 50
200 × 200
Well-balanced Non-wellbalanced Well-balanced Non-wellbalanced
ρ
1.09672e-15 2.41783e-03
1.01388e-15 2.41783e-03
5.44995e-16 1.89444e-03
7.99222e-16 1.84055e-03
2.69286e-15 6.11321e-04
2.69081e-15 6.11321e-04
2.07824e-15 4.63472e-04
2.73340e-15 4.50069e-04
Table 8 L 1 errors for the two-dimensional polytropic atmosphere of Sect. 4.2.3 Cells Scheme vx vy p ρ 50 × 50
200 × 200
Well-balanced Non-wellbalanced Well-balanced Non-wellbalanced
1.02690e-15 2.08336e-03
1.06307e-15 2.08336e-03
5.41545e-16 1.62272e-03
6.48515e-16 2.36791e-03
2.83880e-15 5.19236e-04
2.81328e-15 5.19236e-04
2.07345e-15 3.93168e-04
2.14476e-15 5.81252e-04
well-balanced scheme keeps errors close to machine precision, while the non-wellbalanced scheme introduces significant errors.
4.2.3
Polytropic Atmosphere
Consider the hydrostatic solution (6) with ν = 1.2 as initial condition on the unit square with the gravitational potential φ(x, y) = x + y. We run the simulation on different Cartesian grids to the final time t = 1.0. The results are shown in Table 8 using the L 1 norm. Again, the well-balanced scheme keeps errors close to machine precision, while the non-well-balanced scheme introduces significant errors.
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Fig. 2 Evolution of a small pressure perturbation on an isothermal equilibrium (Lines: dotted at −0.1η, solid at 0.0, dashed at 0.1η)
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Fig. 3 Evolution of a small pressure perturbation on an polytropic equilibrium (Lines: dotted at −0.1η, solid at 0.0, dashed at 0.1η)
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Evolution of a Small Perturbation
In this test, we study the evolution of a small perturbation added to the hydrostatic solution. We test this for the isothermal equilibrium (9) and the polytropic equilibrium (6). In correspondence with [1, 10], we choose the same parameters as in the Sects. 4.2.1 and 4.2.3 respectively and the initial pressure
p(x, y, 0) = p(x, ¯ y) + η exp −100ρ0 (x − 0.3)2 + (y − 0.3)2 / p0 .
(10)
The initial density is ρ(·, ·, 0) = ρ, ¯ the resolution is 64 × 64 cells in each test. The results can be seen in the Figs. 2 and 3. The results are similar for both equilibria: The large perturbation with η = 0.1 is well resolved for both the wellbalanced and the non-well-balanced scheme. When the perturbation is decreased to η = 0.001, the non-well-balanced scheme is not able to resolve it well anymore since the discretization errors of the background start to dominate after some time. The well-balanced scheme shows no problems for the smaller perturbation. The isothermal test case has also been conducted on a curvilinear mesh. The structure of this mesh is shown in Fig. 2f. The result of the test can be seen in Fig. 2e. We see, that the usage of the curvilinear mesh introduces no significant errors in this test. In Fig. 3e and f we can see that even a small perturbation of η = 10−10 or below leads to a well-resolved result, if the well-balanced scheme is used.
References 1. P. Chandrashekar, C. Klingenberg, A second order well-balanced finite volume scheme for Euler equations with gravity. SIAM J. Sci. Comput. 37(3), B382–B402 (2015) 2. S. Chandrasekhar, An Introduction to the Study of Stellar Structure, vol. 2 (Courier Corporation, 1958) 3. P.V.F. Edelmann, Ph.D. thesis, Technische Universität München (2014) 4. M. Hosea, L. Shampine, Method of lines for time-dependent problems. Appl. Numer. Math. 20, 21 (1996) 5. R.J. LeVeque, D.S. Bale, Wave propagation methods for conservation laws with source terms, Hyperbolic Problems: Theory, Numerics, Applications (Birkhäuser, Basel, 1999), pp. 609–618 6. M.-S. Liou, J. Comput. Phys. 214, 137 (2006) 7. F. Miczek, Ph.D. thesis, Technische Universität München (2013) 8. P. Roe, J. Comput. Phys. 43, 357 (1981) 9. C.-W. Shu, S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988) 10. Y. Xing, C.-W. Shu, High order well-balanced WENO scheme for the gas dynamics equations under gravitational fields. J. Sci. Comput. 54(2–3), 645–662 (2013)
A Second-Order Well-Balanced Scheme for the Shallow Water Equations with Topography Christophe Berthon, Raphaël Loubère and Victor Michel-Dansac
Abstract We consider the well-balanced numerical scheme for the shallow water equations with topography introduced in Michel-Dansac et al. (Comput Math Appl 72(3):568–593, 2016, [8]) and its second-order well-balanced extension, which requires two heuristic parameters. The goal of the present contribution is to derive a parameter-free second-order well-balanced scheme. To that end, we consider a convex combination between the well-balanced scheme and a second-order scheme. We then prove that a relevant choice of the parameter of this convex combination ensures that the resulting scheme is both second-order accurate and well-balanced. Afterward, we perform several numerical experiments, in order to illustrate both the second-order accuracy and the well-balanced property of this numerical scheme. Finally, we outline some perspectives in a short conclusion. Keywords Shallow-water equations · Godunov-type schemes Fully well-balanced schemes · Second-order accuracy · Moving steady states
1 Introduction We consider the shallow water system with topography, governed by the following set of equations: C. Berthon Laboratoire de Mathématiques Jean Leray, Université de Nantes, 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France e-mail: [email protected] R. Loubère Institut de Mathématiques de Bordeaux and CNRS, Université de Bordeaux, 351 cours de la Libération, 33405 Talence, France e-mail: [email protected] V. Michel-Dansac (B) Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9, France e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_13
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∂t h + ∂x q = 0, ∂t q + ∂x
q2 h
+ 21 gh 2 = −gh∂x Z ,
(1)
where h(t, x) > 0 is the water height, q(t, x) is the water discharge, Z (x) is the smooth topography, and g is the gravity constant. To shorten the notations, we rewrite this system under the condensed form ∂t W + ∂x F(W ) = S(W ), where we have set ⎛ ⎞ q h 0 2 ⎝ ⎠ . W = ; F(W ) = q ; S(W ) = 1 q −gh∂x Z + gh 2 h 2 In particular, we focus on smooth steady solutions, free from time and governed by q = cst, q + g(h + Z ) = cst. 2h 2
(2)
Well-balanced schemes, i.e., schemes that exactly preserve such steady solutions, have been derived in the past decade (see for instance [2, 3, 5, 8]). Namely, in [8], the authors suggested a well-balanced Godunov-type scheme based on a two-state approximate Riemann solver. We briefly recall the general form of a numerical scheme that falls within this classification (see for instance [10]). In the finite volume framework, the space domain R is discretized into cells, assumed to be of constant width x. The center of the ith cell is denoted by xi and its bounds are labeled xi− 21 and xi+ 21 ; this cell shall henceforth be referred to by its center xi . The approximate solution is piecewise constant, and it is denoted by Win within the cell xi and at time t n . In order to provide a time update of this approximate solution, we note that Riemann problems (i.e., Cauchy problems with discontinuous initial data) are present at each interface between cells. However, the exact solution to such Riemann problems is usually difficult or impossible to compute exactly. To address this issue, an approximate Riemann solver is introduced. More specifically, in [8], the authors develop an approximate Riemann solver satisfying several crucial properties: consistency, well-balanced, and preservation of the water height nonnegativity. The time update of the approximate solution in the cell xi reads Win+1,W B = Win −
t L ,∗ R,∗ n n λi+ 21 Wi+ + λi− 21 Wi− , 1 − hi 1 − hi 2 2 x
(3)
L ,∗ R,∗ where Wi+ are the intermediate states of the approximate Riemann solver, 1 and W i− 21 2 respectively, approximations of the Riemann solutions at the interfaces xi+ 21 and xi− 21 . In addition, λi+ 21 and λi− 21 are approximations of the wave velocities from the exact Riemann solution. The authors of [8] prove that the Godunov-type scheme (3) is consistent, well-balanced, and nonnegativity-preserving. The accuracy of the first-order scheme (3) could be significantly improved by introducing a well-balanced second-order extension. The MUSCL framework is well
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suited to this extension. It consists in providing a piecewise linear approximation, instead of piecewise constant, of the solution in each cell. This is achieved using slope reconstructions, supplemented with slope limiters in order to recover the nonnegativity preservation. For more details on such procedures, the reader is referred for instance to [7]. The resulting MUSCL scheme reads as follows: Win+1,MU SC L = Win −
t Fi + tSi , x
(4)
where Fi and Si are, respectively, second-order approximations of the physical flux F and the source term S in the cell xi . Note that the intermediate states of the scheme (3) are used to define Fi and Si . In addition to the second order in space time update (4), a specific treatment of the steady states is necessary, because this scheme is no longer naturally well-balanced due to the reconstruction procedure. For instance, in [3, 4], the authors suggest a reconstruction based on the steady states, which leads to a well-balanced second-order scheme. However, the downside of this approach is that, in each cell, the nonlinear steady relations (2) have to be solved, thus leading to extra computational cost. In [8], the authors proposed a convex combination, in each cell ci and at time t n+1 , between the well-balanced scheme and a MUSCL reconstruction to recover the wellbalanced property without having to solve nonlinear equations, as follows: Win+1 = θin Win+1,MU SC L + (1 − θin )Win+1,W B .
(5)
On the one hand, for a steady state, the well-balanced scheme is exact, and, therefore, is of order at least two. In this case, we wish to use the well-balanced scheme. On the other hand, for an unsteady state, the well-balanced scheme is of order one, and it should not be used. The MUSCL scheme is second-order accurate in both these cases. Therefore, we wish to use the MUSCL scheme when the approximate solution is unsteady and the well-balanced scheme when it is steady. As a consequence, the convex combination (5) becomes relevant when its parameter θin allows switching between the MUSCL scheme and the well-balanced scheme to ensure both a secondorder accuracy and the well-balanced property. To that end, θin must be equal to 1 in the unsteady case, and it must vanish for a steady state. In [8], this convex combination relied on two heuristic parameters used to define θin with respect to some error to a steady state. The goal of the present manuscript is to propose a parameter-free formula for θin , that ensures both the well-balanced property and the second-order accuracy of the scheme. To that end, we first introduce such an expression of θin . We then prove that the required properties are satisfied by the resulting scheme. Finally, several numerical experiments confirm the second-order accuracy and the well balance of the scheme. A short conclusion outlines several perspectives to this work.
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2 A Second-Order Accurate Convex Combination The goal of this section is to introduce a parameter-free expression of θin such that the convex combination (5) yields a second-order accurate and well-balanced scheme. To that end, let us first define the following potential : =
q2 + g(h + Z ). 2h 2
Note that, for a steady state, we have = cst as well as q = cst, as per (2). Let us then define the spatial errors to a steady state, as follows:
n n eˆin = max |qin − qi−1 |, |qi+1 − qin | ,
n n eˇin = max |in − i−1 |, |i+1 − in | . In order to provide a relevant definition of θin , we make the following remarks: • the well-balanced scheme is stationary (i.e., Win+1,W B = Win ) if and only if the solution is stationary, i.e., eˆin = 0 and eˇin = 0; • the MUSCL scheme is not well-balanced; it can, however, become stationary (i.e., Win+1,MU SC L = Win ), but, in this case, eˆin = 0 and eˇin = 0. These remarks lead us to consider switching between the well-balanced and the MUSCL scheme when the time update of the MUCSL scheme becomes very small. Indeed, we will show that, in this case, the MUSCL scheme approximates a steady solution, and switching to the well-balanced scheme ensures its preservation. The following result states how to define θin in order to make sure that the scheme (5) is both well-balanced and second-order accurate. Theorem 1. We first introduce the following two conditions: (C1 ) (C2 )
eˆin < εm and eˇin < εm , |h in+1,MU SC L − h in | ≤ (eh )in and |qin+1,MU SC L − qin | ≤ (eq )in ,
where the errors (eh )in and (eq )in are defined by t qin t 2 qin x 3 n + e ˇ tx + , i x (h in )3 (h in )2 x 2 (h in )2 n qn t qin 3 qi (eq )in = eˆin tx ni 3 + eˇin tx + x , (h i ) x (h in )2 (h in )3 (eh )in = eˆin tx
(6)
and where εm is a measure of the machine precision, usually taken equal to 10−12 in the numerical simulations. Then, let us define θin as follows: θin
=
0 if (C1 ) or (C2 ) holds, 1 otherwise.
(7)
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The above definition of θin ensures the scheme (5) is well-balanced and second-order accurate. Remark 2. Note that the respective units of (eh )in and (eq )in , as defined by (6), are those of the height and the discharge. In addition, note that eˆin = O(x) and eˇin = O(x), and that x = O(t) because of the CFL condition. As a consequence, we remark that (eh )in = O(x 2 t) and (eq )in = O(x 2 t). The remainder of this section is dedicated to a proof of Theorem 1. First, we prove a preliminary result related to the time update of the well-balanced scheme. Then, this result is used to complete the proof of Theorem 1.
2.1 Time Update of the Well-Balanced Scheme with Respect to the Steady-State Deviation The goal of this section is to prove a result that will be used to complete the proof of Theorem 1. It is an estimation of the time update of the well-balanced scheme with respect to the error to a steady state. Indeed, we know that, if this error vanishes, then so does the time update of the well-balanced scheme. The following statement provides us with such an estimation. Lemma 3. Let us consider the following (almost steady) configuration: n qi−1 = qin + εˆ − , n = qin + εˆ + , qi+1
n i−1 = in + εˇ − , n = in + εˇ + . i+1
(8)
Then the time update of the well-balanced scheme (3) satisfies: Win+1,W B = Win + O(ˆε+ ) + O(ˇε+ ) + O(ˆε− ) + O(ˇε− ).
(9)
Proof. In order to prove Lemma 3, we first provide an estimation of the intermediate states of the approximate Riemann solver involved in the scheme (3) and derived in [8]. This estimation will then act as a stepping stone toward proving Lemma 3, by being used at each interface of the cell xi and injected within the time update (3). We begin by considering two states W L and W R almost defining a steady state, i.e., we assume that there exist small εˆ and εˇ such that q R = q L + εˆ and R = L + εˇ . Let us also define the following quantities: • [X ] = X R − X L denotes the jump of a quantity X , • X a = (X L + X R )/2 its arithmetic mean, and • X h = 2X L X R /(X L + X R ) its harmonic mean. In addition, we introduce β=−
q L2 + gh a . hLhR
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We now consider the intermediate states h ∗L , h ∗R , and q ∗ of the Godunovtype scheme introduced in [8]. Equipped with these assumptions and definitions, these intermediate states are proven to satisfy, after straightforward but tedious computations: hL 1 qL [h] q L [h] + + h − λ βh a h R h λ hLhR εˇ qL h L h R + [h] + O(ˆε2 ) + O(ˆε2 ), + 2βh a λ hL εˆ 1 [h] q L [h] qL h ∗R = h R − + − − 2 λ βh a h R hh λ hLhR εˇ qL h L h R + [h] + O(ˆε2 ) + O(ˆε2 ), − 2βh a λ εˇ h h 1 qL εˆ − 1− q ∗ = qL + + O(ˆε2 ) + O(ˆε2 ), a 2 λh 2 λ h ∗L
εˆ = hL − 2
(10a)
(10b)
(10c)
where λ = −λ L = λ R , as prescribed in [8]. Note that such expressions come from the fact that the scheme is well-balanced. Indeed, for a true steady state, we have εˆ = 0 and εˇ = 0, which correctly yields h ∗L = h L , h ∗R = h R and q ∗ = q L = q R . Now, recall that the time update of the well-balanced scheme from [8] reads as follows: h in+1,W B = h in +
L ,∗ λ+ h + − h in + λ− h −R,∗ − h in ,
(11a)
qin+1,W B
∗
λ+ q+ − qin + λ− q−∗ − qin ,
(11b)
t x t = qin + x
where the subscript ± is a shorter notation for i ± 1/2. Let us assume that the almost steady configuration (8) is satisfied for the cells xi−1 , xi and xi+1 . As a consequence, we can use the formulas (10) to rewrite the update (11) as follows: h in+1,W B
=
h in
qin hi t λ+ − εˆ + 1 + + + 2x β+ h a+ h i+1 εˇ + + (λ+ h i h i+1 + qi [h]+ ) β+ h a+ qin h i−1 λ− + + εˆ − 1 − a β− h − hi εˇ − + h h + q [h] (λ ) − i−1 i i − β− h a−
[h]+ h h+
[h]− h h−
2 2 2 2 + O(ˆε+ ) + O(ˇε+ ) + O(ˆε− ) + O(ˇε− ),
q n [h]+ − i h i h i+1
q n [h]− − i h i−1 h i
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qn qn t h h εˆ + λ+ − ai − εˇ + h + + εˆ − λ− + ai + εˇ − h − 2x h+ h−
2 2 2 2 + O(ˆε+ ) + O(ˇε+ ) + O(ˆε− ) + O(ˇε− ).
As a consequence, the estimation (9) holds, and the proof is achieved.
2.2 Proof of Theorem 1 Proof (Theorem 1). The goal of this proof is to show that, with θin defined by (7), the scheme defined by the convex combination (5) is second-order accurate and well-balanced. More precisely, let W ex (t, x) be a smooth exact solution of the system (1) equipped with suitable initial and boundary conditions. We introduce the notation (W ex )in := W ex (t n , xi ). The expected result is established as soon as we have shown that |h in+1 − (h ex )in+1 | = O(x 2 ) and |qin+1 − (q ex )in+1 | = O(x 2 ) and n n , (W ex )in and (W ex )i+1 define a steady solution, then that, if the states (W ex )i−1 n+1 n = Wi . Wi To that end, we consider the three possible cases: θin = 1, θin = 0 because (C1 ) holds, and θin = 0 because (C2 ) holds. • First, if θin = 1, then the scheme is second-order accurate. Indeed, the contribution of the well-balanced scheme is multiplied to 1 − θin , and the convex combination is, therefore, reduced to contribution of the second-order MUSCL scheme. In addition, neither (C1 ) nor (C2 ) holds, and therefore the exact solution is unsteady. Thus, the well balance property is irrelevant in this case. • Second, if (C1 ) holds, then θin = 0, and the convex combination is reduced to the sole well-balanced scheme. Note that (C1 ) is equivalent to the approximate solution being steady (up to the machine precision). Since only the well-balanced scheme is used in the update (5), the resulting scheme exactly preserves this steady solution, and it is, consequently, at least second-order accurate. • Third, let us assume that (C2 ) holds. As a consequence, θin = 0 and the wellbalanced scheme is used. However, contrary to the case where (C1 ) held, the approximate solution is unsteady. For this third case, we need to prove that the well-balanced scheme is actually second-order accurate. To that end, we prove that (C2 ) necessarily implies that the approximate solution is close to a steady state, up to x 2 . Arguing Lemma 3 will then allow us to conclude that the well-balanced scheme is actually second-order accurate. Using Remark 2, we get that |Win+1,MU SC L − Win | = O(x 2 t). Arguing the expression (4) of the time update Win+1,MU SC L immediately yields
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Fi 2 − S i = O(x ). x
(12)
The above equation is a discretization of the steady relation ∂x F(W ) = S(W ). n n Therefore, since the MUSCL scheme is consistent, the states Wi−1 , Win and Wi−1 are close to a steady state. Thus, there exist small εˆ − , εˆ + , εˇ − and εˇ + such that n qi−1 = qin + εˆ − , n = qin + εˆ + , qi+1
n i−1 = in + εˇ − , n = in + εˇ + . i+1
(13)
Consequently, the above identities are a direct consequence of the condition (C2 ), and they hold as soon as it is true. We now set out to prove that εˆ − , εˆ + , εˇ − and εˇ + are of order O(x 2 ). Once this fact is established, applying Lemma 3 will conclude the proof in this third case. To that end, let us introduce a truly steady state W steady , such that steady steady qi−1 = qi , steady steady , qi+1 = qi
steady steady i−1 = i , steady steady . i+1 = i
Since the MUSCL scheme is consistent and second-order accurate in space, the following estimation holds: steady,MU SC L
|Wi
t
steady
− Wi
|
= O(x 2 ),
steady,MU SC L
where Wi denotes the time update provided by the MUSCL scheme when considering W steady as initial condition. Arguing the expression (4) of this time update, we get Fi steady ) = O(x 2 ). x − Si (W
(14)
Moreover, we show after tedious computations that Fi Fi steady − (W − S − S ) i i x = O(ˆε+ ) + O(ˇε+ ) + O(ˆε− ) + O(ˇε− ), x (15) where the first term of the left-hand side corresponds to the MUSCL scheme applied to the current configuration (13). Plugging the estimation (14) into (15) yields: Fi 2 x − Si = O(x ) + O(ˆε+ ) + O(ˇε+ ) + O(ˆε− ) + O(ˇε− ).
(16)
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Using both (12) and (16), we obtain the result we had set out to prove εˆ − = O(x 2 ) ; εˆ + = O(x 2 ) ; εˇ − = O(x 2 ) ; εˇ + = O(x 2 ).
(17)
Finally, note that since the approximate solution satisfies (13), which is identical to (8). Therefore, we apply Lemma 3, using (17), to conclude that the time update of the well-balanced scheme satisfies the following estimation: Win+1,W B = Win + O(x 2 ).
(18)
In addition, thanks to (17), the configuration (13) becomes n qi−1 = qin + O(x 2 ), n qi+1 = qin + O(x 2 ),
n i−1 = in + O(x 2 ), n i+1 = in + O(x 2 ),
n n , Win , Wi+1 corresponds to a steady state, up to x 2 . Thereand the sequence Wi−1 fore, (18) yields |Win+1,W B − (W ex )in+1 | = O(x 2 ),
i.e., the well-balanced scheme is second-order accurate. Since the convex combination scheme (5) is reduced to the contribution of the well-balanced scheme, it is second-order accurate. Therefore, in all three cases under consideration, the scheme (5) is at least secondorder accurate. In addition, if a steady state is considered, then this scheme is exact. As a consequence, the convex combination (5) yields a well-balanced and second-order accurate scheme, which concludes the proof of Theorem 1.
3 Numerical Experiments In this section, we propose three numerical experiments. The goal of these experiments is to check that the scheme (5) satisfies the required properties. To this end, we first present an experiment dedicated to the computation of the order of accuracy. Then, we check the well-balanced property of the scheme by considering an unsteady state, which, in finite time and after a transient state, converges to a steady state. The last experiment consists in a “dam-break” problem over a non-flat topography. The numerical schemes tested in these experiments are labeled as follows: the first-order well-balanced scheme is called WB, the second-order scheme is labeled MUSCL, and the convex combination (5) is called θ -WB. In addition, the time accuracy of both second-order schemes is improved thanks to Heun’s method.
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Table 1 L 2 -error on for the approximation of a smooth solution N WB MUSCL 25 50 100 200 400
5.46e-01 2.84e-01 1.55e-01 8.11e-02 4.10e-02
— 0.94 0.87 0.94 0.98
2.89e-01 2.84e-02 7.36e-03 1.90e-03 5.15e-04
— 3.34 1.95 1.95 1.88
θ-WB 2.94e-01 2.41e-02 5.99e-03 1.51e-03 4.41e-04
— 3.61 2.01 1.99 1.78
3.1 Order of Accuracy Verification This first experiment consists in the approximation of a smooth solution. This smooth solution is defined on the space domain [0.9, 1.1] by h(x) = 1 + ω
2 (x − 1) , q(x) = 0, 0.05
Z (x) =
1 3 π 2 , + cos π(x + 0.05) + 4 4 4
where we have set ω(y) =
⎧ ⎨ 2−|y| 4 ⎩0
2
(1 + 2|y|) if |y| < 2, otherwise.
The numerical simulation is carried out until the final physical time tend = 0.005 s. In Table 1, we present the errors on in the L 2 -norm, as well as the corresponding orders of accuracy. These errors have been computed using a reference solution, provided by the hydrostatic reconstruction scheme from [1] with 25,600 discretization cells. Note that similar results are obtained by considering the discharge or other norms. These results show that the θ -WB scheme is more accurate than both the WB and the MUSCL scheme, and that is second-order accurate, as expected.
3.2 Well Balance of the Scheme: Capture of a Steady State We now consider the capture of a steady state obtained after a transient state. Such steady states are exactly captured by the WB scheme, and we require the θ -WB scheme to exactly capture them as well. Namely, we focus on the subcritical steady flow presented in [6]. We consider, over the space domain [0, 25], the topography function Z (x) = 0.2 − 0.05(x − 10)2 + . We take initial conditions at rest, given by q(0, x) = 0 and h(0, x) = h 0 − Z (x), where h 0 = 2. The boundary conditions, q(t, 0) = q0 and h(t, 25) = h 0 (with q0 = 4.42), ensure that the solution is a transient state followed by a smooth steady state with nonzero velocity. This steady state is governed by (2).
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Fig. 1 Subcritical flow experiment. Left panel: free surface and topography for the steady state. Right panel: errors on the discharge q and the potential
The numerical experiment is performed using 100 discretization cells and until the final physical time tend = 500 s. The results are presented in Fig. 1, where we note that the steady state is exactly captured, even after the transient state, by the second-order θ -WB scheme. Indeed, the errors between the numerical discharge (resp. potential) and the steady-state discharge (resp. potential) are of the order of the machine precision. Therefore, this scheme is well-balanced in the sense that it is able to exactly capture steady states, even after a transient state.
3.3 Dam Break Experiment This last experiment consists in a “dam-break” problem, on the space domain [0, 1], whose initial data is q L = 5, q R = 5, L = 60, R = 30. Note that the left and right states of this dam break are moving steady states, which satisfy the Eq. (2). As a consequence, they will be exactly preserved by the first-order well-balanced scheme: the goal of this experiment is to display the accuracy gained by the use of the θ -WB scheme. To that end, we take the exact steady solution as boundary conditions, and we display the approximate solution obtained with 100 cells at the final time tend = 0.02 s, as well as a reference solution, in Fig. 2. The left panel of Fig. 2 shows a comparison between the WB scheme, the θ -WB scheme and a reference solution. We observe that the θ -WB scheme is more accurate than the WB scheme, and that the steady areas are exactly preserved. In the right panel, we display the convex combination parameter θin on the space domain. We have added a plot of q/maxi (qi ) to emphasize the steady areas. As expected, we
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Fig. 2 Riemann problem at time t = 0.02 s. Left panel: free surface. Right panel: value of θin for the θ-WB scheme
have θin = 0 away from the waves, and θin = 1 within and close to the waves. This means that the well-balanced scheme is used in the steady areas, while the MUSCL scheme is used within the dam break itself, as expected.
4 Conclusion We have developed a parameter-free, second-order, and well-balanced extension of the scheme presented in [8]. This new scheme is a significant improvement over the second-order scheme suggested in [8], which relied on a heuristic parameter choice. Several perspectives of this work naturally arise. Namely, in [9], the authors propose a well-balanced scheme for the nonlinear Manning friction source term. Due to the nonlinearity, providing a parameter-free, second-order extension of this scheme would be an interesting challenge. Another challenge lies in a two-dimensional extension of this scheme. Indeed, the definitions of the conditions (C1 ) and (C2 ) in Theorem 1 strongly depend on the expression of the one-dimensional scheme. Acknowledgements C. Berthon and V. Michel-Dansac extend their thanks to the ANR-12-IS010004-01 GEONUM for financial support. R. Loubère and V. Michel-Dansac acknowledge the financial support of the ANR-14-CE23-0007 MOONRISE.
References 1. C. Berthon, C. Chalons, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25(6), 2050–2065 (2004) 2. C. Berthon, C. Chalons, A fully well-balanced, positive and entropy-satisfying Godunov-type method for the shallow-water equations. Math. Comput. 85(299), 1281–1307 (2016)
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3. M.J. Castro, A. Pardo Milanés, C. Parés, Well-balanced numerical schemes based on a generalized hydrostatic reconstruction technique. Math. Model. Methods Appl. Sci. 17(12), 2055–2113 (2007) 4. M.J. Castro Díaz, J.A. López-García, C. Parés, High order exactly well-balanced numerical methods for shallow water systems. J. Comput. Phys. 246, 242–264 (2013) 5. U.S. Fjordholm, S. Mishra, E. Tadmor, Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography. J. Comput. Phys. 230(14), 5587–5609 (2011) 6. N. Goutal, F. Maurel, in Proceedings of the 2nd Workshop on Dam-Break Wave Simulation. Technical Report, Groupe Hydraulique Fluviale, Département Laboratoire National d’Hydraulique, Electricité de France (1997) 7. R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics (Cambridge University Press, Cambridge, 2002) 8. V. Michel-Dansac, C. Berthon, S. Clain, F. Foucher, A well-balanced scheme for the shallowwater equations with topography. Comput. Math. Appl. 72(3), 568–593 (2016) 9. V. Michel-Dansac, C. Berthon, S. Clain, F. Foucher, A well-balanced scheme for the shallowwater equations with topography or Manning friction. J. Comput. Phys. 335, 115–154 (2017) 10. E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics. A practical introduction, 3rd edn. (Springer, Berlin, 2009)
A Lagrangian Approach to Scalar Conservation Laws Stefano Bianchini and Elio Marconi
Abstract We provide an informal presentation of the work mainly contained in Bianchini and Marconi, On the structure of L ∞ entropy solutions to scalar conservation laws in one-space dimension, [3]. We consider the entropy solution u of a scalar conservation law in one-space dimension. In particular, we prove that the entropy dissipation is a measure concentrated on countably many Lipschitz curves. This follows from a detailed analysis of the structure of the characteristics. We will introduce a few notions of Lagrangian representations and we prove that characteristics are segments outside a countably 1-rectifiable set. Keywords Conservation laws · Entropy solutions · Shocks · Concentration Lagrangian representation MSC Code 35L65
1 Introduction We are interested in the structure of the entropy solution u to the scalar conservation law in one-space dimension u t + f (u)x = 0,
f : R → R smooth,
(1)
with initial datum u 0 (x) ∈ L ∞ (R). Being an entropy solution, by definition for all convex entropies η it holds in the sense of distributions η(u)t + q(u)x ≤ 0,
(2)
S. Bianchini · E. Marconi (B) SISSA, via Bonomea, 265, 34136 Trieste, Italy e-mail: [email protected] S. Bianchini e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_14
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where q (u) = f (u)η (u) is the entropy flux. In particular, the r.h.s. of (2) is a nonpositive locally bounded measure μ. Moreover, being the divergence of an L ∞ vector field, μ H 1 . For BV solutions, if we denote by J the jump set of u, by Volpert’s formula it holds η(u)t + q(u)x = η (u) Dtcont u + f (u)Dxcont u + μJ = μJ, where D cont u = (Dtcont u, Dxcont u) is the continuous part of the measure Du. This argument immediately applies when the initial datum has bounded variation because u 0 ∈ BV(R) ⇒ u ∈ BVloc ([0+, ∞) × R) and in the case of uniformly convex flux f with general u 0 ∈ L ∞ (R). In fact by Oleinik estimate [8] f ≥ c > 0 ⇒ u ∈ BVloc ((0+, ∞) × R). If the flux f has finitely many inflection points (together with an additional regularity assumption on f around each inflection point) it has been proved in [6] that f ◦ u ∈ BVloc (R+ × R) and in [7] that μ is concentrated on the jump set of f ◦ u. The main result of this presentation is the following. Theorem 1. There exists a 1-rectifiable set J such that for every entropy η the dissipation measure μ is concentrated on J . The flux f is only supposed to be smooth. The result is a consequence of a description of the structure of the solution u, in particular on the behavior of its characteristics. Here Lagrangian representation means an extension of the method of characteristics. In the case of solutions with bounded variation a first formulation appears in [5], then it has been extended to systems in [4]. In this paper, we present a suitable version to deal with the case of L ∞ solutions introduced in [3]. In all the cited works, the strategy is to exhibit a Lagrangian representation for a dense set of solutions, or approximate solutions and pass it to the limit. It turns out that it is possible to decompose the half-plane R+ × R = A ∪ B ∪ C, where 1. A is countably 1-rectifiable, 2. B is open and uB ∈ BVloc , 3. C is the union of segments starting from 0 on which the solution u is constant. Moreover the slope of the segments is given by the characteristic speed f (u). In order to conclude the proof of Theorem 1, it is sufficient to analyze μC. In Sect. 3 we will see how this structure allows to compute μ by exploiting the balance of u and η(u) in the regions delimited by these segments. The most important tools and ideas of the proof of Theorem 1 are presented but details are often omitted or presented in a simplified setting to reduce technicalities to the essential ones. When it is not indicated where to find the details, we implicitly refer to [3].
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2 Lagrangian Representation and Structure of the Solution In this section, we introduce the notion of Lagrangian representation for bounded entropy solutions. In order to motivate Proposition 2, we present two previous formulations in the particular cases of u 0 ∈ BV and u 0 continuous. Once the Lagrangian representation, in the form of family of admissible boundaries, is available, we present how it is possible to deduce a result on the structure of the solution.
2.1 Lagrangian Representation Consider as a motivation the case of a smooth solution: applying the chain rule we have γ˙ (t) = f (u(t, γ (t)))
⇒
d u(t, γ (t)) = u t + f (u)u x = 0 dt
i.e., u is constant along the characteristic γ which is therefore a straight line. So we introduce a flow X : R+ × R → R where X(t, y) denotes the position of the characteristic starting from y at time t. We say that (X, u 0 ) represents the solution in the sense that (Fig. 1) u(t, x) = u 0 (X(t)−1 (x)). Let v = u x , differentiating formally (1) with respect to x we get vt + ( f (u)v)x = 0.
Fig. 1 The solution u at the point (t, x) is determined by the initial datum u 0 at the point X(t)−1 (x), i.e., the starting point of the characteristic passing through (t, x)
t (t, x)
X(t)−1 (x)
x
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Since X is the flow relative to the vector field f (u) it holds v(t)L 1 = X(t) (v0 L 1 ),
where v0 = (u 0 )x
and X(t) (v0 L 1 ) denotes the push-forward of the measure v0 L 1 by the map X. In particular for every ϕ ∈ Cc∞ (R) R
u(t, x)ϕ (x)d x = −
R
ϕ(X(t, y))v0 (y)dy =
R
u 0 (y)D y (ϕ(X(t, y)))dy.
(3) The regularity of X that you have for free by compactness is monotonicity with respect to y and Lipschitz dependence on time. Moreover, in order to represent a rarefaction, it is convenient to renounce the usual assumption, in the linear case, X(0, ·) = Id. This is reflected in the fact that we need an auxiliary function u instead of u 0 in (3). Definition 1. A Lagrangian representation is a pair (X, u) such that 1. X : R+ t × R y → R is Lipschitz with respect to t and non decreasing with respect to y; 2. u : R → R is continuous; 3. for every ϕ ∈ Cc∞ (R) R
u(t, x)ϕ (x)d x =
R
u(y)d D y (ϕ ◦ X(t))(y).
(4)
Since X(t) is monotone for every t, the derivative in the sense of distributions D y (ϕ ◦ X(t)) is a Radon measure and the integral on the r.h.s. of (4) is well defined. We want to prove the existence of a Lagrangian representation in a dense class of solutions and obtain it for a general solution by approximation. We refer to [5] to see how it is possible to construct a Lagrangian representation for solutions with u 0 ∈ BV starting from wave-front tracking approximations and we discuss in which cases we can pass to the limit in the representation formula (4). Suppose (Xn , un ) are a family of Lagrangian representations for the solutions u n with initial datum u n0 . By Kružkov inequality u n0 → u 0 s-L 1
⇒
lim
n→∞ R
u n (t, x)ϕ (x)d x =
R
u(t, x)ϕ (x)d x.
Since Xn can be constructed equi-bounded on compact sets and equi-Lipschitz with respect to y, up to subsequences, D y (ϕ ◦ Xn (t)) D y (ϕ ◦ X(t)) as Radon measures. Therefore, in order to pass to the limit in the r.h.s. of (4), we need un → u uniformly. In particular, it can be done when u 0 is continuous.
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Fig. 2 Interpretation of a characteristic as an admissible boundary
Proposition 1. Every bounded entropy solution with continuous initial datum has a Lagrangian representation. This result is obtained in [2], where it is also shown how it is possible to deduce the rectifiability of the entropy dissipation measures μη . However it is hopeless to represent the entropy solution with u 0 ∈ L ∞ with a continuous u. Therefore we look for a more stable interpretation of Lagrangian representation. It can be proved in the BV setting that a characteristic γ with value w is an admissible boundary for the solution u in the following sense: Definition 2. Let γ : [0, +∞) → R be a Lipschitz curve, w ∈ R and u be an entropy solution of (1). Denote by Ω − = {(t, x) ∈ (0, T ) × R : x < γ (t)},
Ω + = {(t, x) ∈ (0, T ) × R : x > γ (t)}.
Moreover let u − be the solution of (1) in Ω − with initial condition u 0 {x < γ (0)} and boundary datum constant equal to w on {(t, γ (t)) : t ∈ (0, T )} and similarly for u + . We say that (γ , w) is an admissible boundary in (0, T ) for u if u − = uΩ −
and
u + = uΩ + .
The notion of solution for the initial boundary value problem for scalar conservation laws has been introduced in [1] in the BV setting. For a more general treatment see [9] (Fig. 2). Arguing by approximation, for example by wave-front tracking, we get the following result for solutions u with bounded variations. Proposition 2. There exists a family K of admissible boundaries (γ , w) for u and a function T : K → R+ ∪ {+∞} such that the following hold. 1. For every (γ , w), (γ , w ) ∈ K γ (t) ≤ γ (t) ∀t > 0
or
γ (t) ≤ γ (t) ∀t > 0.
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In particular the set Kγ = {γ : ∃w((γ , w) ∈ K )} is ordered. 2. For every (t, x) ∈ R+ × R and every w ∈ conv(u(t, x−), u(t, x+)) there exists an admissible boundary (γ , w) ∈ K with T (γ , w) ≥ t. 3. For every (γ , w) ∈ K and t < T (γ , w), w ∈ conv(u(t, γ (t)−), u(t, γ (t)+)). 4. The characteristic equation holds: for every γ ∈ Kγ , for L 1 -a.e. t > 0 ⎧ ⎨ f (u(t, γ (t))) if u(t) is continuous at γ (t), γ˙ (t) = f (u(t, γ (t)+)) − f (u(t, γ (t)−)) ⎩ if u(t) has a jump at γ (t). u(t, γ (t)+) − u(t, γ (t)−) Cancelations occur in scalar conservation laws; the function T is introduced to take into account this phenomenon: T (γ , w) denotes the time when the value w is canceled along γ . In the next lemma, we state the stability property that we need to pass to the limit in this formulation. Lemma 1. Let (γ n , wn ) be admissible boundaries for entropy solutions u n of (1) and assume that 1. γ n → γ uniformly; 2. wn → w; 3. u n → u strongly in L 1loc (R+ × R). Then (γ , w) is an admissible boundary for u. We can approximate u 0 ∈ L ∞ (R) with a sequence u n0 ∈ BVloc with respect to the strong L 1loc topology. As we already observed it implies the convergence of the relative solutions u n to u in L 1loc . Since the the curves γ n ∈ Kγn satisfy the characteristic equation, they are equi-Lipschitz. So we have the compactness required to apply Lemma 1. What we get in the limit is a priori much less than a representation as in Proposition 2. Monotonicity passes to the limit and we still have enough boundaries to cover the graph of u. Actually the set K of all the limit points of sequences of admissible boundaries is such that Graph u ⊂ U ⊂ {(t, γ (t), w) : (γ , w) ∈ K and T (γ , w) ≥ t}, where U is the Kuratowski limit of the sequence of the graphs of u n . The first inclusion above can be strict and in general U does not identify a unique u ∈ L ∞ , but we will see in the next section that it does up to linearly degenerate components of the flux f , i.e., intervals where f = 0.
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2.2 Structure of the Solution In this section, we see that being admissible boundaries of an entropy solution u, the elements of K enjoy some additional structure. ¯ We distinguish three Let (t¯, x) ¯ ∈ R+ × R and γ¯ ∈ Kγ be such that γ¯ (t¯) = x. situations, see Fig. 3. 1. There exists γ ∈ Kγ and t < t¯ such that γ¯ (t¯) < γ (t¯) and γ¯ (t ) = γ (t ). 2. Condition 1 does not hold and for every (x n ) convergent to x¯ with x n > x¯ and γ n ∈ Kγ with γ n (t¯) = x n , γ n converges uniformly to γ¯ in [0, t¯]. 3. Conditions 1 and 2 do not hold. It is not difficult to prove that the set of points for which conditions 1 and 2 do not hold is contained in the graphs of countably many Lipschitz curves in Kγ . In the next two lemmas, we consider the first two cases. Lemma 2. Let γ¯ and γ be as in Case 1 above. Then the solution u is monotone with respect to x in the region delimited by the two curves: Ω = {(t, x) ∈ (t , t¯) × R : γ¯ (t) < x < γ (t)}. In the following lemma, the linear degeneracy of the flux plays a role so we introduce the following notation: denote by L f the set of maximal closed intervals (eventually singletons) on which f is constant. Lemma 3. Let x n and γ n be as in Case 2 above and let wn be the corresponding values. Then there exists I ∈ L f such that lim dist(wn , I ) = 0
n→∞
and
∀t ∈ (0, t¯) γ¯˙ (t) = f (I ) ,
t t
n
n
t
Case 1
Case 2
Fig. 3 The three possibilities for a point (t¯, γ¯ (t¯))
Case 3
x
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where f (I ) denotes f (w) for one hence any w ∈ I . In particular γ¯ (0, t¯) is a segment. From Lemmas 2 and 3, it follows the announced decomposition R+ × R = A ∪ B ∪ C where 1. A is contained in the union of countably many graphs of curves in Kγ . 2. B is open and uB ∈ BVloc . 3. C is the union of segments starting from 0 with characteristic speed. See Fig. 4. Moreover from the structure of the characteristics we can deduce a result on the structure of the solution u: it is continuous at every point except on countably many Lipschitz curves where it has jump type discontinuities. Everything holds up to linearly degenerate components of the flux. To be more precise consider γ ∈ Kγ , a differentiability point t¯ of γ and r, δ > 0 and let ¯ ¯ ¯ ¯ (r ) := (t, x) ∈ B (r ) : x > γ ( t ) + γ ˙ ( t )(t − t ) + δ|t − t | , Bt¯δ+ ¯ ¯ t ,γ ( t ) ,γ ¯ ¯ ¯ ¯ Bt¯δ− ,γ (r ) := (t, x) ∈ Bt¯,γ (t¯) (r ) : x < γ (t ) + γ˙ (t )(t − t ) − δ|t − t | . Accordingly we define Ut¯δ± (r ) := w ∈ R : ∃t ∈ R+ , (γ , w) ∈ K such that T (γ , w) > t, (t, γ (t)) ∈ Bt¯δ± (r ) . ,γ¯ ,γ¯
Proposition 3. There exist J ⊂ R+ × R contained in the graphs of countably many curves in Kγ and a representative of u such that t A
B
B
C
Bt,+ (r)
x Fig. 4 The partition of the half-plane
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1. For every (t¯, x) ¯ ∈ R+ × R \ J there exists I ∈ L f such that for every ε > 0 there exists r > 0 for which max
(t,x)∈Br (t¯,x) ¯
dist(u(t, x), I ) < ε.
2. For every γ ∈ Kγ , for L 1 -a.e. t > 0, there exist I + , I − ∈ L f such that δ± (r ) ⊂ I ± + (−ε, ε) . ∀δ > 0 ∀ε > 0 ∃r > 0 Ut,γ
3 Concentration of Entropy Dissipation Here we take advantage of the structure of the solution obtained in Proposition 3 to prove Theorem 1. We consider entropies η such that η(0) = 0 so that there exists a constant L > 0 for which |η(u)| ≤ L|u|. This is not a restrictive assumption since μη = μη−η(0) . As we already observed in the introduction, it is sufficient to consider μC. Fix a positive time T . In order to avoid some technicalities we present the proof of Theorem 1 assuming that for every x ∈ R the point (T, x) ∈ C. Even if it is not trivial, it is just a technical issue to implement the following argument in the real setting proving that μC T is concentrated on countably many characteristic segments, where (Fig. 5) C T := {(t, γ (t)) ∈ [0, T ] × R : γ ∈ Kγ , (T, γ (T )) ∈ C}. By Lemma 3 it follows that each γ ∈ Kγ restricted to [0, T ] is a segment. Let ε > 0 be such that 2ε < T . We parametrize the characteristic segments by their position y at time ε, i.e. γ y (ε) = y. By Lemma 3 it also follows that for every y ∈ R there exist I − (y), I + (y) ∈ L f such that the limits of admissible boundaries from the left and the right of γ y are contained in I − (y) and I + (y) respectively. Moreover it is not difficult to prove that I − (y) = I + (y) =: I (y) except at most countably many points. Finally for L 1 -a.e. y there exists U (y) ∈ I (y) such that u(t, γ y (t)) = U (y) for L 1 -a.e. t ∈ (0, T ). Let P : [0, T ] × R → R be such that P(t, x) = y where γ y (t) = x. The goal is to prove that m := P μ is atomic. This immediately implies that μ is concentrated on at most countably many segments and concludes the proof of Theorem 1. The idea is to compute the balances u t + f (u)x = 0,
η(u)t + q(u)x = μη
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t T C(y1 , y2 )
y1
y2
x Fig. 5 A model set of segments parametrized by their position at time ε and a cylinder
for the conserved quantity u and the entropy η(u) on cylindrical regions of the form C(y1 , y2 ) = {(t, x) ∈ (0, T ) × R : γ y1 (t) < x < γ y2 (t)}. The fluxes F and Q of u and η(u) respectively across γ y per unit time are given by F(y) = f (U (y)) − λ(y)U (y),
Q(y) = q(U (y)) − λ(y)η(U (y)),
where λ(y) = f (U (y)) is the slope of the segment γ y . The crucial point is that they do not depend on time. Observe that, since the segments do not intersect in (0, T ) thanks to monotonicity of the family of boundaries, the speed λ(y) is 1/ε-Lipschitz. Therefore the balance for η(u) in C(y1 , y2 ) gives
γ y2 (T )
γ y1 (T )
η(u(T, x))d x −
γ y2 (0)
γ y1 (0)
η(u 0 (x))d x + T (Q(y2 ) − Q(y1 )) = m((y1 , y2 )).
This implies that Q ∈ BV and D y Q = −λ (y)η(U (y))L 1 +
m . T
(5)
In particular F is Lipschitz and for L 1 -a.e. y ∈ R F (y) = −λ (y)U (y). Notice that this is the chain rule corresponding to ( f (u) − f (u)u) y = −( f (u)) y u. The following general lemma links the flux F to the flux Q. See Fig. 6.
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Fig. 6 The illustration of Lemma 4. Since the slope of α is bounded and the first projection of γ is Lipschitz, then γ 2 ∈ SBV and L 1 -a.e. the slope of γ coincides with the slope of α
Lemma 4. Let α = (α 1 , α 2 ) : [−M, M] → R2 be a smooth curve such that there exists a constant L > 0 for which for every w ∈ [−M, M] it holds |α˙ 2 (w)| ≤ L|α˙ 1 (w)|. Let γ = (γ 1 , γ 2 ) : R → R2 be such that γ 1 is Lipschitz, γ 2 has bounded variation and Im γ ⊂ Im α. Then Dγ 2 has no Cantor part and for L 1 -a.e. y ∈ R (γ 2 ) (y)(α 1 ) (w(y)) = (γ 1 ) (y)(α 2 ) (w(y)), for some w(y) such that γ (y) = α(w(y)). We can apply Lemma 4 to the curves
α(w) =
f (w) − f (w)w , q(w) − f (w)η(w)
in fact
α (w) =
γ (y) =
− f (w)w − f (w)η(w)
F(y) Q(y)
and by assumption |η(w)| ≤ L|w|. So we get that D y Q has no Cantor part and for L 1 -a.e. y ∈ R Q (y) = −λ (y)η(U (y)). Therefore, comparing with (5), we get that m is atomic.
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References 1. C. Bardos, A.Y. le Roux, J.-C. Nédélec, First order quasilinear equations with boundary conditions. Commun. Partial. Differ. Equ. 4, 1017–1034 (1979) 2. S. Bianchini, E. Marconi, On the concentration of entropy for scalar conservation laws. Discret. Contin. Dyn. Syst. Ser. S 9, 73–88 (2016) 3. S. Bianchini, E. Marconi, On the structure of L ∞ entropy solutions to scalar conservation laws in one-space dimension, Preprint SISSA 43/2016/MATE 4. S. Bianchini, S. Modena, Quadratic interaction functional for general systems of conservation laws. Commun. Math. Phys. 338(3), 1075–1152 (2015) 5. S. Bianchini, L. Yu, Structure of entropy solutions to general scalar conservation laws in one space dimension. J. Math. Anal. Appl. 428(1), 356–386 (2015) 6. K.S. Cheng, A regularity theorem for a nonconvex scalar conservation law. J. Differ. Equ. 61, 79–127 (1986) 7. C. De Lellis, T. Riviere, Concentration estimates for entropy measures. J. de Mathématiques Pures et Appliquées 82, 1343–1367 (2003) 8. O.A. Ole˘ınik, Discontinuous solutions of non-linear differential equations. Am. Math. Soc. Transl. (2) 26, 95–172 (1963) 9. F. Otto, Initial-boundary value problem for a scalar conservation law. C. R. Acad. Sci. Paris Sér. I Math. 322(8), 729–734 (1996)
On Uniqueness of Weak Solutions to Transport Equation with Non-smooth Velocity Field Paolo Bonicatto
Abstract Given a bounded, autonomous vector field b : Rd → Rd , we study the uniqueness of bounded solutions to the initial value problem for the associated transport equation (1) ∂t u + b · ∇u = 0. This problem is related to a conjecture made by A. Bressan, raised while studying the well-posedness of a class of hyperbolic conservation laws. Furthermore, from the Lagrangian point of view, this gives insights on the structure of the flow of non-smooth vector fields. In this work, we will discuss the two-dimensional case and we prove that, if d = 2, uniqueness of weak solutions for (1) holds under the assumptions that b is of class BV and it is nearly incompressible. Our proof is based on a splitting technique (introduced previously by Alberti, Bianchini and Crippa in J Eur Math Soc (JEMS) 16(2):201–234, 2014, [2]) that allows to reduce (1) to a family of 1-dimensional equations which can be solved explicitly, thus yielding uniqueness for the original problem. This is joint work with S. Bianchini and N.A. Gusev (SIAM J Math Anal 48(1):1–33, 2016), [6]. Keywords Transport equation · Continuity equation · Lipschitz functions Superposition Principle MSC (2010) 35F10 · 35L03 · 28A50 · 35D30
P. Bonicatto (B) SISSA - Scuola Internazionale Superiore di Studi Avanzati, via Bonomea 265, 34136 Trieste, Italy e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_15
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1 Introduction In this paper, we consider the continuity equation ∂t u + div(ub) = 0
(2)
∂t u + b · ∇u = 0,
(3)
and the transport equation where u : I × Ω → R (we write I = (0, T ), T > 0 and Ω ⊂ R2 for an open set) is an unknown scalar function and b : I × Ω → R2 is a given vector field. We study the initial value problems for these equations with the initial condition u(0, ·) = u 0 (·),
(4)
where u 0 : Ω → R is a given scalar field. Our aim is to investigate uniqueness of weak solutions to (2), (4) under weak regularity assumptions on the vector field b. When b ∈ L ∞ (I × Ω) then (2) is understood in the standard sense of distributions: u ∈ L ∞ (I × Ω) is called a weak solution of the continuity equation if (2) holds in D (I × Ω). Concerning the initial condition, one can prove (see e.g. [8]) that, if u is a weak solution of (2), then there exists a u (t, ·) for a.e. t ∈ I and t → u (t, ·) map u ∈ L ∞ ([0, T ] × Ω) such that u(t, ·) = is weakly continuous from [0, T ] into L ∞ (Ω). Thus, we can prescribe an initial condition (4) for a weak solution u of the continuity equation in the following sense: u (0, ·) = u 0 (·). we say that u(0, ·) = u 0 (·) holds if Definition of weak solutions of the transport equation (3) is slightly more delicate. If div b L 2 , being L 2 the Lebesgue measure in the plane, then (3) can be written as ∂t u + div(ub) − udivb = 0, and the latter equation can be understood in the sense of distributions (see e.g. [9] for the details). However, we are here interested in the case when div b is not absolutely continuous. In this case, the notion of weak solution of (3) can be defined for the class of nearly incompressible vector fields. Definition 1. A bounded, locally integrable vector field b : I × Ω → R2 is called nearly incompressible if there exists a function ρ : I × Ω → R (called density of b) and a constant C > 0 such that C −1 ≤ ρ(t, x) ≤ C for a.e. (t, x) ∈ I × Ω and ∂t ρ + div(ρb) = 0
in D (I × Ω).
(5)
It is easy to see that if div b ∈ L ∞ (I × Ω) then b is nearly incompressible. The converse implication does not hold, so near incompressibility can be considered as a weaker version of the assumption div b ∈ L ∞ (I × Ω).
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Definition 2. Let b be a nearly incompressible vector field with density ρ. We say that a function u ∈ L ∞ (I × Ω) is a (ρ–)weak solution of (3) if ∂t (ρu) + div(ρub) = 0
in D (I × Ω).
Thanks to Definition 2, one can prescribe the initial condition for a ρ–weak solution of the transport equation similarly to the case of the continuity equation, which we mentioned above (see again [8] for further details). Existence of weak solutions to the initial value problem for transport equation with a nearly incompressible vector field can be proved by a standard regularization argument [8]. The problem of uniqueness of weak solutions is much more delicate. The theory of uniqueness in the non-smooth framework started with the seminal paper of R.J. DiPerna and P.-L. Lions [9] where uniqueness was obtained as a corollary of the so-called renormalization property for vector fields with Sobolev regularity. This result was improved by Ambrosio in [3], where uniqueness is shown for vector fields of (locally) bounded variation in the space with absolutely continuous, bounded divergence. As we have pointed out, vector fields considered in [3] make up a proper subset of nearly incompressible ones, so it makes sense to wonder whether uniqueness holds in the latter, larger class. This is related to a conjecture, made by A. Bressan in [7], whose 2D statement is the following: Conjecture 1 (Bressan). Uniqueness of weak solutions to (3), (4) holds for any nearly incompressible vector field b ∈ L 1 (I ; BVloc (R2 )). In the two-dimensional, autonomous case, the problem of uniqueness is addressed in the papers [1, 2, 5]. Indeed, in two dimensions and for divergence-free autonomous vector fields, uniqueness results are available under milder assumptions on the regularity of b, because of the underlying Hamiltonian structure. In [2], the authors characterize the autonomous, divergence-free vector fields b on the plane such that the Cauchy problem for the transport equation (3) admits a unique bounded weak solution for every bounded initial datum (4). The characterization they present relies on the so-called Weak Sard Property, which is a (weaker) measure theoretic version of Sard’s Lemma. Since the problem admits a Hamiltonian potential, uniqueness is proved following a strategy based on splitting the equation on the level sets of this function, reducing thus to a one-dimensional problem. This approach requires a preliminary study on the structure of level sets of Lipschitz maps defined on R2 , which is carried out in the paper [1]. The main result of this paper is a partial answer to the Conjecture 1: Theorem 1. Suppose that b : R2 → R2 is a compactly supported, nearly incompressible BV vector field. Then uniqueness of weak solutions to (1), (4) holds. More precisely, we will show that for every bounded initial datum u 0 ∈ L ∞ (R2 ) there exists a unique (ρ-)weak solution u ∈ L ∞ (I × R2 ) to the transport equation
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(3) with the initial condition u(0, ·) = u 0 (·). The main idea behind the proof is to implement a sort of method of characteristics in a non-smooth setting: indeed, we will find a “large” (in a proper sense) family of integral curves γ of the vector field b such that, for any solution u, the function t → u(t, γ (t)) has to be constant: from this the uniqueness easily follows.
2 Preliminaries and Useful Results We collect here some previously known results that will be used in the proof of Theorem 1.
2.1 Ambrosio’s Superposition Principle In [3], L. Ambrosio proved the Superposition Principle. Since we will use it later on, we present here the statement. Theorem 2 (Superposition Principle). Let b : [0, T ] × Rd → Rd be a bounded, Borel vector field and let [0, T ] t → μt be a positive, locally finite, measure-valued solution of the continuity equation
∂t μt + div(bμt ) = 0 μ0 = μ
in D ((0, T ) × Rd ).
(6)
Then there exists a non-negative measure η on the space of continuous curves Γ := C([0, T ]; Rd ) such that μt = et # η
for every t ∈ [0, T ],
(7)
being et : Γ → Rd the evaluation map γ → γ (t). Furthermore, the measure η is concentrated on absolutely continuous solutions of the ordinary differential equation driven by b.
2.2 Level Sets of Lipschitz Functions and Disintegration of Lebesgue Measure Suppose that Ω ⊂ R2 is an open set and H : Ω → R is a compactly supported Lipschitz function. For any h ∈ R, let E h := H −1 (h). We recall the following deep
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Theorem 3 ([1, Theorem 2.5]). For L 1 -a.e. h ∈ H (Ω) the following holds: (1) H 1 (E h ) < +∞ and E h is countably H 1 -rectifiable; (2) for H 1 -a.e. x ∈ E h the function H is differentiable at x with ∇ H (x) = 0; (3) if Conn (E h ) denotes the set of connected components c of E h with H 1 (c) > 0, then Conn (E h ) is countable and every c ∈ Conn (E h ) is a closed simple curve, which admits a parametrization γc with the following properties: • γc : Ic → R2 is Lipschitz and injective (up to the end points), where Ic = R/ Z or Ic = [0, ] for some > 0; • γc (s) is orthogonal to ∇ H (γc (s)) for L 1 -a.e. s ∈ Ic . (4) If E h denotes the union of c ∈ Conn (E h ), then H 1 (E h \ E h ) = 0. In the following, we will call regular the level sets corresponding to values of h satisfying Point (1) of Theorem 3. Using Theorem 3 together with Disintegration Theorem [4, Theorem 2.28], we can characterize the disintegration of the Lebesgue measure restricted to an open set Ω. We have the following: Lemma 1 ([2, Lemma 2.8]). There exist Borel families of measures {σh }h∈R and {κh }h∈R , such that ch H 1 E h + σh dh + κh dζ (h), (8) L 2 Ω = where 1. ch ∈ L 1 (H 1 E h ) and, more precisely, by Coarea formula, we have ch = 1/|∇ H | a.e. (w.r.t. H 1 E h ); E h ∩ {∇ H = 0}; 2. σh , κh are concentrated on 2 3. ζ := H# L (Ω \ h E h ) is singular w.r.t. L 1 .
2.3 The Weak Sard Property Let H : Ω → R be a Lipschitz function as in Sect. 2.2 and let S be the critical set of H , defined as the set of all x ∈ Ω where H is not differentiable or ∇ H (x) = 0. We will be interested in the following property: the push-forward according to H of the restriction of L 2 to S is singular with respect to L 1 , that is H# L 2 S ⊥ L 1 . This property clearly implies the following Weak Sard Property, which is used in [2, Sect. 2.13]: H# L 2 (S ∩ E ) ⊥ L 1 ,
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where the set E is the union over h ∈ H (Ω) of all connected components with positive length of E h . The relevance of the Weak Sard Property in the framework of transport and continuity equation has been completely understood in [2, Theorem 4.7], to whom we refer the reader for further details. Remark 1. Informally, the Weak Sard Property means that the “good” level sets of H do not intersect the critical set S, apart from a negligible set. In terms of the disintegration of the Lebesgue measure (8), it can be proved that H has the weak Sard property if and only if σh = 0 for a.e. h. This is actually the case when, for instance, ∇ H = 0 a.e. (see Point 2 of Lemma 1) or when ∇ H ∈ BV(Ω; R2 ) (see [6]).
3 Proof I: The Local Argument We can now start the proof of Main Theorem. We will split the proof in two steps: first we perform a local argument, finding a suitable covering of the plane in balls and disintegrating the equation inside each ball. Afterwards, we will “glue” together the local results in order to implement the weak method of characteristic. We start here presenting the local argument. Let b : R2 → R2 be an autonomous, nearly incompressible vector field, with b ∈ BV(R2 ) ∩ L ∞ (R2 ); we assume b is compactly supported, defined everywhere and Borel.
3.1 Partition and Curves Let us consider the countable covering B of R2 given by B := B(x, r ) : x ∈ Q2 , r ∈ Q+ . where B(x, r ) is the euclidean ball of center x and radius r . For each ball B ∈ B, we are interested to the trajectories of b which cross B and stay inside B for a positive amount of time. We therefore define, for every ball B ∈ B and for any rational numbers s, t ∈ Q ∩ (0, T ) such that s < t, the sets / B, γ (t) ∈ / B . T B,s,t := γ ∈ Γ : L 1 ({τ ∈ [0, T ] : γ (τ ) ∈ B}) > 0, γ (s) ∈ In this first section, we will work for simplicity with the sets T B := T B,0,T , where B ∈ B (and without any loss of generality we assume T ∈ Q). Remark 2. It is immediate to see that the union over B ∈ B of the sets T B is equal to the set of all non-constant trajectories. The same is true also for the union over B ∈ B and s, t ∈ Q ∩ [0, T ] of the sets T B,s,t .
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By Definition 1, we have that there exists a function ρ : [0, T ] × R2 → R which satisfies continuity Eq. (5) in D ((0, T ) × R2 ). Therefore, by Ambrosio’s Superposition Principle (Theorem 2), there exists a measure η on Γ , concentrated on the set of integral curves of b, such that ρ(t, ·)L 2 = (et )# η,
(9)
where we recall that et : Γ → R2 is the evaluation map γ → γ (t). For a fixed ball B ∈ B, we consider the measure η B := ηT B and we define the density ρ B by ρ B (t, ·)L 2 = (et )# η B . We finally set r B (x) :=
T
ρ B (t, x)dt,
x ∈ B.
(10)
0
Now the following lemma is almost immediate. Lemma 2. It holds div(r B b) = 0 in D (B).
3.2 Local Disintegration of the Equation div(ub) = μ From Lemma 2, we deduce that (being every B ∈ B simply connected) there exists a Lipschitz map H B : B → R such that r B b = ∇ ⊥ H B , where ∇ ⊥ = (−∂2 , ∂1 ). The function H B is well defined (being b compactly supported) and we will often refer to it as the Hamiltonian (relative to the ball B).
3.2.1
Reduction of the Equation on the Level Sets
Fix now a ball B ∈ B and consider the equation div(ub) = μ,
in D (B)
(11)
where u ∈ L ∞ (R2 ) and μ is a Radon measure on R2 . The first step is the disintegration of the equation on the level sets of H : Lemma 3. The Eq. (11) is equivalent to: 1. the disintegration of μ with respect to H has the form μ=
μh dh +
where ζ is defined in Point (3) of Lemma 1; 2. for L 1 -a.e. h
νh dζ (h),
(12)
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P. Bonicatto
3. for ζ -a.e. h
3.2.2
div uch bH 1 E h + div(ubσh ) = μh ;
(13)
div(ubκh ) = νh .
(14)
Reduction on the Connected Components
The next step is to reduce further the analysis of the Eq. (13) on the nontrivial (i.e. with positive length) connected components of the level sets. Lemma 4. The Eq. (13) holds if and only if • for any non-trivial connected component c of E h it holds
• it holds
3.2.3
div uch bH 1 c = μh c,
(15a)
div(ubσh c) = 0;
(15b)
div(ubσh (E h \ E h )) = μh (E h \ E h ).
(16)
Reduction of the Equation on Connected Components in Parametric Form
Finally, we would like to discuss the parametric version of the Eq. (15a). Let γc : Ic → R2 be an injective Lipschitz parametrization of c, where Ic = R/ Z or Ic = (0, ) for some > 0 is the domain of γ . Lemma 5. Equation (15a) holds iff ∂s (
u
ch |
b|) =
μh
(17)
μh = μh c,
u = u ◦ γc ,
ch = ch ◦ γc and
b = b ◦ γc . where (γc )#
3.3 Local Disintegration of a Balance Law We now pass to consider a general balance law associated to the Hamiltonian vector field b, i.e., in D ((0, T ) × B) (18) ∂t u + div(ub) = ν, being u ∈ L ∞ ((0, T ) × R2 ) and ν a Radon measure on (0, T ) × R2 . A reduction on the nontrivial connected components of the level sets of the Hamiltonian H B can be
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still performed, similarly to what we have done for equation div(ub) = μ in Sect. 3.2. Roughly speaking, we present now the time-dependent version of Lemmas 3–5. Lemma 6. The Eq. (18) is equivalent to • for L 1 -a.e. h, the function
u h (t, s) := u(t, γc (s)) solves u h
u h
ch |
b| + ∂s
ch |
b| =
νh , ∂t
in D ((0, T ) × Ic )
(19)
where γh : Ic → R2 is an injective, Lipschitz parametrization of a connected comνh is a measure such that ponent c of the level set E h of the Hamiltonian H and
νh = (γc−1 )# ν; • it holds (20) div(ubσh ) = 0.
3.4 Matching Lemma and Weak Sard Property for HB Before passing to the global argument, we make sure that the Hamiltonians constructed in paragraph before are matching, in the sense given by the following definition: Definition 3. Assume that B1 , B2 ∈ B are such that B1 ∩ B2 = ∅. The Hamiltonians H1 := H B1 and H2 := H B2 match in B1 ∩ B2 if, denoting by C xi denotes the connected component of the level sets Hi−1 (Hi (x)) which contains x (i = 1, 2), it holds C x1 = C x2 for L 2 -a.e. x ∈ B1 ∩ B2 such that the level sets Hi−1 (Hi (x)) are regular. It can be proved that in our setting matching property holds, as a consequence of the fact that ∇ H1 ∇ H2 (being both parallel to b): Lemma 7. (Matching lemma). Let H1 , H2 be defined as above. Then the Hamiltonians H1 and H2 match in B1 ∩ B2 . It is interesting to notice that an application of Lemma 7 yields also the following remarkable piece of information: Lemma 8. For every B ∈ B, the Hamiltonian H B has the Weak Sard Property. Remark 3. If we do not assume BV regularity of b, but b = 0 a.e., then the conclusion of Lemma 8 still holds. After having established the validity of the Weak Sard Property for the Hamiltonians H B , we can finally implement our method of characteristics locally.
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Lemma 9. Let B ∈ B be fixed and consider u ∈ L ∞ ([0, T ] × R2 ) a ρ-weak solution of the problem
∂t u + b · ∇u = 0, u(0, ·) = u 0 (·),
in D ((0, T ) × R2 ).
(21)
Then there exists a negligible set Z = Z B ⊂ R such that for any h ∈ / Z the level set E h = H B−1 (h) is regular and for any nontrivial connected component c of E h with parametrization γc : Ic → R2 solving γc (s) = ∇ ⊥ H (γc (s)), it holds u(t, γc (s + t)) = u 0 (γc (s))
(22)
for any s ∈ Ic and for a.e. t ∈ (0, T ) such that s + t ∈ Ic .
3.5 Level Sets and Trajectories We now turn our attention to the link between the trajectories γ ∈ T B and the level sets of the Hamiltonian H B . As one may expect, the first result one can prove is that η-a.e. γ is contained in a level set. More precisely, we have the following: Lemma 10. Up to a η B negligible set N ⊂ Γ , the image of every γ ∈ T B is contained in a connected component of a regular level set of H B . Interlude: locality of the divergence on a class of L ∞ measure-divergence vector fields Let U ⊂ Rd (for an integer d ≥ 1) be an open set and let E : U → Rd be a vector field. Consider the following set M := x ∈ Rd : E(x) = 0, x ∈ DE and ∇ appr E(x) = 0 , (23) where DE is the set of approximate differentiability points and ∇ appr E is the approximate differential, according to Definition [4, Definition 3.70]. In [6] we have proved the following Proposition 1. Assume that E is approximately differentiable a.e. and let u ∈ L ∞ (U ) be such that div (uE) = λ in the sense of distributions, being λ a Radon measure on U . Then |λ|M = 0. For shortness, we will refer to the property expressed in Proposition 1 as locality of the divergence. Notice that we do not assume any weak differentiability of u or uE, so the conclusion of Proposition 1 does not follow immediately from the standard locality properties of the approximate derivative (see e.g. [4, Proposition 3.73]). Moreover, we point out that in [1] it can be found an example of a bounded vector field whose (distributional) divergence is in L ∞ , is nontrivial but it is supported in the set where the vector field vanishes.
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The property of locality of the divergence expressed in Proposition 1 holds true in our setting, being b ∈ BV, and this provides us with a better description of the relationship between the trajectories γ ∈ Γ B and the level sets of H B , improving Lemma 10. Indeed, let B ∈ B be a fixed ball of the collection and, as usual, let H B denote its Hamiltonian. Thanks to Lemma 10, there exists a η-negligible set N such that for every γ ∈ Γ B \ N the image γ (0, T ) is contained in a connected component c of a regular level set of H B . We now ask the following question: which is the relationship between the trajectory γ ∈ Γ B \ N and the parametrization γc of the corresponding connected component, given by Point 3 of Theorem 3? The answer is in the following Proposition 2. Let N be the set given by Lemma 10 and let γ ∈ T B \ N . Then (a suitable restriction of) γ coincides with γc up to a translation in time.
4 Proof II: The Global Argument We now pass to analyze the second part of the proof of Theorem 1.
4.1 Covering Property of the Regular Level Sets Let us recall that for each ball B ∈ B and for any rational numbers s, t ∈ Q ∩ (0, T ) with s < t we have set / B, γ (t) ∈ / B T B,s,t := γ ∈ Γ : L 1 ({τ ∈ [0, T ] : γ (τ ) ∈ B}) > 0, γ (s) ∈ and we have seen in Remark 2 that the union over B ∈ B and s, t ∈ Q ∩ [0, T ] of the sets T B,s,t coincides with the set of non-constant trajectories. For each B ∈ B, s ∈ Q ∩ (0, T ), t ∈ Q ∩ (s, T ) restricting η to T B,s,t , we can construct the local Hamiltonian H B,s,t as in Sect. 3.1. Setting now E :=
B∈B h∈H B,s,t s,t∈Q∩[0,T ]
E h
(24)
(R2 )
we can prove the following covering property, which shows that a.e. point where b is non zero is contained in a regular level set of one Hamiltonian. Lemma 11. It holds E = {b = 0} mod L 2 .
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4.2 Selection of Appropriate Trajectories We now present the following Lemma 12. Up to an η-negligible set, any non-constant, integral curve γ of the vector field b has the following properties: 1. for any B ∈ B, if γ ∈ T B,s,t then each connected component of γ ([s, t]) ∩ B is contained in a regular level set of H B ; 2. for any τ ∈ (0, T ) there exist a ball B ∈ B, s ∈ Q ∩ (0, τ ) and t ∈ Q ∩ (τ, T ) such that γ ∈ T B,s,t . In view of Lemma 12, for η-a.e. non constant γ we can find a ball of the collection B such that a piece of γ is covered by a regular level set of H B . Now we would apply Lemma 9 to solve equation locally in this ball. Before doing that, we need the following technical Lemma 13. Let Z B,s,t denote negligible set given by Lemma 9. Then η-a.e. γ ∈ Γ which is non-constant satisfies H B,s,t (γ ([0, T ])) ∩ Z B,s,t = ∅.
(25)
Furthermore, the end points γ (0) and γ (T ) are contained in regular level sets of some Hamiltonians γ (T ) ∈ E . (26) γ (0) ∈ E and Using Lemmas 12 and 13, we have been removing η-negligible sets of trajectories of b. Let us summarize in the following proposition some properties of the remaining ones: Proposition 3. For a.e. non-constant γ ∈ Γ and any τ ∈ [0, T ] there exists δ > 0 and a constant w such that the function ξ → u(ξ, γ (ξ )) is equal to w for a.e. ξ ∈ (τ − δ, τ + δ) ∩ [0, T ]. Moreover, if τ = 0 then the constant w is equal to u 0 (γ (0)).
4.3 Conclusion: Solutions Are Constant Along η-a.e. Trajectory Now we are in a position to recover the method of characteristics in our weak setting. The following is the global analog of Lemma 9 and is an immediate consequence of Proposition 3 and the compactness of [0, T ]: Lemma 14. Let u ∈ L ∞ ([0, T ] × R2 ) be a ρ-weak solution of the Problem 21. Then for η-a.e. γ ∈ Γ for a.e. t ∈ [0, T ] it holds that u(t, γ (t)) = u 0 (γ (0)).
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Proof. Call N the set given by Proposition 3. Let γ ∈ Γ \ N be a non-constant trajectory. For any τ ∈ [0, T ] there exists δ > 0 such that the function t → u(t, γ (t)) is equal to some constant wτ for a.e. t ∈ (τ − δ, τ + δ) ∩ [0, T ]. Moreover, if τ = 0 then wτ = u 0 (γ (0)). It remains to extract a finite covering of [0, T ].
References 1. G. Alberti, S. Bianchini, G. Crippa, Structure of level sets and Sard-type properties of Lipschitz maps. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12(4), 863–902 (2013) 2. G. Alberti, S. Bianchini, G. Crippa, A uniqueness result for the continuity equation in two dimensions. J. Eur. Math. Soc. (JEMS) 16(2), 201–234 (2014) 3. L. Ambrosio, Transport equation and Cauchy problem for BV vector fields. Invent. Math. 158(2), 227–260 (2004) 4. L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems (Oxford Science Publications, Clarendon Press, 2000) 5. S. Bianchini, N.A. Gusev, Steady nearly incompressible vector fields in two-dimension: chain rule and renormalization. Arch. Ration. Mech. Anal. 222(2), 451–505 (2016) 6. S. Bianchini, P. Bonicatto, N.A. Gusev, Renormalization for autonomous nearly incompressible BV vector fields in two dimensions. SIAM J. Math. Anal. 48(1), 1–33 (2016) 7. A. Bressan, An ill posed Cauchy problem for a hyperbolic system in two space dimensions. Rend. Semin. Mat. Univ. Padova 110, 103–117 (2003) 8. C. De Lellis, Notes on hyperbolic systems of conservation laws and transport equations, Handbook of Differential Equations: Evolutionary Equations, vol. III (Elsevier/North-Holland, Amsterdam, 2007), pp. 277–382 9. R.J. DiPerna, P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547 (1989)
Johnson–Segalman–Saint-Venant Equations for a 1D Viscoelastic Shallow Flow in Pure Elastic Limit Sébastien Boyaval
Abstract The shallow-water equations of Saint-Venant, often used to model the long-wave dynamics of free-surface gravity flows governed by inertia and hydrostatic pressure, can be generalized to account for the elongational rheology of nonNewtonian fluids too. We consider here 1D shallow-water equations generalized to viscoelastic fluids using the Johnson–Segalman model in pure elastic limit (i.e., at infinitely-large Deborah number, when source terms vanish). The quasilinear system of first-order equations is hyperbolic when the slip parameter is small: ζ ≤ 21 (ζ = 1 is the corotational case and ζ = 0 the upper-convected Maxwell case). It is naturally endowed with a mathematical entropy (a physical free-energy), and it is strictly hyperbolic when vacuum is excluded. Then, for any initial data, we construct the unique solution to the Riemann problem under Lax admissibility conditions. The standard Saint-Venant case is recovered for small data in the non-elastic limit G → 0. Keywords Riemann problem · Hyperbolic system · Viscoelastic shallow flow Generalized Saint-Venant equations
1 Setting of the Problem The well-known one-dimensional shallow-water equations of Saint-Venant
∂t (hu) + ∂x
∂t h + ∂x (hu) = 0 hu 2 + gh 2 /2 = 0
(1) (2)
model the dynamics of the mean depth h(t, x) > 0 of a perfect fluid flowing with mean velocity u(t, x) on a flat open channel with uniform cross section along a straight axis e x , under gravity (perpendicular to e x , with constant magnitude g). S. Boyaval (B) Laboratoire d’hydraulique Saint-Venant (Ecole des Ponts ParisTech – EDF R& D – CEREMA), Université Paris-Est, INRIA Paris MATHERIALS, EDF’lab 6 quai Watier, 78401 Chatou Cedex, France e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_16
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Now, following the interpretation of (1)–(2) as an approximation of the depthaveraged Free-Surface Navier–Stokes (FSNS) equations governing Newtonian fluids, and starting depth-averaging from the FSNS/Upper-Convected-Maxwell(UCM) system of equations for (linear) viscoelastic fluids [3, 4], one can in fact derive a generalized Saint-Venant (gSV) system of shallow-water-type equations
∂t (hu) + ∂x
∂t h + ∂x (hu) = 0 hu + gh 2 /2 + h N = 0 2
(3) (4)
where the normal stress difference term in the momentum balance N = τzz − τx x is function of additional extra-stress variables τzz (t, x), τx x (t, x) governed by, e.g., τx x + λ(∂t τx x + u∂x τx x − 2τx x ∂x u) = ν∂x u τzz + λ(∂t τzz + u∂x τzz + 2τzz ∂x u) = −ν∂x u
(5) (6)
i.e., depth-averaged UCM equations modeling elongational viscoelastic effects. When the relaxation time is small λ → 0 (i.e., the Deborah number, when λ > 0 is non-dimensionalized with respect to a time scale characteristic of the flow) the system (3)–(6) converges (formally) to standard viscous Saint-Venant equations with viscosity ν ≥ 0. When the relaxation time and the viscosity are equivalently large λ ∼ ν → +∞, the system (3)–(6) converges to elastic Saint-Venant equations (in Eulerian formulation, see, e.g., [8]) with elasticity G = ν/(2λ) ≥ 0, which coincides with the homogeneous version of the system (7)–(8) (i.e., when the source term vanish) ∂t σx x + u∂x σx x − 2σx x ∂x u = (1 − σx x )/λ
(7)
∂t σzz + u∂x σzz + 2σzz ∂x u = (1 − σzz )/λ
(8)
obtained after rewriting (5)–(6) using τx x,zz = G(σx x,zz − 1), N = G(σzz − σx x ). More general evolution equations of differential rate-type for the extra-stress, the Johnson–Segalman (JS) equations with slip parameter ζ ∈ [0, 2], can also be coupled to FSNS before depth-averaging. In fact, (7)–(8) arise in the specific case ζ = 0 (upper-convected Gordon–Schowalter derivative) for gSV system (3)–(10) ∂t σx x + u∂x σx x + 2(ζ − 1)σx x ∂x u = (1 − σx x )/λ
(9)
∂t σzz + u∂x σzz + 2(1 − ζ )σzz ∂x u = (1 − σzz )/λ
(10)
that accounts for linear viscoelastic elongational effects standardly established for, e.g., polymeric liquids [1]. The gSV system with JS is already an interesting starting point for mathematical studies, although it could still be further complicated to account for more established physics; we refer to [1] for details. In the following, we consider the Cauchy problem on t ≥ 0 for the quasilinear gSV system (3)–(10) when it is supplied by an initial condition with small total variation. Weak solutions with bounded variations (BV) and small data can be constructed for
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the strictly hyperbolic quasilinear systems, in particular when characteristic fields are genuinely nonlinear or linearly degenerate [5, 7]. First, we show that gSV is strictly hyperbolic for all h ≥ 0, σx x > 0, σzz > 0 when ζ ≤ 21 and provided h > 0, or G > 0 and, ζ > 0 or σx x = σzz . Next, assuming ζ ≤ 21 and h > 0, we construct univoque gSV solutions guided by the dissipation rule −1 ∂t F + ∂x (u(F + P)) ≤ Gh 2 − σx x − σx−1 x + 2 − σzz − σzz /(2λ)
(11)
as admissibility criterion with F the mathematical entropy of [3] when ζ = 0 2F/ h = u 2 + gh + G (σx x + σzz − ln σx x − ln σzz − 2)/(1 − ζ ) ,
(12)
denoting P = gh 2 /2 + h N . Smooth gSV solutions satisfy the equality (11). When h, σx x , σzz > 0, gSV reads as a system of conservation laws rewriting (9)–(10) ∂t h log(h 2(1−ξ ) σx x ) + ∂x hu log(h 2(1−ξ ) σx x ) = h(σx−1 (13) x − 1)/λ, 2(ξ −1) 2(ξ −1) −1 ∂t h log(h σzz ) + ∂x hu log(h σzz ) = h(σzz − 1)/λ. (14) But whereas F is convex in, e.g., (h, hu, hσx x , hσzz ), see [3] when ζ = 0, it cannot be convex with respect to any variable V = (h, hu, hX (σx x h 2(1−ζ ) ), hZ (σzz h 2(ζ −1) )) + using smooth X , Z ∈ C 1 (R+ , R ) such that the system rewrites ∂t V + ∂x F(V ) = 0 ,
(15)
F(V )=(hu, hu 2 + gh 2 /2+Ghσzz − Ghσx x , huX (σx x h 2(1−ζ ) ), huZ (σzz h 2(ζ −1) )). Now, whereas univoque solutions to quasilinear (possibly non-conservative) systems can be constructed using (convex) entropies [7], conservative formulations alone (without admissibility criterion) are not enough. This is why we carefully investigate the building-block of unique BV solutions: univoque solutions to Riemann initial-value problems for a quasilinear system (16) in well-chosen variable U ∂t U + A(U )∂x U = S(U )
(16)
in the homogeneous case S ≡ 0 (obtained in the limit λ → ∞). Precisely, when ζ ≤ 21 and G > 0 we build the unique weak solutions U (t, x) admissible under Lax condition to Riemann problems for (16) with piecewise-constant initial conditions Ul x < 0 + U (t → 0 , x) = (17) Ur x > 0 given any states Ul , Ur ∈ U in the strict hyperbolicity region U = {h > 0, σx x > 0, σzz > 0} ⊂ Rd . Our Riemann solutions satisfy the conservative system (15) in the distributional sense on (t, x) ∈ R+ × R and are consistent with the standard SaintVenant case when G → 0. These Riemann solutions are a key tool to define weak BV solutions to the Cauchy problem for gSV which are unique within the admissible BV solutions’class modulo some restriction on oscillations, see, e.g., [7, Chap. 10].
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However, note that the vacuum state h = 0 shall never be reached as a limit state by any sequence of admissible Riemann solutions when G > 0, as opposed to the standard Saint-Venant case G = 0 (like in the famous Ritter problem for instance). This is in fact related to well-posedness in the large (i.e., for any initial condition Ul , Ur ∈ U ) when G > 0, as opposed to the standard Saint-Venant case G = 0 (so the latter case is some kind of singular limit): when G > 0, gSV impulse blows up as h → 0, so vacuum cannot be reached, while Hugoniot curves in turn span the whole range U . Also, consistently with the occurrence of vacuum when G = 0, the standard Saint-Venant case can be recovered when G → 0 for small initial data only (otherwise, the intermediate state in Riemann solution may blow up).
2 Hyperbolic Structure of the System of Equations Given g > 0, G ≥ 0, consider first the gSV system (3)–(10) written in the nonconservative quasilinear form (16) using the variable U = (h, u, σx x , σzz ) ∈ U . One easily sees that λ0 := u is an eigenvalue with multiplicity two for the matrix A, associated with the linearly degenerate 0-characteristic field (i.e. r 0 · ∇U λ0 = 0) ⎛
⎛ ⎞ ⎞ Gh Gh ⎜ ⎜ ⎟ 0 ⎟ 0 0 ⎜ ⎟ ⎟ r 0 ∈ Span{r10 , r20 } r10 := ⎜ ⎝(gh + N )⎠ r2 := ⎝ ⎠ 0 0 −(gh + N )
(18)
with Riemann invariants u, P (i.e. r 0 · ∇U P = 0). Moreover, as long as ζ ≤ 21 , holds ∂h P|σx x h 2(1−ζ ) ,σzz h 2(ζ −1) = gh + G(σzz − σx x ) + 2G(1 − ζ )(σzz + σx x ) > 0
(19)
for h, σx x , σzz > 0 so, after computations, the two other eigenvalues of A are real λ± := u ±
∂h P|σx x h 2(1−ζ ) ,σzz h 2(ζ −1)
(20)
and define two genuinely nonlinear fields (denoted by + and −) spanned by ⎞ h ⎜± ∂h P|σ h 2(1−ζ ) ,σ h 2(ζ −1) ⎟ xx zz ⎟ r ± := ⎜ ⎠ ⎝ 2(ζ − 1)σx x 2(1 − ζ )σzz ⎛
(21)
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with σx x h 2(1−ζ ) , σzz h 2(ζ −1) as Riemann invariants; note in particular for ζ ∈ [0, 21 ] r ± · ∇ U λ± = ±
3gh + 2G(3 − 2ζ )(2 − ζ )σzz + 2Gζ (1 − 2ζ )σx x ≷ 0. 2 ∂h P|σx x h 2(1−ζ ) ,σzz h 2(ζ −1)
(22)
Univoque piecewise-smooth solutions to Cauchy–Riemann problems for (3)–(10) U (t, x) = U˜ (x/t) with U˜ (ξ ) piecewise differentiable solution on R ξ to ξ U˜ = A(U˜ )U˜
U˜ −→ Ul , U˜ −→ Ur ξ →−∞
(23)
ξ →+∞
having finitely-many discontinuities ξm (m = 0 . . . M) shall next be constructed for any initial condition Ul , Ur ∈ U using elementary waves satisfying U˜ ∈ Span r ± , ξ = λ± therefore U˜ = r ± /(r ± · ∇U λ± ), or U˜ = 0, and an admissibility criterion.
3 Elementary Waves Solutions For all Ul , Ur ∈ U , unique solutions to (16)–(17) shall be constructed in the form
U˜ (ξ ) =
⎧ ⎪ Ul ≡ U˜ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨U˜ 1 (ξ )
ξ < ξ0 ξ0 < ξ < ξ1
··· ⎪ ⎪ ⎪ ξ M−1 < ξ < ξ M U˜ M (ξ ) ⎪ ⎪ ⎪ ⎩U ≡ U˜ r M+1 ξ M < ξ
(24)
using M differentiable states U˜ m to connect Ul , Ur ∈ U through elementary waves.
3.1 Contact Discontinuities and Shocks Elementary waves solutions (24) with a single discontinuity (M = 1) shall be 0contact discontinuities when, denoting Υl (resp. Υr ) the left (resp. right) value of Υ , ξ0 = u l = u r Pl = Pr
(25) 1+2(1−ζ )
or ±-shocks when, denoting Pk = gh 2k /2 + G Z −1 h k
1+2(ζ −1)
− G X hk
ξ0 (h r − h l ) = (h r u r − h l u l ) , ξ0 (h r u r − h l u l ) = (h r u r2 + Pr − h l u l2 − Pl ) ,
, hold (26) (27)
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S. Boyaval 2(ζ −1)
with 2 constants Z −1 = σzz,k h k
ur = ul ±
2(1−ζ )
> 0, X = σx x,k h k
> 0 (k ∈ {l, r }), thus
(h l−1 − h r−1 )(Pr − Pl )
(28)
on combining (26), (27). Both waves satisfy Rankine–Hugoniot (RH) relationships ξ0 (Vr − Vl ) = Fr − Fl
(29)
and thus define standard weak solutions to (16) in the conservative variable V (t, x) = V˜ (x/t). Moreover, the entropy dissipation (11) in the elastic limit λ → ∞ E := −ξ0 (Fr − Fl ) + u(F + gh 2 /2 + h N )|r − u(F + gh 2 /2 + h N )|l ≤ 0 (30) can be checked for contact discontinuities (as an equality), and for the weak shocks in the genuinely nonlinear fields λ± (i.e. shocks with small enough amplitude) which satisfy Lax admissibility condition, see [5, 6] and [7, (1.24) Chap. 6]: Lemma 1. Right and left states Vr , Vl can be connected through an admissible • −-shock if u r = u l − (h l−1 − h r−1 )(Pr − Pl ), h r ≥ h l • +-shock if u r = u l − (h l−1 − h r−1 )(Pr − Pl ), h r ≤ h l Indeed, it is enough that F| X,Z is convex in (h, hu) to discriminate against nonphysical (weak) shocks, or equivalently, that F| X,Z / h is convex in (h −1 , u) [2]. Proof. 2 Fh = u 2 + gh + ∂h2−2 | X,Z
G 1−ζ
(σx x + σzz − ln σx x σzz − 2) is convex in (h −1 , u) if
2g F = 3 + 2Gh 2 ((2(1 − ζ ) − 1)σx x + (2(1 − ζ ) + 1)σzz ) h h
is positive, which holds when ζ ∈ [0, 1/2].
3.2 Rarefaction Waves Elementary waves with two discontinuities (M = 2) which are not a combination of two elementary waves with one discontinuity each shall be, on noting k ∈ {l, r }, • a +-rarefaction wave if h l = h 0 < h r = h 2 such that for all ξ ∈ (ξ0 ≡ λl+ , ξ2 ≡ λr+ ) h (ξ ) 1
3gh + (4ζ 2 − 14ζ + 12)G Z −1 h 2(1−ζ ) + 2ζ (1 − 2ζ )G X h 2(ζ −1) hk 2h gh + (1 + 2(1 − ζ ))G Z −1 h 2(1−ζ ) − (1 + 2(ζ − 1))G X h 2(ζ −1) h (ξ ) 1 u 1 (ξ ) = u k + dh gh −1 + (1 + 2(1 − ζ ))G Z −1 + h −2ζ − (1 + 2(ζ − 1))G X h 2(ζ −2) , ξ = λ+ k +
hk
dh
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• a −-rarefaction wave if h l = h 0 > h r = h 2 such that for all ξ ∈ (ξ0 ≡ λl− , ξ2 ≡ λr− )
h 1 (ξ )
3gh + (4ζ 2 − 14ζ + 12)G Z −1 h 2(1−ζ ) + 2ζ (1 − 2ζ )G X h 2(ζ −1) hk 2h gh + (1 + 2(1 − ζ ))G Z −1 h 2(1−ζ ) − (1 + 2(ζ − 1))G X h 2(ζ −1) h 1 (ξ ) u 1 (ξ ) = u k − dh gh −1 + (1 + 2(1 − ζ ))G Z −1 h −2ζ − (1 + 2(ζ − 1))G X h 2(ζ −2) . ξ = λ− k −
dh
hk
4 Solution to the General Riemann Problem The general Riemann problem can be solved by combining elementary waves [6]. Solutions (24) to systems with 3 characteristic fields require 3 backward characteristics through all points in t > 0, except on discontinuities, that are: ξ0 ≤ ξ1 associated with the −-field, ξ2 ∈ (ξ1 , ξ3 ) associated with the 0-field, and ξ3 ≤ ξ4 associated with the +-field. So finally, a solution to the Riemann problem is characterized by X l = X 1 = X 2 , Z l = Z 1 = Z 2 u 2 = u 3 , P2 = P3 X r = X 4 = X 3 , Z r = Z 4 = Z 3
(31) −1 −1 with a (h 2 , u 2 )-locus given by u 2 = u l − (h l − h 2 )(P2 − Pl ) when h 2 ≥ h l and
h 1 (ξ )
u 2 ← u 1 (ξ ) = u l −
hl
dh gh −1 + (1 + 2(1 − ζ ))G Z l−1 h −2ζ − (1 + 2(ζ − 1))G X l h 2(ζ −2)
3gh + (4ζ 2 − 14ζ + 12)G Z l−1 h 2(1−ζ ) + 2ζ (1 − 2ζ )G X l h 2(ζ −1) hl 2h gh + (1 + 2(1 − ζ ))G Z l−1 h 2(1−ζ ) − (1 + 2(ζ − 1))G X l h 2(ζ −1) when h 2 ← h 1 (ξ ) ≤ h l ; a (h 3 , u 3 )-locus given by u 3 = u r + (h r−1 − h −1 3 )(P3 − Pr ) on ξ ≥ λl −
h 1 (ξ )
dh
the other hand when h 3 ≥ h r and, when h 3 ← h 4 (ξ ) ≤ h r , u 3 ← u 4 (ξ ) = u r +
h 4 (ξ )
dh gh −1 + (1 + 2(1 − ζ ))G Z r−1 h −2ζ − (1 + 2(ζ − 1))G X r h 2(ζ −2)
hr
ξ ≤ λr +
h 4 (ξ )
dh hr
3gh + (4ζ 2 − 14ζ + 12)G Z r−1 h 2(1−ζ ) + 2ζ (1 − 2ζ )G X r h 2(ζ −1) . 2h gh + (1 + 2(1 − ζ ))G Z r−1 h 2(1−ζ ) − (1 + 2(ζ − 1))G X r h 2(ζ −1)
Theorem 1. Given ξ ∈ 0, 21 , g > 0, G > 0, the Riemann problem for gSV admits a unique admissible weak solution in U for all Ul , Ur ∈ U ; this solution is piecewise continuous and differentiable with at most 5 discontinuity lines in (x, t) ∈ R × R+ .
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−-shock ζ2
U2
ζ4
U3
Ul ≡ U0
+-shock
ζ3
ζ1 ζ0
P
U5 ≡ Ur
+-rarefaction x
−-rarefaction u
Proof. It suffices to show that there exists one unique solution satisfying (31) for all Ul , Ur ∈ U . Now, it holds ∂h P > 0 and one can use (u, P, X, Z ) ∈ R × R × 1 + + R+ × R as parametrization of the state space U (with P ∈ R when ξ = 2 ), see −1 figures above. Moreover, ∂ P u = (∂h P) ∂h u is negative along the (h 2 , u 2 )-locus, strictly except at (h l , u l ), and positive along the (h 3 , u 3 )-locus, strictly except at (h r , u r ). This is indeed easily established using ∂h u = (∂ζ h)−1 ∂ζ u for rarefaction P −P+h 2 ∂ P(h −1 −h −1 ∗ ) part; ∂h u = ± ∗√ −1 h −1 for shock part, where ∂h P > 0 and P is monotone 2
(h ∗ −h
)(P−P∗ )
increasing while h −1 is monotone decreasing thus P ≥ P∗ , h −1 ≤ h −1 ∗ when h ≥ h ∗ with ∗ = l/r . So finally, since (u 3 | X r ,Z r − u 2 | X l ,Z l ) → −∞ as h = h 2 = h 3 → 0+ and (u 3 | X r ,Z r − u 2 | X l ,Z l ) → +∞ as h = h 2 = h 3 → +∞, there exists one, and only one, P = P2 = P3 zero of the continuous strictly non-decreasing function (u 3 | X r ,Z r − u 2 | X l ,Z l ). Note that it is not clear yet whether the unique Riemann solutions constructed above under Lax admissibility condition always satisfy the entropy dissipation (11). Classically, this is ensured for weak shocks only, using the asymptotic expansion of the convex entropy F as usual (see, e.g., [7, Chap. 6]) like in Saint-Venant case G = 0 with small initial data. Interestingly, the latter limit case can be recovered in the limit G → 0+ also for small initial data only. Corollary 1. When G → 0+ one recovers the usual Riemann solution to the standard Saint-Venant system G = 0 (X, Z then being “passive tracers”) only for initial data such that Ul , Ur are close enough within U . In particular, it is not possible to reach piecewise continuous and differentiable Riemann solutions with a vacuum state h = 0 as the limits of bounded continuous sequences of Riemann solutions when G > 0, as opposed to the standard Saint-Venant case G = 0. Proof. The limit h → 0 can only be reached through rarefaction waves. When G = 0, this necessarily occurs for large initial data. But when G > 0, the integrals defining the rarefaction waves are not well-defined (bounded) as h 1 → 0 (−-field) or h 4 → 0 (+-field), so this cannot occur for bounded (continuous sequences of) solutions.
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References 1. R.B. Bird, C.F. Curtiss, R.C. Armstrong, O. Hassager, Dynamics of Polymeric Liquids, Vol. 1: Fluid Mechanics (Wiley, New York, 1987) 2. F. Bouchut, Nonlinear Stability Of Finite Volume Methods for Hyperbolic Conservation Laws and Well-balanced Schemes for Sources, Frontiers in Mathematics (Birkhäuser Verlag, Basel, 2004). MR MR2128209 (2005m:65002) 3. F. Bouchut, S. Boyaval, A new model for shallow viscoelastic fluids. M3AS 23(8), 1479–1526 (2013) 4. F. Bouchut, S. Boyaval, Unified derivation of thin-layer reduced models for shallow free-surface gravity flows of viscous fluids. Eur. J. Mech. B/Fluids 55, Part 1, 116–131 (2016) 5. C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, vol. GM 325 (Springer, Berlin, 2000) 6. P.D. Lax, Hyperbolic systems of conservation laws ii. Commun. Pure Appl. Math. 10(4), 537– 566 (1957) 7. P.G. LeFloch, Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves, Lectures in Mathematics (ETH Zürich, Birkhäuser, Basel, 2002). MR 1927887 (2003j:35209) 8. M. Picasso, From the free surface flow of a viscoelastic fluid towards the elastic deformation of a solid. Comptes Rendus Mathmatique 1195(1), 1–92 (2016)
On the Exact Dimensional Splitting for a Scalar Quasilinear Hyperbolic Conservation Law Michael D. Bragin and Boris V. Rogov
Abstract A dimensional splitting scheme is applied to a multidimensional scalar homogeneous quasilinear hyperbolic equation (conservation law). It is analytically proved that the splitting error is zero. The proof is presented for the above partial differential equation in an arbitrary number of dimensions. A numerical example is given that illustrates the proved accuracy of the splitting scheme. In the example, the grid convergence of split (locally one-dimensional) compact and bicompact difference schemes and unsplit bicompact schemes combined with high-order accurate time stepping schemes (namely, Runge–Kutta methods of order 3, 4, and 5) is analyzed. The errors of the numerical solutions produced by these schemes are compared. It is shown that the orders of convergence of the split schemes remain high, which agrees with the conclusion that the splitting error is zero. Split compact and bicompact schemes are compared by their accuracy and computation speed. Keywords Hyperbolic equations · Conservation laws · Dimensional splitting Compact and bicompact schemes
1 Introduction The class of hyperbolic conservation laws is a major one in the theory of nonlinear partial differential equations and in numerous scientific and engineering applications [4, 6, 13]. Frequently considered within this class are homogeneous nonstationary quasilinear first-order equations [4, 6, 13] M. D. Bragin (B) Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudnyi, Moscow region 141700, Russia e-mail: [email protected] B. V. Rogov Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, Moscow 125047, Russia e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_17
215
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Ln (U ) ≡ ∂t U +
n
∂xi f i (U ) = 0, 0 < t ≤ T ;
i=1
U |t=0 = u 0 (x),
(1)
where U = U (x, t) and u 0 (x) are smooth scalar functions of time t and spatial coordinates x = (x1 , . . . , xn ) ∈ Rn , n ≥ 1; and the symbols ∂t and ∂xi denote ∂/∂t and ∂/∂ xi respectively. The numerical solution of multidimensional quasilinear equations by applying multidimensional finite difference schemes is a time-consuming process. Accordingly, various splitting procedures have been proposed (see, e.g., [9, 12, 17]), which reduce Cauchy problem (1) to a sequence of simpler subproblems. An example of such a procedure is the splitting of the multidimensional evolutionary problem (1) into a system of one-dimensional differential equations (see, [9, p. 129], [12]). This splitting approach is known as dimensional splitting [17]. To implement dimensional splitting, the time interval 0 < t ≤ T is divided into M intervals with the splitting step τsplit = T /M. Consider, for example, the first of these M intervals. Divide it into n subintervals of equal length. Let ti = iτsplit /n (i = 0, 1, . . . , n). The multidimensional evolutionary Cauchy problem (1) is split into a sequence of n one-dimensional problems L1,i (u) =
1 ∂t u + ∂xi f i (u) = 0, ti−1 < t ≤ ti i = 1, 2, . . . , n. n
(2)
The one-dimensional operators L1,i are chosen so that Ln =
n
L1,i .
i=1
Representation (2) means that, on each time interval ti−1 < t ≤ ti , we solve a onedimensional equation for which initial data at t = ti−1 are taken from the solution of the preceding Cauchy problem. For the first of these problems, the initial condition is the same as for problem (1). Splitting into a sequence of Eqs. (2) is also known as the method of fractional steps [17]. A widely used technique is based on splitting into problems defined on the entire interval [9, p. 133]. In the case of homogeneous equation (1), this splitting technique has the same properties as the method of fractional steps. Assume that Eq. (1) and system (2) were precisely integrated over the interval 0 < t ≤ τsplit and the solutions of the corresponding Cauchy problems at t = τsplit were found to be U (x, τsplit ) and u(x, τsplit ), respectively. The difference between these solutions is called the local splitting error [5, 6]: E split = u(x, τsplit ) − U (x, τsplit ).
(3)
On the Exact Dimensional Splitting for a Scalar …
217
If the splitting error is zero, then the splitting is exact. Solutions of Eq. (2) are usually found approximately by applying, for example, a finite difference scheme. In this context, the given scheme for the numerical solution of Eq. (1) is called a dimensional splitting scheme [17], or a locally one-dimensional scheme [12]. The total error of the numerical solution produced by the splitting scheme is the sum of splitting error (3) and the discretization error of the difference scheme [5, 6]. The order of accuracy of a stable splitting scheme coincides with its order of approximation at a zero splitting error. If all the flux functions f i (U ) in (1) are linear, i.e., f i (U ) = ai U, ai = const, then Eq. (1) becomes linear as well and the error caused by splitting (2) is equal to zero [6]. For the nonlinear scalar equation (1) with nonlinear flux functions, the splitting error in most publications (see [3, 15] and references therein) is analyzed in the case when its solution becomes discontinuous (generalized [4], weak [6]) due to the intersection of characteristics. It is assumed that a generalized solution exists and satisfies the entropy condition [4, 6]. Majorant estimates for the splitting error are obtained in [3, 15]. When nonlinear equation (1) has a classical (smooth) solution [4], the splitting error (3) can be expanded in a series for small values of τsplit . In [1, 5] for a nonlinear evolution equation of the form (4) ∂t U = A(U ) the nonlinear operator A(U ) was split into two components: A(U ) = A1 (U ) + A2 (U ). In the case of consecutive splitting of the Cauchy problem for Eq. (4) into two subproblems defined on the entire splitting interval, namely, ∂t U = Ai (U ), i = 1, 2, 0 < t ≤ τsplit
(5)
2 ) of the splitting error were derived formulas for the leading term of order O(τsplit in [1, 5] by expanding solutions of problems (4), (5) in Taylor series at t = 0. It was found that the leading term is proportional to the quantity
[A2 A1 − A1 A2 ]U |t=0 ,
(6)
where Ai is the Fréchet derivative of the nonlinear operator Ai (U ). In the case of quasilinear equation (1), the nonlinear operators Ai are given by Ai (U ) =
∂ f i (U ) ∂xi U. ∂U
(7)
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Simple calculations show that the quantity defined by (6) is zero for operators (7). 3 ), i.e., the order of accuracy Therefore, the local splitting error (3) at n = 2 is O(τsplit of splitting (2) is at least 2. In this paper, first, we analytically prove that the consecutive splitting (2) is exact for any n ≥ 2. Second, using an example of the multidimensional quasilinear equation (2) and particular high-order accurate difference integration schemes (compact and bicompact) for one-dimensional Eq. (2), we demonstrate that the order of accuracy of the splitting scheme coincides with that of these difference schemes. Third, we make a detailed comparison between locally one-dimensional compact and bicompact schemes by their accuracy and computation speed.
2 Exact Dimensional Splitting Theorem 1. If τ ≡ τsplit < t∗ , where t∗ is the time up to which the characteristics of Eq. (1) do not intersect, then splitting error (3) of the fractional step method (2) is equal to zero. Proof. It easy to show (see [10]) that the exact solution U = U (x, t) of problem (1) is determined by the relation U = u 0 (x1 − f 1 (U )t, . . . , xn − f n (U )t). Specifically, at t = τ , we have Ue = u 0 (x1 − f 1 (Ue )τ, . . . , xn − f n (Ue )τ ),
(8)
where Ue ≡ U (x, τ ). For notational convenience, we introduce the spatial functions u i (x) ≡ u(x, ti ), ti =
iτ , i = 1, 2, . . . , n. n
In what follows, the values of these functions at the point x are denoted for brevity by u i , i.e., the argument is dropped. The exact solution u = u(x, t) of problem (2) is given by the relation [10] u = u i−1 (x1 , . . . , xi−1 , xi − n f i (u)(t − ti−1 ), xi+1 , . . . , xn ). Clearly, the profile u i (x) satisfies the algebraic equation u i = u i−1 (x1 , . . . , xi−1 , xi − f i (u i )τ, xi+1 , . . . , xn ).
(9)
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219
Our goal is to compare Ue from (8) with u n from (9) at i = n. For this purpose, we pass from xi in (9) to X i = xi − f i (u i )τ : u i (x1 , . . . , xi−1 , X i + f i (u i )τ, xi+1 , . . . , xn ) = u i−1 (x1 , . . . , xi−1 , X i , xi+1 , . . . , xn )
(u = u i−2 (x1 , . . . , xi−2 , xi−1 − f i−1 i−1 (x 1 , . . . , xi−1 , X i , xi+1 , . . . , x n ))τ,
(u (x , . . . , x X i , xi+1 , . . . , xn ) = u i−2 (x1 , . . . , xi−2 , xi−1 − f i−1 i 1 i−1 , X i + f i (u i )τ,
xi+1 , . . . , xn ))τ, X i , xi+1 , . . . , xn ). (10)
In the last equation in (10), we used the first equality from the same chain. The second equality is a consequence of (9) with substitutions i → (i − 1) and xi → X i . Now, in the left- and rightmost parts in (10), we go back from X i to xi to obtain (u i )τ, xi − f i (u i )τ, xi+1 , . . . , xn ). u i = u i−2 (x1 , . . . , xi−2 , xi−1 − f i−1
(11)
(u i )τ and from xi to X i yields Passing in (11) from xi−1 to X i−1 = xi−1 − f i−1 (u i )τ, X i + f i (u i )τ, xi+1 , . . . , xn ) u i (x1 , . . . , xi−2 , X i−1 + f i−1 = u i−2 (x1 , . . . , xi−2 , X i−1 , X i , xi+1 , . . . , xn ) = u i−3 (x1 , . . . , xi−3 , xi−2 − f i−2 (u i−2 (x1 , . . . , xi−2 , X i−1 , X i , xi+1 , . . . , xn ))τ, X i−1 , X i , xi+1 , . . . , xn ) = u i−3 (x1 , . . . , xi−3 , xi−2 − f i−2 (u i (x1 , . . . , xi−2 , X i−1 + f i−1 (u i )τ, X i + f i (u i )τ, xi+1 , . . . , xn ))τ, X i−1 , X i , xi+1 , . . . , xn ). (12)
To write (12), we used the same techniques as in (10). Retaining only the left- and rightmost parts in (12), we go back from X i−1 , X i to xi−1 , xi , respectively, to obtain u i = u i−3 (x1 , . . . , xi−3 , xi−2 − f i−2 (u i )τ, xi−1 − f i−1 (u i )τ, xi − f i (u i )τ, xi+1 , . . . , xn ).
(13)
Continuing the “descent” to x1 in the same manner, we finally obtain (for analogy, see (11), (13)) u i = u 0 (x1 − f 1 (u i )τ, . . . , xi − f i (u i )τ, xi+1 , . . . , xn ).
(14)
Setting i = n in (14) yields the desired equation for the profile u n (x): u n = u 0 (x1 − f 1 (u n )τ, . . . , xn − f n (u n )τ ).
(15)
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M. D. Bragin and B. V. Rogov
It is easy to see that Eqs. (8) and (15) completely coincide. In other words, for every x, the quantities Ue (x) and u n (x) are solutions of the same equation ϕ(z) ≡ z − u 0 (x1 − f 1 (z)τ, . . . , xn − f n (z)τ ) = 0.
(16)
Because τ < t∗ , Eq. (16) has a unique solution. It follows that Ue (x) = u n (x) ∀x ∈ Rn and splitting (2) is exact.
3 Locally One-Dimensional Compact and Bicompact Schemes Let us consider the following numerical example, which vividly illustrates the result proved in Sect. 2. As problem (1), we use a two-dimensional initial-boundary value problem with flux functions U2 U3 , f2 =
= f 1 f1 = 2 3 in the computational domain D = {(x, y) | 0 < x < 1, 0 < y < 1}. The maximum time is T = 0.5. The initial condition is f 1 2 P7 (4r ) 1 2 x− U (x, y, 0) = 1 + , r= + y− , 16 4 4 f
where the function Pq (x) ∈ C q−1 (R) is defined as Pqf (x) f
=
(1 − x 2 )q if |x| ≤ 1, 0 if |x| > 1.
Plots of functions Pq (x) for q = 1, 2, 7 are given on Fig. 1. The boundary conditions are specified as a constant: u|∂ D = 1. The exact solution of this problem is well known and can be computed to any prescribed (say, machine) accuracy; it remains smooth (of class C 6 ) for all 0 ≤ t ≤ T . The one-dimensional Eq. (2) are integrated by applying difference schemes based on the method of lines. Specifically, difference approximations of the spatial derivatives are used to obtain evolutionary systems of ordinary differential equations (ODEs), which are then integrated in time. For each of the spatial approximations used, we considered three integration schemes for ODE systems based on the method of lines. These are the third-, fourth-, and fifth-order accurate schemes from [14]
On the Exact Dimensional Splitting for a Scalar … 1
Fig. 1 Plots of f functions Pq (x) for q = 1, 2, 7
0.8
Pqf
221
q=1 q=2 q=7
0.6 0.4 0.2 0 −1.5
−1
−0.5
0
x
0.5
1
1.5
occupying positions 2, 6, and 9, respectively, in Table 1 in [14]. They are Lstable stiffly accurate singly diagonally implicit Runge–Kutta (SDIRK) methods. The Butcher tableaus for them are given by formulas (17), (26), and (27) in [14]. These methods are denoted by SDIRK3, SDIRK4, and SDIRK5, respectively. The spatial derivatives were approximated using a classic fourth-order three-point compact scheme [16] and a fourth-order bicompact scheme [2, 11]. Along with the locally one-dimensional (LOD) schemes, an unsplit bicompact scheme [11] was used together with SDIRK3, SDIRK4, and SDIRK5 for the integration of Eq. (1). Table 1 presents results concerning the convergence (in the L ∞ norm) of the locally one-dimensional bicompact (LOD-BiC) and classic compact (LOD-C) schemes and the unsplit bicompact (BiC) scheme with SDIRK3, SDIRK4, and SDIRK5 time stepping (altogether nine schemes) on uniform meshes. For schemes with SDIRK3 and SDIRK4, the mesh was refined so that N x = N y = Nt , where N x and N y are the numbers of mesh cells in D in the x and y directions, respectively, and Nt is the number of mesh cells on the time interval [0, T ]. The time step is denoted by τ = T /Nt . For schemes with SDIRK5, the mesh was refined so that Nt was doubled, 4 /Nt5 = const. The initial step sizes were the same as for schemes with while N x,y SDIRK3 and SDIRK4. Let E ∞ denote the absolute error of the numerical solution in the L ∞ norm, and let k∞ be the local order of convergence of the schemes, which is given by the formula k∞ (2Nt ) = log2
E ∞ (Nt ) . E ∞ (2Nt )
The data from Table 1 shows that the actual order of accuracy of the locally one-dimensional splitting schemes is close to the order of accuracy in time of the difference schemes. Interestingly, for SDIRK4 and SDIRK5, the accuracy of BiC is higher than that of LOD-C and LOD-BiC. For SDIRK3, on the contrary, the accuracy of LOD-BiC and LOD-C is higher than that of BiC except the coarsest mesh (for LOD-C).
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M. D. Bragin and B. V. Rogov
Table 1 Errors in the L ∞ norm and local orders of convergence Nt SDIRK3 LOD-C LOD-BiC E∞ k∞ E∞ k∞ 50 100 200 400 800 Nt
50 100 200 400 800 Nt
50 100 200 400 800
6.47 · 10−3 1.52 · 10−3 2.76 · 10−4 3.28 · 10−5 2.94 · 10−6 SDIRK4 LOD-C E∞ 6.10 · 10−3 1.34 · 10−3 2.35 · 10−4 2.35 · 10−5 1.54 · 10−6 SDIRK5 LOD-C E∞ 6.08 · 10−3 9.40 · 10−4 8.18 · 10−5 3.24 · 10−6 9.33 · 10−8
2.27 · 10−3 2.09 2.47 3.07 3.48
k∞ 2.19 2.50 3.32 3.94
k∞ 2.69 3.52 4.66 5.12
6.51 · 10−4 1.26 · 10−4 2.01 · 10−5 2.63 · 10−6 LOD-BiC E∞ 2.26 · 10−3 5.19 · 10−4 6.58 · 10−5 5.36 · 10−6 3.49 · 10−7 LOD-BiC E∞ 2.33 · 10−3 3.05 · 10−4 2.05 · 10−5 7.73 · 10−7 2.33 · 10−8
BiC E∞
k∞ · 10−3
1.80 2.37 2.66 2.93
k∞ 2.12 2.98 3.62 3.94
k∞ 2.94 3.89 4.73 5.05
4.00 1.54 · 10−3 3.77 · 10−4 7.18 · 10−5 1.03 · 10−5
BiC E∞ 1.97 · 10−3 3.80 · 10−4 4.92 · 10−5 4.23 · 10−6 2.79 · 10−7 BiC E∞ 2.28 · 10−3 2.92 · 10−4 1.89 · 10−5 7.41 · 10−7 2.33 · 10−8
1.38 2.03 2.39 2.80
k∞ 2.37 2.95 3.54 3.92
k∞ 2.96 3.95 4.68 4.99
In addition, this numerical example gives an opportunity to compare classic LODC schemes and LOD-BiC schemes by their accuracy and computation speed. As can be clearly seen from Table 1, for SDIRK4 and SDIRK5, LOD-BiC schemes are approximately 4 times more accurate than LOD-C schemes with the same Nt . Let us also consider the time costs of computing convergence tests for these schemes. The corresponding runtimes1 are given in Table 2, as well as the runtimes for BiC schemes. According to the data, LOD-BiC schemes are roughly 2 times slower than LOD-C schemes and up to 17% slower than BiC schemes. Thus, for SDIRK4 time stepping and N x = N y = Nt we have E ∞ (Nt ; LOD-C) = 4, E ∞ (Nt ; LOD-BiC)
(17)
calculations on a 1 core of Intel CoreTM i7-4770K CPU @ 3.50 GHz × 4, g++ 4.6.3 compiler, OS Ubuntu 64-bit 12.04 LTS.
1 Sequential
On the Exact Dimensional Splitting for a Scalar … Table 2 Convergence tests runtimes Time stepping LOD-C runtime, sec SDIRK4 SDIRK5
Obviously,
4157 20377
223
LOD-BiC runtime, sec BiC runtime, sec 8649 42962
8265 36801
1 Runtime(Nt ; LOD-C) = . Runtime(Nt ; LOD-BiC) 2
(18)
E ∞ (Nt ; LOD-C) = O(Nt−4 ),
(19)
Runtime(Nt ; LOD-C) = O(Nt3 ).
(20)
Let N˜ t be a number such that E ∞ ( N˜ t ; LOD-C) = E ∞ (Nt ; LOD-BiC).
(21)
From (17), (19), (21) we obtain the following relation: √ √ 4 N˜ t = Nt 4 = Nt 2.
(22)
Finally, Eqs. (18), (20), (22) yield √ N˜ 3 Runtime( N˜ t ; LOD-C) = t 3 = 2 ≈ 1.41. Runtime(Nt ; LOD-BiC) 2Nt For SDIRK5 time stepping one can similarly derive 43/5 Runtime( N˜ t ; LOD-C) = ≈ 1.15. Runtime(Nt ; LOD-BiC) 2 Therefore, if a LOD-C scheme and a LOD-BiC scheme both have the same temporal approximation (at least fourth order) and the same accuracy (namely, E ∞ ), then the LOD-BiC scheme will be up to 1.4 times faster than the LOD-C scheme. It is important to note that bicompact schemes have other beneficial properties besides only being faster than compact ones. For instance, bicompact schemes retain their high order of approximation even on arbitrary nonuniform grids, unlike three-point compact schemes. Furthermore, bicompact schemes give more physically adequate solutions for problems with discontinuities. This may be illustrated by computing the moving contact discontinuity problem (test 6) from [8] with the schemes without any limiters, filters, etc. The results are given on Fig. 2. The plot shows that SDIRK3 bicompact scheme produces only two small “features”, while SDIRK3 compact scheme generates well-known nonlocal spurious oscillations in the whole
224 Fig. 2 Nonmonotonicities produced by three-point compact and bicompact schemes with SDIRK3 without any limiters in the moving contact discontinuity problem
M. D. Bragin and B. V. Rogov
1.4 1.3
ρ 1.2 O(τ 3 , h4 ) O(τ 3 , h4 )
1.1 1.0 0
0.2
0.4
x
0.6
0.8
1
area behind the discontinuity. Such a difference between bicompact and traditional compact schemes is due to the fact, that the bicompact scheme belongs to “fast type” schemes, while the compact one belongs to “slow type” schemes. “Fast type” means that the scheme has positive dispersion error, and “slow type” means that the scheme has negative dispersion error [7].
4 Conclusions It was analytically proved that LOD splitting is exact as applied to a multidimensional scalar homogeneous quasilinear hyperbolic conservation law. A numerical example was considered that demonstrates the accuracy of this splitting procedure for a number of difference schemes. It was shown that LOD bicompact schemes are superior to LOD compact schemes: the former are faster up to 1.4 times at the same accuracy and generate lesser nonmonotonicities near discontinuities. Acknowledgements This research was supported by the Russian Foundation for Basic Research (project no. 14-01-00775).
References 1. A.V. Bobylev, T. Ohwada, The error of the splitting scheme for solving evolutionary equations. Appl. Math. Lett. 14(1), 45–48 (2001) 2. M.D. Bragin, B.V. Rogov, Minimal dissipation hybrid bicompact schemes for hyperbolic equations. Comput. Math. Math. Phys. 56(6), 947–961 (2016) 3. H. Holden, K.H. Karlsen, K.-A. Lie, N.H. Risebro, Splitting Methods for Partial Differential Equations with Rough Solutions (EMS, Zurich, 2010) 4. A.G. Kulikovskii, N.V. Pogorelov, AYu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems (Chapman and Hall/CRC, London, 2001) 5. R.J. LeVeque, Ph.D. thesis, Report No. STAN-CS-82-904. Stanford University, Stanford (1982) 6. R.J. LeVeque, Numerical Methods for Conservation Laws (Birkhäuser, Berlin, 1992)
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7. X. Li, D. Fu, Y. Ma, Optimized group velocity control scheme and DNS of decaying compressible turbulence of relative high turbulent Mach number. Int. J. Numer. Methods 48(8), 835–852 (2005) 8. R. Liska, B. Wendroff, Comparison of several difference schemes on 1D and 2D test problems for the Euler equations. SIAM J. Sci. Comput. 25(3), 995–1017 (2003) 9. G.I. Marchuk, Splitting Methods (Nauka, Moscow, 1988). [in Russian] 10. A.D. Polyanin, V.F. Zaitsev, A. Moussiaux, Handbook of First-Order Partial Differential Equations (Taylor and Francis, London, 2002) 11. B.V. Rogov, High-order accurate running compact scheme for multidimensional hyperbolic equations. Dokl. Math. 86(1), 582–586 (2012) 12. A.A. Samarskii, The Theory of Difference Schemes (Marcel Dekker, New York, 2001) 13. V.D. Sharma, Quasilinear Hyperbolic Systems, Compressible Flows, and Waves (CRC, New York, 2010) 14. L.M. Skvortsov, Diagonally implicit Runge–Kutta FSAL methods for stiff and differentialalgebraic systems. Mat. Model. 14(2), 3–17 (2002) 15. Z.-H. Teng, On the accuracy of fractional step method for conservation laws. SIAM J. Numer. Anal. 31(1), 43–63 (1994) 16. A.I. Tolstykh, Compact Finite Difference Schemes and Application in Aerodynamic Problems (Nauka, Moscow, 1990). [in Russian] 17. N.N. Yanenko, The Method of Fractional Steps: The Solution of Problems of Mathematical Physics in Several Variables (Springer, Berlin, 1971)
On the Derivation of Newtonian Gravitation from the Brownian Agitation of a Regular Lattice Yann Brenier
Abstract The Vlasov–Monge–Ampère model is a nonlinear correction of the classical Vlasov-Poisson model of classical gravitation. We show how it can be derived from the elementary model of a lattice subject to Brownian agitation. Keywords Vlasov equation · Monge–Ampère equation · Large deviations Heat equation · Pilot wave
1 Introduction On a periodic domain such as Td = (R/Z)d , Newtonian gravitation is commonly described in terms of the density of probability f (t, x, ξ ) to find gravitating matter at time t, position x ∈ Td , and velocity ξ ∈ Rd , subject to the Vlasov–Poisson equation ∂t f (t, x, ξ ) + ∇x · (ξ f (t, x, ξ )) − ∇ξ · (∇x ϕ(t, x) f (t, x, ξ )) = 0, x ϕ(t, x) =
Rd
f (t, x, ξ )dξ − 1,
(t, x, ξ ) ∈ R × Td × Rd ,
where ϕ is the gravitational potential. Notice that the averaged density, say 1, has been substracted out from the right-hand side of the Poisson equation, due to the periodicity of the spatial domain. (This is a common feature of computational cosmology.) The Vlasov–Poisson system can be seen as an “approximation” to the more nonlinear Vlasov–Monge–Ampère (VMA) system ∂t f (t, x, ξ ) + ∇x · (ξ f (t, x, ξ )) − ∇ξ · (∇x ϕ(t, x) f (t, x, ξ )) = 0
Y. Brenier (B) CNRS, CMLS, Ec ole Polytechnique, 91128 Palaiseau, France e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_18
227
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det(I + Dx2 ϕ(t, x)) =
Rd
f (t, x, ξ )dξ,
(t, x, ξ ) ∈ R × Td × Rd ,
where the fully nonlinear Monge–Ampère equation substitutes for the linear Poisson equation of Newtonian gravitation. Indeed, for “weak” gravitational potential, by expanding the determinant about the identity matrix I, we get det(I + Dx2 ϕ(t, x)) ∼ 1 + trace(Dx2 ϕ(t, x)) = 1 + x ϕ(t, x) and recover the Newtonian model approximately (and exactly as d = 1). In this paper, we will speak of “Monge–Ampère gravitation” (“MAG” in short). The Vlasov– Monge–Ampère system has been introduced in [6], studied in [1], and related to the Vlasov–Poisson system [6]. It can also be solved numerically thanks to efficient Monge–Ampère solvers recently designed by Mérigot [10]. It has been argued in [4] that the MAG may also be seen as an approximation of Newtonian gravitation for which the “Zeldovich approximation” [12] (see [7, 9]), popular in computational cosmology, becomes exact. In a recent paper [5], we have claimed that MAG (and, therefore, Newtonian gravitation, through a further asymptotic analysis) can be derived from the very elementary stochastic model of a brownian point cloud. However, the derivation was obtained through a double application of the large deviation principle (LDP), through a purely formal use of the Vencel–Freidlin theory. The main purpose of the present paper is to explain how such a derivation can be made substantially more rigorous (but not completely) by substituting for one of the applications of the LDP a PDE method inspired by the famous concept of “onde pilote” introduced by Louis de Broglie at the early stage of quantum mechanics.
2 The Stochastic Model of a Lattice with Brownian Agitation We consider a cubic lattice {A(α), α = 1, · · ·, N } of N points in Rd . (It would be more consistent to consider a periodic lattice rather than a finite one, but this would make the presentation—slightly—more complicated.) We assume each point of this lattice to be subject to Brownian agitation for times t ≥ 0. Its position at time t is given by √ Yt (α) = A(α) + ε Bt (α), α = 1, · · ·, N , where the Bt (α) are independent normalized brownian curves in Rd and ε monitors the (common) level of noise. At a given time T > 0, the density of probability for the point cloud {YT (1), · · ·, YT (N )} to be observed at location X T = (X T (α), α = 1, · · ·, N ) ∈ Rd N , up to a permutation σ ∈ S N of the labels α, is easy to compute. We find
On the Derivation of Newtonian Gravitation from the Brownian Agitation …
ρ(T, X ) = or, in short,
1
√ Nd N ! 2π εT
N
exp(−
σ ∈S N α=1
229
|X T (α) − A(σ (α)|2 ), 2εT
||X T − Aσ ||2 1 ), exp(− √ Nd 2εT N ! 2π εT σ ∈S N
where | · |, || · || = denote the Euclidean norm, respectively, in Rd and R N d and Aσ = {A(σ (α)) , α = 1, · · ·, N }. This was the starting point of the discussion made in [5], using a double large deviation principle. In the present paper, we rather turn to a PDE viewpoint, where ρ can be just seen as the solution of the heat equation in Rd N ε ∂ρ (t, X ) = ρ(t, X ) ∂t 2 with, as initial condition, the delta measure located at Aσ = (A(σ (1), · · ·, A(σ (N )) ∈ Rd N and symmetrized with respect to σ ∈ S N , namely ρ(t = 0, X ) =
N 1 1 δ(X − Aσ ) = δ(X (α) − A(σ (α))). N! N! α=1 σ ∈S N
σ ∈S N
(In some sense, we have solved the heat equation in the space of “point clouds” (Rd ) N /S N , with initial position A = (A(α) ∈ Rd , α = 1, · · ·, N ), defined up to a permutation σ ∈ S N of the labels α = 1, · · ·, N .)
3 “Surfing” the “Heat Wave” After solving the heat equation in the space of “clouds” (Rd ) N /S N , ε ∂ρ (t, X ) = ρ(t, X ), ∂t 2
ρ(t = 0, X ) =
1 δ(X − Aσ ), N! σ ∈S N
we introduce the companion ODE in the space R N d d Xt ε = vε (t, X t ), vε (t, X ) = − ∇ X log ρ(t, X ), dt 2
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or, more explicitly vε (t, X ) =
1 2t
σ || (X − Aσ ) exp(− ||X −A ) 2εt 2
σ ∈S
N
=
σ || exp(− ||X −A ) 2εt 2
σ ∈S
N
1 (X − 2t
σ ∈S
N
σ ∈S
σ )) Aσ exp( ((X,A ) εt N
σ )) exp( ((X,A ) εt
),
where ((·, ·)) denotes the inner product in R N d . We may solve this ODE for arbitrarily chosen position X t0 ∈ R N d and initial time t0 > 0. In other words, we let the set of N “particles” X t = (X t (α), α = 1, · · ·, N ) ∈ R N d “surf” the “heat wave” generated by the lattice subject to Brownian agitation! By doing that, we just mimic the idea of quantum particles driven by the “onde pilote”, as imagined by Louis de Broglie at the early stage of Quantum Mechanics. [In that case, we would use the same ODE with v = ε∇I m log ψ, ψ solving the Schrödinger equation. For instance, we could consider the following free Schrödinger equation instead of the heat equation: ε (i∂t + )ψ = 0, 2
ψ(0, X ) =
exp(−||X − Aσ ||2 /a 2 ),
σ
with initial condition chosen according to “bosonic statistics”. However, in the quantum case, the analysis gets substantially more difficult, due to the possible vanishing of the wave function ψ during the evolution.]
4 Zero Noise Limit of the “Heat Wave” ODE Through the change of time t = e2θ (so that θ ranges from −∞ to +∞, while t ranges from 0 to ∞), the “heat wave” ODE becomes d Xθ = Xθ − dθ and can also be written
σ ∈S N
σ )) Aσ exp( ((Xεeθ ,A ) 2θ
σ ∈S N
σ )) exp( ((Xεeθ ,A ) 2θ
)
d Xθ = X θ − ∇Φε (θ, X θ ), dθ
where Φε (θ, X ) = εe2θ log
σ ∈S N
exp(
((X, Aσ )) ), εe2θ
is a time-dependent convex function of X , Lipschitz continuous, uniformly in ε, whose limit, as ε ↓ 0 is just the convex Lipschitz function Φ(X ) = sup ((X, Aσ )), σ ∈S N
On the Derivation of Newtonian Gravitation from the Brownian Agitation …
231
which no longer depends on time θ . It is very easy to pass to the limit ε ↓ 0 thanks to the well-established theory of maximal monotone operators (see [8]) and obtain (in the sense of max. monotone operators) d+ Xθ = X θ − ∇Φ(X θ ), dθ where d + /dθ stands for the right derivative with respect to θ and ∇Φ(X ) is just a notation for the “gradient” of a Lipschitz convex function Φ at point X , namely the unique element of its subdifferential ∂Φ at X with minimal euclidean norm. This formulation makes the initial value problem well posed (i.e., existence and uniqueness of solutions, with stability with respect to initial conditions). Notice, however, that Φ is not smooth (in contrast with Φε ) and only Lipschitz continuous, with a set N (of zero Lebesgue measure) of non-differentiability which makes unavoidable the use of the generalized gradient ∇Φ.
5 Large Deviations of the “Heat Wave” ODE Let us go back to the “heat wave” ODE and add a white noise of intensity η > 0: d Xθ √ d Bθ = X θ − ∇Φε (e2θ , X θ ) + η , dθ dθ where Φε (e2θ , X ) = εe2θ log
σ ∈S N
exp(
((X, Aσ )) ). εe2θ
In other words, our “surfers” are now subject to some additional brownian agitation, while surfing on the heat wave generated by the lattice already under brownian agitation! Since, for fixed ε > 0 and t > 0, Φε is a very smooth potential, with bounded first- and second-order derivatives, we may pass to the limit η → 0, while ε > 0 is kept fixed. The large deviation theory tells us that the probability to join point X θ0 at θ = θ0 and point X θ1 at later time θ = θ1 behaves as exp(− where Aε [X ; θ0 , θ1 ] =
1 2
inf X Aε [X ; θ0 , θ1 ] ), η ↓ 0 η
θ1 θ0
||
d Xθ − X θ + ∇Φε (e2θ , X θ )||2 dθ dθ
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and the infimum is taken over all smooth paths joining X θ0 at θ = θ0 to X θ1 at time θ = θ1 . In addition, as η ↓ 0, the noisy trajectory converges to a minimizer of the action with prescribed end points. (As a matter of fact, this has to be stated much more carefully, in precise probabilistic terms.) Subsequently, we will call Aε the Freidlin–Vencel action (although it is usually called “good rate function”).
6 Γ −Limit of the Freidlin–Vencel Action It turns out that, due to its gradient structure, the Freidlin-Vencel action Aε has a definite Γ −limit A , as ε ↓ 0, according to a personal communication of Luigi Ambrosio. This limit simply reads 1 2
θ1 θ0
||
d Xθ − X θ + ∇Φ(X θ )||2 dθ, dθ
Φ(X ) = sup ((X, Aσ )), σ ∈S N
which is nothing but the L 2 norm of the first-order ODE we got in the previous section by passing directly to the limit ε ↓ 0 in the “heat wave” ODE. So, one can rigorously obtain an effective action to describe the double limit limε↓0 limη↓0 . Unexpectedly, this action is exactly the one previously suggested by the author in [4] to include dissipative phenomena (such as sticky collisions in one-space dimension) in the Monge–Ampère gravitational model! It can also be shown to be equivalent to the following action A [X ; θ0 , θ1 ] =
θ1
θ0
1 d Xθ 2 1 || + ||X θ − ∇Φ(X θ )||2 }dθ. { || 2 dθ 2
7 Application of the Least Action Principle Since Φ(X ) = supσ ∈S N ((X, Aσ )), we observe that the points X where Φ is differentiable are those for which the supremum is reached by a unique optimal permutation σopt so that ∇Φ(X ) is nothing but Aσopt . For such points X , we get ||X − Aσopt ||2 ||X ||2 + ||Aσopt ||2 ||X − ∇Φ(X )||2 = = − ((X, Aσopt )) 2 2 2 =
||X ||2 + ||A||2 − Φ(X ) 2
(by definition of Φ and using that ||Aσ || = ||A|| for any σ ∈ S N ), while, on the set N of non-differentiability of Φ, we rather have
On the Derivation of Newtonian Gravitation from the Brownian Agitation …
233
||X ||2 + ||A||2 ||X − ∇Φ(X )||2 < − Φ(X ). 2 2 So the action we have obtained in the previous section, namely, A [X ; θ0 , θ1 ] =
θ1
θ0
1 d Xθ 2 1 { || || + ||X θ − ∇Φ(X θ )||2 }dθ. 2 dθ 2
bounds from below A + [X ; θ0 , θ1 ] =
θ1
θ0
1 d X θ 2 ||X θ ||2 + ||A||2 || + − Φ(X θ )}dθ. { || 2 dθ 2
The second action is definitely strictly larger than the first one for those curves θ → X θ , which take values in N (where Φ is not differentiable) on a set of times θ ∈ [θ0 , θ1 ], which is not negligible for the Lebesgue measure. So, the least action principle may provide different optimal curves, depending on the action we choose. However, if a curve is optimal for A and almost surely takes value outside of N , then it must also be optimal for A + . Clearly, it is much easier to get the optimality equation for such a curve, by working with A + rather than with A . By varying action A + , we get, as optimality equation, d 2 X θ (α) = X θ − ∇Φ(X θ ) = X θ (α) − A(σopt (α)) , dθ 2 σopt = Arginf σ ∈S N
N α=1
X θ (α) ∈ Rd , α = 1, · · ·, N
|X θ (α) − A(σ (α))|2 .
Of course, these equations have to be suitably modified for those curves which are optimal for action A but not for A + because they takes values in N for a nonnegligible amount of time. At this stage, we do not know how to do it. However, at least in the one-dimensional case d = 1, we believe that such modifications are tractable and should correspond to sticky collisions as X t (α) = X t (α ) occurs for different “particles” of labels α = α and during interval of times of strictly positive Lebesgue measure.
8 Obtention of the Vlasov–Monge–Ampère System In this last section, we f or mally pass to the limit N → ∞ in the equations we have derived from the least action principle with action A + (rather than A ), namely d 2 X θ (α) = X θ (α) − A(σopt (α)) , dθ 2
X θ (α) ∈ Rd , α = 1, · · ·, N ,
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σopt = Arginf
N σ ∈S N
|X θ (α) − A(σ (α))|2 .
α=1
We assume, as N → ∞, the “empiric measure” N 1 d X θ (α) ) δ(x − X θ (α))δ(ξ − N α=1 dθ
to converge to a density of probability f (t, x, ξ ) in phase space (x, ξ ) ∈ R2d , while N 1 δ(x − X θ (α)) converges to f (θ, x, ξ )dξ. N α=1 Rd The discrete optimal transport problem σopt = Arginf
N σ ∈S N
|X θ (α) − A(σ (α))|2 ,
α=1
leads (as well known in optimal transport theory [2, 3, 11]) to the Monge–Ampère equation det(I + Dx2 ϕ(θ, x)) =
Rd
f (θ, x, ξ )dξ,
while the dynamical system leads to the Vlasov equation ∂θ f (θ, x, ξ ) + ∇x · (ξ f (θ, x, ξ )) − ∇ξ · (∇x ϕ(θ, x) f.(θ, x, ξ )) = 0. So we have finally obtained, as announced, the Vlasov–Monge–Ampère system. Acknowledgements This work has been partly supported by the ANR Grant “ISOTACE”. The author thanks Luigi Ambrosio for his help concerning Sect. 6.
References 1. L. Ambrosio, W. Gangbo, Hamiltonian ODE in the Wasserstein spaces of probability measures. Commun. Pure Appl. Math. 61, 18–53 (2008) 2. Y. Brenier, Décomposition polaire et réarrangement monotone des champs de vecteurs, C. R. Acad. Sci. Paris I Math. 305, 805–808 (1987) 3. Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44, 375–417 (1991) 4. Y. Brenier, A modified least action principle allowing mass concentrations for the early universe reconstruction problem. Confluentes Mathematici 3, 361–385 (2011)
On the Derivation of Newtonian Gravitation from the Brownian Agitation …
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5. Y. Brenier, A double large-deviation principle for Monge–Ampère gravitation. Bull. of the Inst. Math. Acad. Sin. (2016) 6. Y. Brenier, G. Loeper, A geometric approximation to the Euler equations: the Vlasov–Monge– Ampère equation. Geom. Funct. Anal. 14, 1182–1218 (2004) 7. Y. Brenier, U. Frisch, M. Hénon, G. Loeper, S. Matarrese, R. Mohayaee, A. Sobolevskii, Reconstruction of the early universe as a convex optimization problem. Mon. Not. R. Astron. Soc. (2002) 8. H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, vol. 5, North-Holland Mathematics Studies (North-Holland, 1973) 9. U. Frisch, S. Matarrese, R. Mohayaee, A. Sobolevski, A reconstruction of the initial conditions of the Universe by optimal mass transportation. Nature 417, 260–262 (2002) 10. Q. Mérigot, A multiscale approach to optimal transport. Comput. Graph. Forum 30(5), 1583– 1592 (2011) 11. C. Villani, Topics in Optimal Transportation, vol. 58, Graduate Studies in Mathematics (AMS, Providence, 2003) 12. Y. Zeldovich, Gravitational instability: an approximate theory for large density perturbations. Astron. Astrophys. 5, 84–89 (1970)
Traffic Flow Models on a Network of Roads Alberto Bressan
Abstract Macroscopic models of traffic flow on a network of roads can be formulated in terms of a scalar conservation law on each road, together with boundary conditions, determining the flow at junctions. Some of these intersection models are reviewed in this note, discussing the well posedness of the resulting initial value problems. From a practical point of view, one can also study traffic patterns as the outcome of many decision problems, where each driver chooses his departure time and route to destination, in order to minimize the sum of a departure and an arrival cost. For the new models including a buffer at each intersection, one can prove: (i) the existence of a globally optimal solution, minimizing the total cost to all drivers, and (ii) the existence of a Nash equilibrium solution, where no driver can lower his own cost by changing his departure time or the route taken to reach destination. Keywords Traffic flow · Road network · Conservation law · Riemann solver
1 Introduction Macroscopic models of traffic flow based on conservation laws, introduced in the classical papers [23, 24], have been widely used in the mathematical literature. To extend these models from a single road to an entire network of roads, one needs to introduce additional boundary conditions satisfied at intersections. Various papers [1, 11, 12, 14, 17–19, 22] have been concerned with modeling, analysis, and control of traffic flow, on a network of roads. A different approach, often found in engineering literature [15], is to look at vehicular traffic in connection with decision problems [4–6]. Traffic patterns are determined by the choices of a large number of individual drivers, choosing the departure time and the route to reach destination in an optimal way, for a given a cost criterion. In this direction, it is of interest to study globally optimal solutions, A. Bressan (B) Department of Mathematics, Penn State University, University Park, PA 16802, USA e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_19
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minimizing the sum of the costs to all drivers, and Nash equilibrium solutions, where no driver can lower his own cost by changing his departure time or the route taken to reach destination. In many situations, the drivers’ turning preferences (roughly speaking, the percentage of drivers that will turn left or right at a given intersection) can be regarded as constant in time. However, when studying an optimal decision problem, these turning preferences must be determined as part of the solution itself. Surprisingly, the counterexamples discovered in [10] show that in this case several models based on Riemann solvers can be ill posed, admitting two solutions for the same initial data. This motivated the introduction of new models [7] including buffers at each intersection, which are very robust with respect to perturbations of the initial data. This note provides a brief survey of these recent developments. In Sect. 2, we review the main ideas in the construction of Riemann solvers, and the cases when they can be ill posed. Sect. 3 introduces an intersection model with buffer, and states the main existence-uniqueness results in [7], for fully general L∞ initial data. Finally, in Sect. 4, we review the concepts of globally optimal and Nash equilibrium solutions, for a decision problem on a network of roads, and state the main existence theorems proved in [8].
2 Modeling Traffic Flow at a Road Intersection Consider an intersection, say with m incoming roads i ∈ {1, . . . , m} = I and n outgoing roads j ∈ {m + 1, . . . , m + n} = O. Along each road k ∈ I ∪ O, following [23, 24] the traffic density is described by a scalar conservation law ρt + (ρ vk (ρ))x = 0.
(1)
Here, ρ = ρ(t, x) is the density of cars at time t at the point x along the road. The decreasing function vk (ρ) describes the velocity of cars on the kth road, depends on the traffic density. We use the space variable x ∈] − ∞, 0] for incoming roads and x ∈ [0, +∞[ for outgoing roads. Throughout the following, we assume that the flux functions f k (ρ) = ρvk (ρ) satisfy fk ∈ C 2 ,
f k < 0,
jam
f k (0) = f k (ρk
) = 0,
(2)
jam
where ρk is the maximum density of cars allowed on the kth road. This corresponds to bumper-to-bumper packing, where no car can move. Next, to model an intersection one needs to introduce suitable conditions satisfied by the m + n boundary values ρi (t, 0−), i ∈ I ,
and
ρ j (t, 0+),
j ∈ O.
(3)
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These conditions will depend on • The fraction θi j of drivers arriving from road i that wish turn into road j. • The relative priority ci given to drivers arriving from road i. For example, if the intersection is governed by a crosslight, the constant ci would correspond to the fraction of time when drivers from road i get green light. It is reasonable to assume that, at the time of departure, each driver chooses a path to destination, and follows it throughout the trip. Calling ρi = ρi (t, x), the density of drivers on road i ∈ I and ρi j = θi j ρi the density of drivers that will eventually turn into road j ∈ O, the conservation of the total number of these drivers yields the conservation laws (ρi )t + (ρi vi (ρi ))x = 0,
(ρi j )t + (ρi j vi (ρi ))x = 0.
(4)
Comparing the two above equations, one obtains (θi j )t + vi (ρi ) (θi j )x = 0,
(5)
showing that θi j is passively transported along the flow. In a realistic model, it turns out that there is no simple formula relating the boundary values (3). An alternative way to proceed is to introduce a Riemann solver, i.e., a rule specifying how to construct the solution in the special case where all the initial data are constant on each incoming and outgoing road: ρi (0, x) = ρ¯i , i ∈ I ,
ρ j (0, x) = ρ¯ j ,
j ∈ O.
(6)
The underlying idea behind this approach is that general solutions of conservation laws can be obtained as limits of piecewise constant, front tracking approximations [2, 13]. By specifying how to construct the solution in the piecewise constant case, in turn, one can construct solutions to the Cauchy problem with general initial data. We remark that all conservation laws (1) have finite propagation speed. Hence, by solving the initial value problem in a neighborhood of each intersection and patching together all these solutions, one can construct a solution on the entire network of roads. A theory of Riemann solvers was developed in [12, 18, 22]. See also [3, 17] for a survey. The main idea is as follows. Let constant values ρ¯i , ρ¯ j for the densities on incoming and outgoing roads be given. Regarding each road as independent from the others (Fig. 1, left), these densities determine an upper bound for the fluxes f i , f j that can arrive or leave the intersection along each roads, say f i (ρ¯i ) fi ≤
i ∈ I,
fj ≤ f j (ρ¯ j ),
j ∈ O.
(7)
Let θi j ∈ [0, 1] be the fraction of drivers arriving from road i that wish to turn into road j. Calling f i the flux of drivers arriving from road i, the m + n constraints (7) determine an admissible set of fluxes (Fig. 1, right)
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f2 f2 f1
f2
P2
f4
P1 Ω
f3 0
f1
f1
Fig. 1 Constructing a Riemann solver
. Ω=
( f1 , . . . , fm ) ;
f i ∈ [0, f i (ρ¯i )] θi j f i ≤ f j (ρ¯ j )
∀i ∈ I ,
(8)
∀j ∈ O .
i∈I
At this stage, to define a Riemann solver we need a rule which selects one point P = ( f¯1 , . . . , f¯m ) inside this feasible region Ω. In turn, the outgoing fluxes are then determined by the identities f¯j =
f¯i θi j
j ∈ O.
i∈I
Clearly, the choice of a point P ∈ Ω will be motivated by some modeling assumptions. For example, consider an intersection with two incoming and two outgoing roads. If drivers coming from road 2 have a stop sign, then all cars from road 1 will proceed right away. If there is any space left, this will be filled by cars from road 2. To model this situation, we shall thus select the point P1 = ( f¯1 , f¯2 ) ∈ Ω in Fig. 1, right, with f¯2 = max {s ; ( f¯1 , s) ∈ Ω}.
f¯1 = max {s ; (s, 0) ∈ Ω},
On the other hand, if the intersection is regulated in order to maximize the total flux passing through, then our model should select the point P2 = ( f¯1 , f¯2 ) = argmax f 1 + f 2 ; ( f 1 , f 2 ) ∈ Ω . More generally, for the Riemann solver [12] maximizing the weighted total flux ( f¯1 , . . . , f¯m ) = argmax
m i=1
ci f i ; ( f 1 , . . . , f m ) ∈ Ω
(9)
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for some given constants c1 , . . . , cm > 0, one can show that (i) If the turning preferences θi j do not change in time, then under generic assumptions the fluxes f¯i depend Lipschitz continuously on the Riemann data (6). (ii) However, the Riemann solver (9) can be discontinuous with respect to changes in the θi j . If the turning preferences θi j remain constant, then the solution of the Riemann problem constructed by the Riemann solver (9) depends continuously on the data (6), with respect to the L1 distance. In this case, the analysis in [17] shows that for any initial data with bounded variation, the Cauchy problem for traffic flow around an intersection has a unique entropy weak solution. One way to preserve continuity of the solution also with respect to changes in the θi j , suggested in [10], is to maximize the product of the incoming fluxes (instead of the sum). Namely ( f¯1 , . . . , f¯m ) = argmax
m
fi ; ( f1 , . . . , fm ) ∈ Ω .
(10)
i=1
Unfortunately, the continuity of the Riemann solver does not guarantee the well posedness of the general Cauchy problem. Indeed, consider an intersection with two incoming and two outgoing roads, with the following natural assumptions: • If all drivers arriving at the intersection can immediately move on to the outgoing road of their choice, they do so. • If the two incoming roads are both congested (i.e., if drivers line up in a queue in front of the intersection), then the inflow of cars from road 1 is a fixed rational multiple of the inflow from road 2. Under these assumptions, the counterexamples in [10] show that (i)
There exists an initial datum ρi (0, x) = ρ¯i ,
(ii)
(iii)
ρ j (0, x) = ρ¯ j
θi j (0, x) = θ¯i j (x),
i ∈ I , j ∈ O,
where the densities ρ¯k are constant but the turning coefficients θ¯i j : ] − ∞, 0] → [0, 1] have unbounded variation, such that the initial value problem has two distinct entropy-admissible solutions. There is a network (containing 8 roads) and an initial datum ρk (0, x) = ρ¯k (x) on each road, having arbitrarily small total variation, such that at timet = 1 a configuration as in (i) is reached, and two distinct solutions are possible for t > 1. Even for a single intersection with one incoming road and two outgoing roads, the travel time needed by each driver to reach destination may not depend continuously on the initial data, in the topology of weak convergence.
In view of (i), one may conjecture that uniqueness can still be achieved within a class of solutions with small total variation. However, by (ii) this is not possible.
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Indeed, in a general network of roads, the total variation of the coefficients θi j can become infinite within finite time. We remark that (iii) renders the model unsuitable for the study of optimal decision problems. Given cost criterion, to obtain an optimal solution one can follow the direct method in the calculus of variations and construct a minimizing sequence of nearly optimal solutions. However, since the total variation of these solutions may well be infinite, in general, we are only guaranteed the existence of a weakly convergent subsequence. But in this case, by (iii) we have no estimate on the cost of the weak limit. Indeed, lower semicontinuity does not hold, in general.
3 Intersection Models with Buffers To readdress this situation, and achieve the uniqueness and continuous dependence of solutions for fully general initial data ρk , θi j ∈ L∞ , a different intersection model was proposed in [7]. As in [16, 21], it is here assumed that the intersection contains a buffer with finite capacity (say, a traffic circle). Incoming drivers are admitted to the intersection at a rate depending on the number of cars already present inside the buffer. After entering the intersection, drivers flow out to the road of their choice at the fastest possible rate (Fig. 2). To formulate this model more precisely, let M > 0 be the size of the buffer, describing the maximum number of cars that can occupy the intersection at any given time, and let ci > 0, i ∈ I , be constants which account for priorities given to different incoming roads. Moreover, for j ∈ O denoted by q j (t) ∈ [0, M] the number of cars already inside the intersection, that wish to proceed to road j. Given f i (ρi (t, 0−)), fj = f j (ρ j (t, 0+)) be the the m + n boundary densities (3), let fi = maximum fluxes that can exit from road i ∈ I or enter into road j ∈ O, respectively. Junction Model with Buffer. In the above setting, at each time t the incoming fluxes f¯i are determined by
Fig. 2 Modeling an intersection with a buffer
1 4 2 3
q 4 q
5
5
Traffic Flow Models on a Network of Roads
f¯i = min
⎧ ⎨ ⎩
243
f i , ci M −
j∈O
⎫
⎬ qj
⎭
,
i ∈I.
(11)
Moreover, the outgoing fluxes f¯j satisfy
if q j > 0, then f¯j = fj, ¯ ¯ if q j = 0, then f j = min , fj, θ f i i j i∈I
j ∈O.
(12)
In turn, the conservation of total number of cars implies that the queues q j inside the buffer evolve in time according to the ODEs q˙ j =
f¯i θi j − f¯j
j ∈O.
(13)
i∈I
Notice that, by (11), the rate at which cars can enter the intersection is bounded in terms of the empty space M − j q j (t) still available within the buffer. The main result proved in [7] provides global existence and uniqueness of solutions, with arbitrary measurable initial data (possibly with unbounded variation). Theorem 1. Assume that all flux functions f k (ρ) satisfy (2), and consider initial data ⎧ jam ♦ k ∈I ∪O, ⎪ ⎨ ρk (0, x) = ρk (x) ∈ [0, ρk ] ⎪ ⎩
θi j (0, x) = θi♦j (x) ∈ [0, 1] q j (0) =
q ♦j
i ∈ I, j ∈ O,
j ∈O,
satisfying ρk♦ ∈ L∞ ,
θi♦j ∈ L∞ ,
j∈O
θi♦j (x) = 1 ,
q ♦j < M.
j∈O
Then, the conservation laws (1) together with the transport equations (5) and the junction conditions (11)–(13) admit a unique solution, globally defined for all times t ≥ 0. The proof is based on a fixed point argument. If queue sizes q j (t) are given, using a Lax-type variational formula one can construct the traffic densities ρk (t, x), separately on each incoming and each outgoing road. In turn, these densities determine the incoming and outgoing fluxes f i (t, 0−) and f j (t, 0+), for all i ∈ I and j ∈ O. By (13), we can then define the updated queue sizes q j (t)
=
q ♦j
+
t 0
i∈I
f i (s, 0−)θi j (s, 0−) − f j (s, 0+) ds .
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Fig. 3 Constructing the solution of the the Riemann problem, according to the limit Riemann solver (LRS), with two incoming and two outgoing roads. The vector ( f¯1 , f¯2 ) = γ (¯s ) of incoming fluxes is the largest point on the curve γ that satisfies the two constraints i∈I γi (s)θi j ≤ f j , j = 1, 2
f2 _ γ (s)
(c1s, c2s)
γ
Ω
f1 Working on a sufficiently small time interval [0, T ], one can show that the map (q j ) → (q j ) is a strict contraction in C 0 ([0, T ]; IR n ). Its unique fixed point yields the desired solution. The same construction can then be repeated on successive time intervals [T, 2T ], [2T, 3T ], etc. An interesting question is what happens if the size M of the buffer approaches zero. The recent analysis in [9] shows that, for Riemann initial data, the limits of solutions with vanishing buffer can be described in terms of a specific Riemann solver. More precisely, for any ε > 0 consider a buffer with size εM, replacing (11) by ⎧ ⎫ ⎨ ⎬
ci εM − fi , qj , i ∈I. (14) f¯i = min ⎩ ⎭ ε j∈O
Then, for any Riemann data, as ε → 0+ the limit of solutions to the conservation laws with buffer converge to the solution determined by the following limit Riemann solver (Fig. 3): (LRS) Consider Riemann data as in (6), together with drivers’ turning preferences fi = f i (ρ¯i ) and fj = f j (ρ¯ j ) be the maximum possible θi j , i ∈ I , j ∈ O. Let fluxes at the boundary of the incoming and outgoing roads, as in (7). Consider the one-parameter curve s → γ (s) = (γ1 (s), . . . , γm (s)),
with
. γi (s) = min{ci s , f i }.
Then, for t > 0, the Riemann problem is solved by the incoming fluxes f¯i = γi (¯s ) where
i ∈ I,
(15)
Traffic Flow Models on a Network of Roads
s¯ = max
s ≥ 0;
245
γi (s) θi j ≤ fj ∀j ∈ O .
(16)
i∈I
In turn, the outgoing fluxes are given by f¯j =
f¯i θi j
j ∈O.
(17)
i∈I
For a detailed proof, we refer to [9].
4 Global Optima and Nash Equilibria Consider a general network of roads, with nodes A1 , . . . , A N and arcs γk connecting various nodes. Along γk we again assume that the flow of traffic is described by the conservation law (18) ρt + [ρ vk (ρ)]x = 0 . Here, t is time and x ∈ [0, L k ] is the space variable along γk . We consider N groups of drivers traveling on the network, distinguished by their departure and arrival nodes, and by their cost functions. More precisely, • All drivers of the kth group depart from a node Ad(k) and arrive at a node Aa(k) , but can choose different paths to reach destination. • Any driver of the kth group, departing at time τ d and arriving at destination at time τ a , will incur in the total cost ϕk (τ d ) + ψk (τ a ). Here, ϕk is a cost for early departure, while ψk is a cost for late arrival. For k ∈ {1, . . . , N }, let G k be the total number of drivers in the kth group. We denote by Vk be the set of all viable paths (i.e., concatenations of arcs) for the k-drivers, connecting Ad(k) with Aa(k) . Each driver can decide on • the departure time, • the path chosen to reach destination. Let u k, p (·) the rate of departure of drivers of the kth group who choose a particular b path Γ p to reach destination. Hence, a u k, p (t)dt is the total number of such drivers who depart during the time interval [a, b]. We say that a family of departure rates {u k, p } is admissible if u k, p ≥ 0 and p
u k, p (t) dt = G k ,
k = 1, . . . , N .
(19)
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Call τ p (t) the arrival time of a driver departing at time t and following the path Γ p . With this notation, the definition of globally optimal and of Nash equilibrium solution can be formulated as follows. Definition 1. An admissible family of departure rates {u k, p } is a globally optimal solution if it provides a global minimum to the functional
. ϕk (t) + ψk (τ p (t)) u k, p (t) dt . J =
(20)
k, p
Definition 2. An admissible family of departure rates {u¯ k, p } is a Nash equilibrium solution if no driver can lower his total cost by changing departure time or the route taken to reach destination. Equivalently, {u¯ k, p } is a Nash equilibrium iff there exist constants c1 , . . . , c N such that (i) For almost every time t in the support of u¯ k, p , one has ϕk (t) + ψk (τk, p (t)) = ck .
(21)
(ii) For every t ∈ IR there holds ϕk (t) + ψk (τk, p (t)) ≥ ck .
(22)
Notice that (i) states that all k-drivers bear the same cost ck , regardless of the path Γ p that they take to reach destination. Moreover, (ii) implies that no k-driver can achieve a cost < ck by choosing any other departure time t ∈ IR. We observe that, given the departure rates u k, p , the corresponding arrival times τ p (t) may depend on the overall traffic pattern on the entire network, in a highly complicated way. Globally, optimal solutions and Nash equilibrium solutions have been studied under a natural set of assumptions: (A1) The flux functions f k satisfy the conditions (2). (A2) The cost functions ϕk , ψk are continuously differentiable and satisfy ϕk < 0 ,
ψk , ψk > 0 ,
ϕk (x) + ψk (x) = +∞ . lim ϕk (x) = lim x→−∞ x→+∞
(23) The main results proved in [8] can be summarized as follows. Theorem 2. Consider a general network of roads, where each intersection is modeled by a buffer, as in (11)–(13). If the assumptions (A1)–(A2) hold, then there exists at least one globally optimal solution {u k, p }. If, in addition, all travel times admit a uniform upper bound, then a Nash equilibrium {u¯ k, p } exists.
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The proof is achieved by finite-dimensional approximations, combined with a topological argument. It heavily relies on the continuity of the travel time with respect to weak convergence of the departure rates. For this reason, it is essential to adopt an intersection model with buffer: models based on Riemann solvers would not guarantee the continuous dependence on data. It is worth noting that the globally optimal and the Nash equilibrium solutions may not be unique, in general. We remark that the assumption that the travel time of all drivers has a uniform upper bound seems very reasonable in practice. However, it is a relevant open problem to determine conditions which guarantee that this holds. For an example of initial data and of intersection conditions that lead to “stuck traffic”, see [11]. Acknowledgements This work was partially supported by NSF with grant DMS-1411786: “Hyperbolic Conservation Laws and Applications”.
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17. M. Garavello, B. Piccoli, Traffic Flow on Networks. Conservation Laws Models, AIMS Series on Applied Mathematics (Springfield, Mo, 2006) 18. M. Garavello, B. Piccoli, Traffic flow on complex networks. Ann. Inst. H. Poincaré, Anal. Nonlinear 26, 1925–1951 (2009) 19. M. Gugat, M. Herty, A. Klar, G. Leugering, Optimal control for traffic flow networks. J. Optim. Theory Appl. 126, 589–616 (2005) 20. M. Herty, J.P. Lebacque, S. Moutari, A novel model for intersections of vehicular traffic flow. Netw. Heterog. Media (2009) 21. M. Herty, S. Moutari, M. Rascle, Optimization criteria for modeling intersections of vehicular traffic flow. Netw. Heterog. Media 1, 275–294 (2006) 22. H. Holden, N.H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads. SIAM J. Math. Anal. 26, 999–1017 (1995) 23. M. Lighthill, G. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. R. Soc. Lond. Ser. A 229, 317–345 (1955) 24. P.I. Richards, Shock waves on the highway. Oper. Res. 4, 42–51 (1956)
Chemotaxis and Haptotaxis on Cellular Level A. Brunk, N. Kolbe and N. Sfakianakis
Abstract Chemotaxis and haptotaxis have been a main theme in the macroscopic study of bacterial and cellular motility. In this work, we use a successful model that describes cellular motility and investigate the influence these processes have on the shape and motility of fast migrating cells. We note that, despite the biological and modelling differences of chemotaxis and haptotaxis, the cells exhibit many similarities in their migration. In particular, after an initial adjustment phase, the cells obtain a stable shape, similar in both cases, and move with constant velocity. Keywords Cell motility · Lamellipodium · Chemotaxis · Haptotaxis
1 Introduction In the biology of many diseases and, in particular, in the growth and metastasis of cancer, the processes of chemotaxis and haptotaxis play a fundamental role, see e.g. [11]. Chemotaxis is the directed motion of biological organisms (cells in particular) as response to an extracellular chemical signal. Due to their size, cells can identify A. Brunk · N. Kolbe · N. Sfakianakis (B) Institute of Mathematics, Johannes Gutenberg-University, Staudingerweg 9, 55128 Mainz, Germany e-mail: [email protected] A. Brunk e-mail: [email protected] N. Kolbe e-mail: [email protected] N. Sfakianakis Institute of Applied Mathematics, University of Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_20
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spatial gradients of the chemical ingredient along their membrane and adjust their migration accordingly. Haptotaxis, on the other hand, can be described as the directed cell motion as response to a gradient of extracellular adhesion sites or substratebound chemo-attractant/repellents. The cells attach on the adhesion sites by use of specialized transmembrane proteins like the integrins. Mathematically, chemotaxis and haptotaxis are often studied in the spirit of Keller–Segel systems [4, 10, 12]. In such approach, the involved quantities are represented by their densities. This has created a gap between the mathematical investigations —at least from a macroscopic point of view— and the experimental biological/medical sciences, where most of the knowledge/understanding refers to single cells and their properties. The current work is an effort to shed some light in this research direction. In more detail, we consider very motile cells, like fibroblast, keratocyte or even cancer cells, that migrate over adhesive substrates. These cells develop thin protrusions, called lamellipodia, see Fig. 1 and [13]. The lamellipodium can be found at the leading edge of the cells, and is comprised of a network of actin filaments, which are highly dynamic linear biopolymers [14]. Intra- or extracellular reasons might lead to polarizations of the lamellipodium and to cell motion that resembles to “crawling” [3, 15]. For the modelling of the lamellipodium and the ensuing cell motility, we follow the approach proposed in [9] and later extended in [6], and consider the Filament-Based Lamellipodium Model (FBLM); a two-dimensional continuum model that describes the dynamics of the actin meshwork and results to the motility of the cell. The FBLM distinguishes between two different families of filaments and takes into account the interactions between them and the Extracellular Matrix (ECM). Numerically, we use a problem specific Finite Element Method (FEM) that we have previously developed and that allows for efficient investigation of the FBLM, see [7].
Fig. 1 a NIH3T3 cell during migration. The lamellipodium is located in the light-coloured front of the cell; (License: Cell Image Library, CIL:26542). b The inside of the lamellipodium as reconstructed by realistic experimental data
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The main aim of this paper is to investigate, the influence chemotaxis and haptotaxis have on the shape and the motion of migrating cell, when this is modelled and simulated by the FBLM and the corresponding FEM. In more detail: in Sect. 2, we present the main components of the FBLM and of the FEM. In Sects. 3 and 4, we elucidate on the way that chemotaxis and haptotaxis are incorporated in our study. In Sect. 5, we present comparative results between the two motility scenarios.
2 The Model and the Method This section is devoted in the brief presentation of the FBLM and the corresponding FEM. Model. The main assumption behind the model is that the lamellipodium is a twodimensional structure comprised of actin filaments organized in two locally parallel families, see [6, 9] for details. The filaments of each family (denoted as F± ) are labelled by an index α ∈ [0, 2π), they have, attime t, length L ± (α, t), and can be parametrized with respect to their arclength as F± (α, s, t) : −L ± (α, t) ≤ s ≤ 0 ⊂ IR2 , where the membrane of the cell corresponds to s = 0. The two families define identical membranes
F+ (α, 0, t) : 0 ≤ α < 2π = F− (α, 0, t) : 0 ≤ α < 2π ,
(1)
which, along with the inextensibility assumption ∂s F± (α, s, t) = 1
∀ (α, s, t) ,
(2)
constitute additional constraints for the unknowns F± . The FBLM reads for the family F = F+ as (and similarly for F = F− ):
0 = μ A η Dt F − ∂s (ηλinext ∂s F) + ∂s p(ρ)∂α F⊥ − ∂α p(ρ)∂s F⊥ adhesion
in-extensibility
pressure
S Dt F − D − F− T (φ − φ )∂ F⊥ + ηη − μ + μ B ∂s2 η∂s2 F ± ∂s ηη − μ 0 s t bending
twisting
(3)
stretching
with F⊥ = (F1 , F2 )⊥ = (−F2 , F1 ). The function η(α, s, t) represents the number density of filaments of the family F with length at least −s at time t with respect to α. The first term in the (3) is responsible for the interaction of the intra- and the extracellular environment, and in particular for the momentum transfer between the cell and the ECM. It is the prominent term of (3) and, in this sense, the model
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(3) is advection dominated. The polymerization speed of the filaments is given by v(α, t) ≥ 0, and the material derivative Dt F := ∂t F − v∂s F,
(4)
describes the velocity of the actin material relative to the substrate. The inextensibility term follows from the constraint (2) with a Lagrange multiplier λinext (α, s, t). The pressure term models the electrostatic repulsion between filaments of the same family. The filament bending is modelled according to Kirchhoff’s bending theory. The last two terms in (3) model the interaction between the two families caused by elastic cross-link junctions. They resist against twisting away from the equilibrium angle φ0 of the cross-linking molecule, and against the stretching between filaments of the two families. The system (3) is subject to boundary conditions; at s = 0 they describe the tethering forces of the filaments at the membrane, and at s = −L the contraction effect of the actin–myosin interaction with the interior region of the cell. Numerical method. For the numerical treatment of the FBLM, we have developed a problem-specific FEM. We present here its main characteristics and refer to [7] for details. The maximal filament length varies around the lamellipodium, hence, the computational domain B(t) = {(α, s) : 0 ≤ α < 2π , −L(α, t) ≤ s < 0} is non-rectangular. The orthogonality of the domain is recovered using the coordinate transformation (α, s, t) → (α, L(α, t)s, t). This gives rise to the orthogonal domain s). B0 := [0, 2π) × [−1, 0) (α, Na Ns −1 We discretize B0 as B0 = i=1 j=1 C i, j , with C i, j = [αi , αi+1 ) × [s j , s j+1 ) 2π where αi = (i − 1)Δα, Δα = Nα , and s j = −1 + ( j − 1)Δs, Δs = Ns1−1 . The conforming finite element space we consider is V :=
2 F ∈ Cα [0, 2π]; Cs1 ([−1, 0]) such that F Ci, j (·, s) ∈ IP1α , F Ci, j (α, ·) ∈ IP3s for i = 1, . . . , Nα ; j = 1, . . . , Ns − 1 ,
(5)
and includes the, per direction and per cell Ci, j , shape functions i, j
i, j
i, j
i, j
i, j
i, j
Hk (α, s) = L 1 (α)G k (s), for k = 1 . . . 4 Hk (α, s) = L 2 (α)G k−4 (s), for k = 5 . . . 8 where, for (α, s) ∈ Ci, j holds
,
(6)
Chemotaxis and Haptotaxis on Cellular Level i, j
L 1 (α) = i, j
L 2 (α) =
αi+1 −α , Δα i, j 1 − L 1 (α),
i, j
253 3(s−s j )2 2(s−s )3 + Δs 3j Δs 2 2(s−s )2 (s−s )3 s − s j − Δs j + Δs 2j i, j 1 − G 1 (s) i, j −G 2 (s j + s j+1 − s)
G 1 (s) = 1 − i, j
G 2 (s) = i, j
G 3 (s) = i, j
G 4 (s) =
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
,
(7)
i, j
and where Hk (α, s) = 0 for (α, s) ∈ / Ci, j . Accordingly, the weak formulation of (3) (neglecting the boundary conditions) reads as
η μ B ∂s2 F · ∂s2 G + L 4 μ A Dt F · G + L 2 λinext ∂s F · ∂s G d(α, s) 0= B 0
T (φ − φ )∂ F⊥ · ∂ G d(α, s) + Dt F − ηη − L 4 μS Dt− F− · G ∓ L 2 μ 0 s s B 0 1 p( ) L 3 ∂α F⊥ · ∂s G − ∂s F⊥ · ∂α (L 4 G) d(α, s) , (8) − L B0
1 2 for F, G ∈ H (0, 2π); H (−1, 0) and the modified material derivative D t = ∂t − α s
s∂t L v + L ∂s and in-extensibility constraint |∂s F(α, s, t)| = L(α, t). L
3 Cellular-Level Chemotaxis-Driven Cell Migration We prescribe the sensing of the extracellular chemical signal, directly on the membrane of the cell, using the function S = S0 + S1 (x cos(φca ) + y sin(φca )) , where (x, y) traverses the membrane, φca denotes the relative direction of the chemical signal with respect to the cell, and S0 and S1 the strength of the signal. We introduce a cut-off value c on the relative signal intensity S to account for the sensitivity of the cell to the low- chemical ingredient densities. The higher the value of c the smaller is the part of the cell that “senses” the chemical ingredient. Accordingly, the polymerization rate vref is adjusted between a minimum and a maximum value vmin and vmax . Furthermore, the polymerization rate is adjusted by the signed local curvature κ of the membrane as 2vref (9) v= κ 1 + e κref where κref is the reference membrane curvature related to the local intensity of the chemical signal. In its turn, the polymerization rate v influences the length of the lamellipodium as follows:
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κcap,eff L=− + κsev
κ2cap,eff κ2sev
+
2v η(s = 0) log , κsev ηmin
cf. with Table 1 and [7] for a biological justification of ηmin . Experiment 3.1. [Less sensitive cells]: The cell moves over a uniform substrate with μ A = 0.4101. The sensitivity cut-off value is set to c = 0.33. The rest of the parameters are given in Table 1. For this, and for the rest experiments in this paper, the resolution of the mesh is set to snodes = 7 and αnodes = 36. The initial conformation of the cell is assumed to be circular with a uniform-sized lamellipodium. This experiment is visualized in Fig. 2, where we see the creation of a “tail” on the retracting side of the cell. After an initial transition phase, the cell continues its migration with a constant shape, see also Fig. 8.
Experiment 3.2. [More sensitive cells]: With similar parameters as in Experiment 3.1, and for a sensitivity cut-off set at c = 0.5 instead of c = 0.33, we obtain the results exhibited in Fig. 3. Due to the higher value of c, a smaller part of the membrane senses the chemical, and the cell exhibits a longer tail than in the c = 0.33
t = 0.0015
t = 0.0165
t = 9.0015
t = 18.4365
Fig. 2 Chemotaxis Experiment 3.1 with c = 0.33. The combination of the internal myosin retraction and the threshold value of c give rise to the particular shape of the “tail” that the cell develops
t = 0.0165
t = 9.0015
t = 18.4365
Fig. 3 Chemotaxis Experiment 3.2 with c = 0.5. The stronger chemical signal from the “east” leads to a higher polymerization rate, wider lamellipodium and more effective pulling force
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case. The cell is robust and after an initial adjustment phase it attains a stable moving shape. Further experiments on the effect of chemotaxis on the motility and shape of the cells, using the FBLM-FEM, can be found in [6].
4 Cellular-Level Haptotaxis-Driven Cell Migration Here, we study the influence that haptotaxis has in the motility of the cells, in the absence of a chemotaxis influence. We take haptotaxis into account by the density of the ECM fibres and the coefficient μ A in (3). Variations in the density of the ECM are introduced by a spatial non-uniform adhesion coefficient μ A . The polymerization rate v is considered constant, but is locally adjusted by the curvature of the membrane as in (9). There is no influence on v by the density or any other characteristic of the ECM. Biologically, the condition of the ECM is a strong indication of health. For cancer, in particular, the interaction of the cancer cells with the ECM is of fundamental importance for the invasion and the metastasis steps of the disease. For that reason, we investigate three particular cases: a “normal tissue” where the ECM is smooth and exhibits small gradients, a “tissue repair” experiment where the ECM is smooth and exhibits larger gradients, and a “damaged tissue” experiment where the ECM exhibits discontinuities. Moreover, we consider an experiment where the ECM is “pulled” below the cell and accordingly the cell is dragged along with it. Experiment 4.1. [Normal tissue]: We consider an ECM that varies spatially in a nonlinear but smooth manner as follows: 0.1, x 0: (u(y, t) − u(x, t))+ ≤
(y − x)+ mt
For a nonuniformly convex flux, the previous one-sided Lipschitz condition becomes a one-sided Hölder condition: Inequality (5) at the end of this section. Before, a precise definition of a nonlinear flux is needed. A power-law type of nonlinear degeneracy is considered as in [6]. The following condition is enough for a strictly convex flux with some regularity and power-law flux for instances. Definition 1. Let f ∈ C 1 (K , R), where K is a closed interval of R. We say that the degeneracy of f on K is at least q > 0 if the continuous derivative a(u) = f (u) satisfies: |a(u) − a(v)| > 0, (3) inf (u,v)∈(K ×K )D K |u − v|q where D K = {(u, v) ∈ (K × K ) | u = v}. The lowest real number q, if there exists, is called the degeneracy of f on K and denoted p. Inequality 3 is equivalent to: ∃m > 0, ∀(u, v) ∈ K 2 , |a(u) − a(v)| ≥ m |u − v|q .
(4)
In particular, (4) implies that a : K → R is strictly monotonic, since a is injective and continuous. Example 1. If f (u) = |u|1+ p , p ≥ 1, then p is the degeneracy of f on any interval which contains 0. q Remark 1. (1) Suppose that for all u, v ∈ K , |a(u) − a(v)| ≥ m |u − v| and that a a(u)−a(u 0 ) is differentiable at u 0 . Then q ≥ 1, since ∞ > u−u 0 ≥ m |u − u 0 |q−1 . (2) If f is smooth on K , then f has a degeneracy p which is a positive integer. [6]
Replacing f (u) by − f (−u) if necessary, we will assume subsequently that a(u) = f (u) is strictly increasing, so f is strictly convex. Now, one-sided condition (2) on the velocity is interpreted as the following onesided Hölder condition for almost all x, y: s (y − x)+ , (u(y, t) − u(x, t))+ ≤ K ts
(5)
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where s = 1p , K = m −s , (v)+ = max(v, 0) since m(u − v) p = m((u − v)+ ) p ≤ a(u) − a(v) for u ≥ v. Again, the exponent s naturally appears to estimate the positive variation of the entropy solution u. Now, we can state the smoothing effect for entropy solutions. Theorem 1. Let u 0 belongs to L ∞ (R) function, K be the convex hull of u 0 (R), f a C 1 strictly convex flux with degeneracy p on K . Then the associated entropy solution of the conservation law (1) have got the following regularity in space for all positive s−ε,1/s s (R) ∩ BVloc (R). time t and for all ε > 0: u(., t) ∈ Wloc s−ε,1 (R) was conjectured in [22] and proved in [18]. The regularity The regularity in Wloc s−ε,1/s s in BV was first proved in [6]. The regularity in Wloc is the consequence of the s BVloc regularity. The optimality in Sobolev spaces can be found in [17] and in the fractional BV spaces in [8]. The main originality of this note is, hence, to give new and simpler proofs for this smoothing effect. Our proofs are based on the key BV estimate of the velocity a(u) for positive time given by (2). This is obviously a nonlinear regularity estimate on the entropy solution u which can easily be translated into more traditional regularity estimates. The spaces BV s are in particular well adapted for this, leading to a very simple proof in the next section. From this BV s regularity, it is straightforward to deduce fractional Sobolev regularity as well. But of course one can also prove directly the Sobolev regularity; we give an example of such a proof in the last section.
3
BV s Smoothing Effect
We can define BV+s as W+s,1 and try to adapt the previous proof from Sect. 4.3. Unfortunately, the equality: L ∞ ∩ BV+s = BV s for 0 < s < 1 is an open problem. It is only known for s = 1 and, fortunately, it is enough to get the optimal BV s regularity.
3.1
BV s Spaces
We recall briefly the definition and the main properties of fractional BV spaces. Definition 2. Let I be an non-empty interval of R and let S (I ) be the set of subdivisions of I : S (I ) = {(x0 , x1 , ..., xn ), n ≥ 1, xi ∈ I, x0 < x1 < ... < xn }. For 0 < s ≤ 1 set T V u[I ] = sup s
n
S (I ) i=1
1
|u(xi ) − u(xi−1 )| s ,
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then BV s (I ) := {u, T V s u[I ] < ∞}. For s = 1, we recover the space BV of functions of bounded variation. Functions in BV s have always left and right traces, like in BV [23]. The exponent s is related to the fractional Sobolev derivative: BV s ⊂ W s−ε,1/s for all ε > 0 [6]. The space BV s is also called the space of functions of bounded p -variation, with p = 1s .
3.2 The Short Proof of the BV s Smoothing Set T V+ u :=
n sup (u(xi ) − u(xi−1 ))+ and BV+ := {u, T V+ u < ∞} n ∈ N∗ i=1 x0 < x1 < ... < xn
then a(u(., t)) ∈ BV+ from inequality (2). According to the Maximum principle a(u(x, t)) ∈ L ∞ as the initial data u 0 . Since BV+ ∩ L ∞ = BV , then a(u(., t)) ∈ BV . Moreover, the velocity a() has at most a power-law degeneracy: |a(u) − a(v)| ≥ m |u − v| p , thus u ∈ BV s , where s = 1p . This proof is very short and shortens the proof given in [6]. Moreover, it gives more information about the singularity of u. For instance, if the flux is a convex power law f (u) = |u|1+ p and the convex hull K of u 0 (R)does not contain the singular point 0, then the entropy solution associated to the initial date u 0 belongs to BV . That means that the BV s regularity is due to bigger oscillations around the state u = 0. Notice that since there are only a finite number of oscillations with any given positive strength, thus the oscillations near state u = 0 have to be smaller and smaller and with infinitely many oscillations as optimal examples given in [8, 17].
4 Optimal Smoothing Effect in Sobolev Spaces W s,1 The best smoothing effect in Sobolev spaces W s,1 was suggested in [22] with the Lax–Oleinik formula, bounded in [17] and proved in [18] with a kinetic formulation and a BV assumption on the velocity. In this short note, an another proof is proposed. We recall a classical result for W+1,1 and BV+ . Then, a similar result in W+s,1 is proved and used to get the maximal smoothing effect for conservation laws.
4.1 Usual Results in W 1,1 W+1,1 (R)
is the set of functions u such that the semi-norm
finite, where (v)+ = max(v, 0).
|u|+ 1
=
R
(∂x u)+ d x is
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Lemma 1. W+1,1 (R) ∩ L 1 (R) ⊂ BV (R). Notice that W+1,1 (R) ∩ L 1 (R) is bigger than W 1,1 (R): for instance, consider the BV function u(x) = x χ[0,1](x) where χ I is the
indicator function of the set I . The lemma follows from the equality R |∂x u| d x = 2 |u|+ 1 , which is
valid for any |∂ |x| − x and then smooth compactly supported function, since = 2 x + R x u| d x =
2 R (∂x u)+ d x − R ∂x u d x = 2 R (∂x u)+ d x. The space BV is better fitted through the control of the positive variation since BV+ (R) ∩ L ∞ (R) = BV (R) where BV+ (R) is the space of function u such that T V+ u =
sup
n
n∈N,x0 y |x−y|1+s d x d y, which is a priori singular. Consider for ε > 0 the well-defined integrals:
Fractional Spaces and Conservation Laws
Iε+ = Iε− =
x>y
Iε = x>y Jε = x>y
x>y
291
[u(x) − u(y)]+ dx dy ε + |x − y|1+s , [u(x) − u(y)]− d x d y ε + |x − y|1+s
|u(x) − u(y)| d x d y = Iε+ + Iε− ε + |x − y|1+s . u(x) − u(y) + − d x d y = I − I ε ε ε + |x − y|1+s
Since Jε =
u(x)
=
x>y u(x)
=
x>y u(x) h>0
1 1 d y d x − u(y) dx dy 1+s ε + |x − y| ε + |x − y|1+s x>y 1 1 d y d x − u(x) dy dx 1+s 1+s ε + |x − y| , x0 |x|h>0
|u|+ σ,loc ≤ 2 AC
which is finite if and only if σ < s.
h 0 >h>0
h s−σ −1 dh
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Finally, the one-sided Holder condition for an entropy solution is already given by Inequality (5) and the Sobolev regularity follows since an entropy solution is bounded σ,1 σ,1 ∩ L ∞ ⊂ Wloc . in L ∞ as its initial data and W+,loc Acknowledgements We thank the support of the project SlowDyn (team leader Bruno Lombard): an interdisciplinary CNRS project based on the LMA (CNRS, UPR 7051, Marseille). P-E Jabin acknowledges the support of NSF Grants DMS 1312142 and 1614537 and by NSF Grant RNMS (Ki-Net) 1107444.
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19. S. Junca, High frequency waves and the maximal smoothing effect for nonlinear scalar conservation laws. SIAM J. Math. Anal. 46(3), 2160–2184 (2014) 20. C. Laurence, Partial Differential Equations (American Mathematical Society, Evans, 1998) 21. P.D. Lax, Hyperbolic systems of conservation laws II. Commun. Pure Appl. Math. 10, 537–566 (1957) 22. P.-L. Lions, B. Perthame, E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Am. Math. Soc. 7, 169–192 (1994) 23. J. Musielak, W. Orlicz, On generalized variations. I. Stud. Math. 18, 11–41 (1959) 24. O. Oleinik, Discontinuous solutions of nonlinear differential equations. Usp. Mat. Nauk. 12, 3–73 (1957). (English transl. in Am. Math. Soc. Transl. Ser. 2(26), 95–172 (1963) 25. E. Yu, Panov, Existence of strong traces for generalized solutions of multidimensional scalar conservation laws. J. Hyperbolic Differ. Equ. 2(4), 885–908 (2005) 26. E.Yu, Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws. J. Hyperbolic Differ. Equ. 4(4), 729–770 (2007)
Jacobian-Free Incomplete Riemann Solvers Manuel J. Castro, José M. Gallardo and Antonio Marquina
Abstract The purpose of this work is to present some recent developments about incomplete Riemann solvers for general hyperbolic systems. Polynomial Viscosity Matrix (PVM) methods based on internal approximations to the absolute value function are introduced, and they are compared with Chebyshev-based PVM solvers. These solvers only require a bound on the maximum wave speed, so no spectral decomposition is needed. Moreover, they can be written in Jacobian-free form, in which only evaluations of the physical flux are used. This is particularly interesting when considering systems for which the Jacobians involve complex expressions. Some numerical experiments involving the relativistic magnetohydrodynamic equations are presented, both in one and two dimensions. The obtained results are in good agreement with those found in the literature and show that our schemes are robust and accurate, running stable under a satisfactory time step restriction. Keywords Hyperbolic systems · Incomplete Riemann solvers Jacobian-free methods · Relativistic magnetohydrodynamics
This research has been partially supported by the Spanish Government Research projects MTM2015-70490-C2-1R and MTM2011-28043. The numerical computations have been performed at the Laboratory of Numerical Methods of the University of Málaga. M. J. Castro · J. M. Gallardo (B) Dept. Análisis Matemático, E.I.O. y Matemática Aplicada, Universidad de Málaga, Campus de Teatinos s/n, 29080 Málaga, Spain e-mail: [email protected] M. J. Castro e-mail: [email protected] A. Marquina Dept. Matemática Aplicada, Universidad de Valencia, Avda. Dr. Moliner 50, 46100 Valencia, Spain e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_24
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1 Preliminaries Consider a hyperbolic system of conservation laws ∂t w + ∂x F(w) = 0,
(1)
where w(x, t) takes values on an open convex set O ⊂ R N and F : O → R N is a smooth flux function. We are interested in the numerical solution of (1) using finite volume methods of the form win+1 = win −
Δt (Fi+1/2 − Fi−1/2 ), Δx
(2)
where win denotes the approximation to the average of the exact solution on the cell Ii = [xi−1/2 , xi+1/2 ] at time t n = nΔt (unless necessary, dependence on time will be dropped). The numerical flux is assumed to have the form Fi+1/2 =
F(wi ) + F(wi+1 ) 1 − Q i+1/2 (wi+1 − wi ), 2 2
(3)
where Q i+1/2 denotes the numerical viscosity matrix, which determines the numerical diffusion of the scheme. It is worth noticing that Roe’s method [19] can be written in the form (3) with viscosity matrix Q i+1/2 = |Ai+1/2 |, where Ai+1/2 is a Roe matrix for the system. This remark has originated several numerical methods in the literature (see, e.g., [10, 11, 22] and the references therein), for which the corresponding viscosity matrix consists of some approximation to the absolute value matrix |Ai+1/2 |. A systematic way to build such approximations by means of polynomial or rational functions has recently been introduced in [7, 8], and further extended in [9]. For the sake of completeness, we give an overview of such methods in the next section.
2 PVM-Type Riemann Solvers: A Review Polynomial Viscosity Matrix (PVM) Riemann solvers were introduced in [7]. They are based on the idea of approximating the absolute value of the Roe matrix Ai+1/2 by means of a suitable polynomial evaluation of such matrix. If P(x) is some polynomial approximation of |x| in the interval [−1, 1], and λi+1/2,max is the eigenvalue of Ai+1/2 with maximum modulus (or an upper bound of it), the numerical flux of the PVM method associated to P(x) is given by (3) with viscosity matrix Q i+1/2 = |λi+1/2,max |P(|λi+1/2,max |−1 Ai+1/2 ),
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which provides an approximation to |Ai+1/2 |, the viscosity matrix of Roe’s method. In turn, notice that the best P(x) approaches |x|, the closer the behavior of the associated PVM scheme will be to that of Roe’s method. It is important to make clear that no spectral decomposition of the matrix Ai+1/2 is needed to build a PVM method, but only a bound on its spectral radius. This feature makes PVM methods greatly efficient and applicable to systems in which the eigenstructure is not known or difficult to obtain. A number of well-known schemes in the literature can be viewed as particular cases of PVM methods: Lax–Friedrichs, Rusanov, HLL, FORCE, Roe, etc. (see [7] for details). In the cases in which a Roe matrix is not available or is difficult to compute, Ai+1/2 can be taken as the Jacobian matrix of the system evaluated at some average state. Regarding this, notice that some systems have a Roe matrix which is not the Jacobian evaluated at some Roe state (e.g., the ideal MHD equations [6]). The stability of a PVM scheme strongly depends on the properties of the basis polynomial P(x). In particular, it must verify the stability condition |x| ≤ P(x) ≤ 1, ∀ x ∈ [−1, 1].
(4)
Of course, a standard CFL restriction has also to be imposed. The technique for constructing PVM methods has been further extended in [8] to the case of rational functions, which has originated new families of very precise incomplete Riemann solvers. Moreover, in [9] the authors have introduced the socalled approximate DOT (Dumbser–Osher–Toro) solvers, which combine the technique of PVM methods with the universal Osher-type solvers proposed in [13]. These methods can be viewed as simple and efficient approximations to the classical Osher– Solomon method [18], sharing most of its interesting features and being applicable to general hyperbolic systems, unlike the original Osher–Solomon method. With respect to the choice of the basis function, in [8, 9] Chebyshev polynomials (which provide optimal uniform approximations to |x|) were considered. An advantage of these methods is that they can be implemented in a recursive way using only vector operations. Another property of Chebyshev-based methods is that they admit a Jacobian-free implementation (see the Appendix in [9]). This means that the numerical flux can be constructed using only evaluations of the physical flux F at different states, thus avoiding the computation of the Jacobian. This point is particularly interesting for systems with complex physical fluxes (as, for example, the equations of RMHD), for which the computation of the corresponding Jacobian may be a difficult or costly task.
3 Internal Polynomial Approximations to |x| As it is well known (see, e.g., [21]), Roe’s method needs an entropy fix to handle sonic flow correctly, in order to avoid entropy violating solutions. PVM solvers share the same problem when the basis function crosses the origin, as is the case
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Fig. 1 Internal polynomial approximations (5)
of Chebyshev approximations. Moreover, Chebyshev functions do not satisfy the stability condition (4) strictly, which may cause the scheme to be unstable under certain conditions (see [8]). For these reasons, it would be interesting to consider another family of polynomial approximations to |x| satisfying the stability condition (4) and not crossing the origin. An example of such a family of polynomials can be iteratively constructed as follows (see Fig. 1): p0 (x) ≡ 1,
pn+1 (x) =
1 2 pn (x) − pn (x)2 + x 2 , n = 0, 1, 2, . . . 2
(5)
Some straightforward properties of pn (x) are listed below: • • • • •
pn (x) is even (only powers of two are involved) and deg( pn ) = 2n . |x| < pn (x) < 1 for x ∈ (−1, 1), and pn (±1) = 1. pn (1) = 1 and pn (−1) = −1. min−1≤x≤1 pn (x) = pn (0) > 0. The sequence { pn (x)}n∈N converges uniformly to |x|.
4 Jacobian-Free Implementation In this section, we build Jacobian-free PVM solvers associated to the internal approximation pn (x) introduced in the previous section. First of all, it should be noted that the recursive form (5) is not suitable for that purpose due to the term pn (x)2 . For this reason, the explicit form of pn (x) combined with Horner’s method will be considered instead. On the other hand, notice that it will not be necessary to compute the viscosity matrix Q i+1/2 explicitly, but only the vector Q i+1/2 (wi+1 − wi ) appearing in the numerical flux (3).
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Table 1 Coefficients of the internal approximation pn (x) for n = 1, 2, 3, 4 n α0 α1 α2 α3 α4 α5 α6 1
1 2
2
− 18
3 4
3 8
3
1 − 128
3 32
23 − 64
31 32
4
1 − 32768
3 4096
59 − 8192
169 4096
α7
α8
4807 4096
8463 32768
1 2
39 128 2635 − 16384
1693 4096
5891 − 8192
To illustrate the procedure, consider the polynomial p2 (x) = α0 x 4 + α1 x 2 + α2 = x 2 (α0 x 2 + α1 ) + α2 , where the coefficients αi can be found in Table 1. Let A ≡ A(w) be the Jacobian matrix of F evaluated at an intermediate state w, and let v be an arbitrary state; for simplicity, assume that λmax = 1. Then, as stated in Sect. 2, the following approximation holds: |A|v ≈ p2 (A)v = (A2 (α0 A2 + α1 I ) + α2 I )v. The above expression can be computed using Horner’s algorithm: • Define v0 = v and compute v0 = A2 v0 . v0 + α1 v0 and v1 = A2 v1 . • Calculate v1 = α0 v1 + α2 v0 . Then, |A(w)|v ≈ p2 (A)v = v2 . • Compute v2 = The product A(w)v can be approximated using the finite difference formulation A(w)v ≈
F(w + εv) − F(w) , ε
which leads to F w + F(w + εv) − F(w) − F(w) A(w) v ≈ ≡ ε (w; v). ε 2
In practice, the value ε has to be chosen small relative to the norm of w. Finally, combining the above results, the vector |A(w)|v can be approximated using the following steps, in which only vector operations and evaluations of the physical flux F are involved: • Define v0 = v and compute v0 = ε (w; v0 ). v0 + α1 v0 and v1 = ε (w; v1 ). • Calculate v1 = α0 • Compute v2 = v1 + α2 v0 . Then, |A(w)|v ≈ v2 .
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5 Numerical Results The equations of relativistic ideal magnetohydrodynamics (RMHD) have been chosen here to analyze the behavior of the proposed schemes, mainly due to the complex form of the Jacobian of the system. In this case, our Jacobian-free schemes seem to be a very advantageous approach. It will be assumed throughout this section that the speed of light is normalized to c = 1. The vector of primitive variables, measured in the laboratory frame, is UP = (ρ, v, P, B)t , where ρ is the rest-mass density, v = (vx , v y , vz ) is the velocity field, B = (Bx , B y , Bz ) is the magnetic field, and P is the total pressure. On the other hand, the conserved variables are given by D = ρW, Sα = ρhW 2 vα − b0 bα , τ = ρhW 2 − P − b02 − D,
(6)
being W = (1 − |v|2 )−1/2 the Lorenz factor, h the specific enthalpy, b0 = W v · B, and bα = Bα /W + b0 vα , for α = x, y, z. We shall also assume an ideal gas equation of state P ≡ P(ρ, ε) = ρε(γ − 1), where γ is the adiabatic index and ε is the specific internal energy, which verifies the relationship h = 1 + ε + P/ρ. Written in conservative form, the RMHD equations read as ∂t U +
∂α Fα = 0,
α
where the vector U of conserved variables is defined by U = (D, S, τ, B)t and the flux in the α-direction is given by ⎛
Dvα
⎞
⎜ b B ⎟ ⎜ Sβ vα + Pδαβ − β α ⎟ ⎜ W ⎟ Fα = ⎜ ⎟ ⎜ (τ + P)v − b0 Bα ⎟ α ⎠ ⎝ W vα Bβ − vβ Bα where the indices α and β run in {x, y, z} and δαβ is Kronecker’s delta function. Additionally, the magnetic field must satisfy the divergence-free constraint ∇ · B = 0.
(7)
An inherent difficulty of RMHD is that the vectors UP and U of primitive and conserved variables are related in a nonlinear way through (6). In particular, the recovery of UP from U is done here using an iterative process, as described in [16].
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Table 2 Initial conditions for test problems Sects. 5.1.1 and 5.1.2 Test ρ vx vy vz P Section 5.1.1. L R Section 5.1.2. L R
1.000 0.125 1.0 1.0
0.0 0.0 0.999 −0.999
0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0
1.0 0.1 0.1 0.1
Bx
By
Bz
0.5 0.5 10.0 10.0
1.0 −1.0 7.0 −7.0
0.0 0.0 7.0 −7.0
5.1 One-Dimensional Test Problems In this section, we have chosen two one-dimensional tests that constitute standard references in RMHD (see, e.g., [1, 12, 14]). The initial conditions for these Riemann problems are listed in Table 2; details on the exact solutions can be found in [14]. The adiabatic coefficient in Test 1 is γ = 2, while for Test 2 it is γ = 5/3. The tests have been computed using 800 cells, a Courant number of 0.8, and a final time of t f = 0.4. To save space, only the results for the density component will be shown; similar comments and results apply for the other variables. The numerical experiments have been performed with the Jacobian-free versions of the following methods: • PVM-Cheb-12 and PVM-int-8: PVM methods based, respectively, on the Chebyshev approximation of degree 12 and the internal approximation of degree 8. The intermediate matrix Ai+1/2 has been taken as the Jacobian of the flux evaluated at the mean state 21 (wi + wi+1 ). • DOT-Cheb-12 and DOT-int-8: approximate DOT solvers using the same polynomials as above and a Gauss–Legendre quadrature with q = 3 points. The results have also been compared with the classical Harten–Lax–van Leer (HLL) method. In this case, the minimum and maximum speeds of propagation have been taken as −1 and 1, respectively, so HLL reduces to Rusanov’s method. Finally, with respect to the higher order schemes, the third-order PHM method [15] has been considered in space, combined with a third-order TVD Runge–Kutta method [20] for time stepping.
5.1.1
Relativistic Brio–Wu Problem
This problem is a relativistic version of the classical test proposed by Brio and Wu [5] in Newtonian MHD. The solution consists of a fast rarefaction and a compound wave traveling to the left, a contact discontinuity, a slow shock, and a fast rarefaction wave going to the right. Figure 2 shows the density component of the solution, as well as a closer view of the compound wave. The results obtained are in good agreement with those presented in [1, 12]. As it can be observed, for the first-order schemes, the more precise results
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Fig. 2 Density component of the solution in test Sect. 5.1.1. Zoom of the compound wave. Left: first order. Right: third order Table 3 Test Sect. 5.1.1: Relative CPU times with respect to the first-order HLL solver Method First order Third order HLL PVM-int-8 PVM-Cheb-12 DOT-int-8 DOT-Cheb-12
1.00 2.25 2.95 4.55 6.39
5.55 10.01 11.54 16.86 18.07
are obtained with PVM-Cheb-12 and DOT-Cheb-12, followed by the PVM-int-8 and DOT-int-8; on the other hand, HLL is too diffusive to provide a good resolution of the compound wave. A similar behavior is found when the third-order schemes are applied, although in this case the differences between methods are much smaller. In any case, PVM and DOT solvers give very similar results. Taking into account the CPU times given in Table 3 it can be concluded that, at least for this test, PVM-based methods are a better option than DOT-based ones.
5.1.2
Relativistic Shock Reflection Problem
In this section, we consider the relativistic MHD analog of the Noh test problem proposed in [1]. Initially, there are two streams approaching each other with a Lorenz factor of 22.366, which makes this problem an extremely strong relativistic one. The solution, shown in Fig. 3, has two very strong fast shocks propagating outward symmetrically in opposite directions. Moreover, two slow shocks traveling in opposite directions are formed, which are also properly computed. The post-shock oscillations at the fast shocks are minimal and can be greatly damped by reducing the Courant number in the computation. At the point of symmetry x = 0.5, there is a spurious density undershoot (see Fig. 3, bottom) due to the numerical pathology known as wall heating, which is produced by an undesired
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Fig. 3 Density component of the solution in test Sect. 5.1.2. Zoom of the central zone. Left: first order. Right: third order
Fig. 4 Test Sect. 5.1.2: L 1 errors for ρ (in logarithmic scale) obtained with meshes with 200, 400, 800, 1600 and 3200 cells. Left: first order. Right: third order
accumulation of entropy in a few zones around the point of symmetry. The numerical error around the undershoot is about 4.44%, which is quite acceptable (see [1, 12]). Figure 4 shows the L 1 -errors obtained with different meshes. As it can be seen, in first order the best results are obtained with the PVM-Cheb-12 scheme, followed by DOT-Cheb-12 and PVM-int-8 (which gives very similar results to DOT-int-8). In third order, DOT-Cheb-12 provides the more precise results, closely followed by PVM-int-8 and DOT-int-8 (with very similar errors).
5.2 Two-Dimensional Test Problems Due to the challenging nature of the two-dimensional tests considered here, we have considered second-order TVD versions of the schemes, which seem robust enough to resolve the complex features of the solutions accurately. Notice that, as no analytical solutions are available in this context, the comparisons with the results in the literature
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Fig. 5 The rotor problem. Solution at time t = 0.4. Left: density. Right: pressure
have to be done at a rather qualitative level. In general, the results obtained with the PVM-int-8 and PVM-Cheb-12 schemes are very similar. For this reason, in order to save space only the results obtained with the PVM-int-8 scheme will be shown. On the other hand, DOT-type schemes are not considered here as they produce similar results as PVM-type schemes, but at a higher computational cost. The divergence-free constraint (7) on the magnetic field has to be imposed in order to ensure the accuracy and stability of the numerical schemes. This has been accomplished by using the projection technique introduced in [4].
5.2.1
Relativistic Rotor Problem
Our first two-dimensional test is the relativistic version of the MHD rotor problem, originally proposed in [2], which has been studied by several authors (see, e.g., [12, 23, 24]). Initially, a disk of radius R0 = 0.1 with higher density ρ = 10 is rotating at high relativistic speed, with angular frequency ω = 8.5. This rotor is placed at the center of the computational domain [−0.6, 0.6] × [−0.6, 0.6] and is embedded in an ambient fluid at rest with density ρ = 1. As in [12, 23], no taper function has been applied to the rotor for smoothing the initial condition. The pressure and the magnetic field are assumed to be uniform everywhere, with P = 1, Bx = 1, and B y = Bz = 0. The adiabatic index has been taken as γ = 4/3. The spinning of the rotor induces a complicated pattern of shocks and torsional Alfvén waves that transfer angular momentum from the cylinder to the ambient background. The problem has been solved up to time t = 0.4 using a 512 × 512 mesh, with CFL = 0.5. Transmissive boundary conditions have been imposed. The computed solutions in Fig. 5 are in good agreement with those reported in the literature, which shows that our schemes are robust and precise enough to resolve smooth waves with high accuracy and avoiding numerical oscillations at shocks.
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Fig. 6 Orszag–Tang vortex. Solutions at time t = 4. Left: density. Right: pressure
5.2.2
Relativistic Orszag–Tang Problem
This test constitutes the relativistic version of the Orszag–Tang vortex problem [17], which is a well-known model for testing the transition to supersonic MHD turbulence. The initial conditions are given by (ρ, vx , v y , vz , P, Bx ,B y , Bz ) =
v0 v0 1, − √ sin(y), √ sin(x), 0, 1, −B0 sin(y), B0 sin(2x), 0 , 2 2 with v0 = 0.75, B0 = 1, and adiabatic index γ = 4/3. The problem has been solved up to time t = 4 in the computational domain [0, 2π ] × [0, 2π ] using a 512 × 512 grid, with CFL=0.5. Periodic boundary conditions have been considered. Figure 6 shows the results obtained with the second-order PVM-int-8 scheme for the density and pressure components at times t = 0.5, 2, 4. At a qualitative level, our results are in good agreement with those presented in [3, 23, 24]. This demonstrates the ability of our schemes for solving two-dimensional problems involving complex features.
6 Conclusions In this work, new classes of incomplete Riemann solvers have been introduced. They are constructed following the PVM/RVM and approximate DOT techniques (earlier introduced in [7–9], respectively), taking as basis functions polynomial internal approximations to the absolute value function. This approach has several advantages. First, internal approximations do not cross the origin and satisfy the stability condition needed to ensure the robustness and convergence of the schemes. Second, the structure of internal polynomials allows a Jacobian-free writing of the associated schemes. Thus, only evaluations of the physical flux and vector operations are involved in the
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implementation of the numerical schemes. This is particularly important in those situations in which the Jacobian of the system is difficult or computationally expensive to calculate. A number of one- and two-dimensional tests in relativistic magnetohydrodynamics have been performed, showing the good performances of the proposed schemes. After extensive numerical experiments, our conclusion is that for problems involving complex features, high-order PVM and related solvers may be more competitive than a high-order HLL solver. We have not addressed this issue here (mainly due to lack of space), but we refer the interested reader to our recent article [9].
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A Finite-Volume Tracking Scheme for Two-Phase Compressible Flow Christophe Chalons, Jim Magiera, Christian Rohde and Maria Wiebe
Abstract We propose a finite-volume tracking method in multiple space dimensions to approximate weak solutions of the hydromechanical equations that allow twophase behavior. The method relies on a moving mesh ansatz such that the phase boundary is represented as a sharp interface without any artificial smearing. At the interface, an approximate solver is applied, such that the exact Riemann solution is not required. From precedent work, it is known that the method is locally conservative and recovers planar traveling wave solutions exactly. To demonstrate the efficiency and reliability of the new scheme, we test it on various situations for liquid–vapor flow. Keywords Sharp-interface resolution · Compressible two-phase flow Moving mesh method · Finite-volume scheme · Phase transition Subject Classifications 35L65 · 76M25 · 76T10 · 65M50
1 Introduction We consider weak solutions of the Euler equations for compressible two-phase flow. This system is equipped with an entropy–entropy flux pair such that the entropy is C. Chalons Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 78035 Versailles, France e-mail: [email protected] J. Magiera · C. Rohde · M. Wiebe (B) Institute for Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany e-mail: [email protected] J. Magiera e-mail: [email protected] C. Rohde e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_25
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strictly convex in a state space that is split into two disjoint open subsets—the liquid and the vapor bulk. The separating set is called spinodal region. As a consequence, one has strict hyperbolicity in the complete state space. Phase boundaries are considered as shock waves that connect states in different phases in a subsonic way. In this way, the spinodal region can be avoided. Phase boundaries still have to satisfy the entropy inequality, thus being consistent with the second law of thermodynamics. Nevertheless, well-posedness in the setting of two disjoint state spaces must be restored with additional constraints, e.g., so-called kinetic relations [17], which are put on the phase boundary. The numerical approximation of problems with phase boundaries is a challenging issue: To avoid approximate solutions with values outside of the state space, advanced techniques like the precise tracking of the interface are required. In this contribution, we use the approach [5], extended to the two-phase Euler system. It relies on the tracking of the phase boundary using a moving mesh and exploiting the exact dynamics across phase boundaries. The moving mesh approach in this paper is different from standard uses where the mesh is changed globally to reduce the error or to get aligned with appropriate transport directions. Here, we intend to track the mesh only locally around the discrete interface and try to avoid any global changes of the mesh that affect the bulk domains. In the paper at hand, we present a specific approach for the two-phase Euler equations. In particular, we present pertinent numerical tests in one and two space dimensions. For the general approach and for analytical results, we refer to [6]. Another approach for the numerical treatment of phase boundaries in compressible liquid–vapor flow is the ghost fluid method [10, 11]. Mixed-phase volumes for compressible multiphase flow are allowed in [8] where also the moving-mesh approach is used.
2 Isothermal Euler Equations The isothermal Euler equations with non-monotone pressure function govern the dynamics of compressible liquid–vapor flow. Assume that a time T ∈ (0, ∞) and a time–space domain DT = Rd × (0, T ) are given. Then, the isothermal Euler equations read as ρ ρv +· = 0 in DT . (1) ρv t ρv ⊗ v + p(ρ)I The unknowns are the density ρ = ρ(t, x) and the momentum ρv, where v = (v1 (t, x), . . . , vd (t, x))T denotes the velocity of the fluid. With u = (ρ, ρv) and a suitable flux function f, system (1) can be rewritten as ut + · f(u) = 0. The given pressure function p is chosen in van der Waals form
A Finite-Volume Tracking Scheme for Two-Phase Compressible Flow Fig. 1 Pressure function p = p(ρ), which defines the phases of a fluid by the domains where p is monotone increasing
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p( )
spinod liq
p : (0, B −1 ) → R+ , ρ → Rθ
spinod vap
ρ − Aρ 2 , 1 − Bρ
(2)
with positive constants A, B, θ, R > 0, where the fixed temperature θ is chosen in the subcritical regime such that p is non-monotone (see Fig. 1). We denote by spinod spinod < ρvap the extreme values of the interval where the pressure function p is ρliq spinod
decreasing. The density must not take values in the interval (ρliq spinod
ρ : [0, T ) × Rd → (0, ρliq
spinod
, ρvap ), i.e.,
spinod ] ∪ [ρvap , B −1 ).
Therefore, we define liquid and vapor bulk states according to spinod spinod × Rd , P+ = ρvap , B −1 × Rd , P− = 0, ρliq
(3)
and define the state space U as the union of both sets U = P− ∪ P+ ⊂ Rm . We distinguish the phases by a mapping π given as π : U → {−, +}, u →
− if u ∈ P− , + if u ∈ P+ .
(4)
In the following, we will consider an initial state u0 = (ρ0 , ρ0 v0 ), such that (ρ, ρv)(0, ·) = (ρ0 , ρ0 v0 ).
(5)
A function u ∈ L ∞ ((0, T ) × Rd , U ) is called a weak solution of the initial value problem (1), (5) in DT if
T 0
Rd
uφt + f(u) · φ dV dt = −
Rd
u0 φ(0, x) dV
holds for all φ ∈ C0∞ ([0, T ) × Rd , R). The system (1) is equipped with an entropy–entropy flux pair (η, q) : U → Rd+1 . The canonical entropy–entropy flux pair for (1) is given by
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η(ρ, m) = ρΨ (ρ) +
|m|2 m , q(ρ, m) = (q1 (ρ, m), . . . , qd (ρ, m)) = (η + p(ρ)), 2ρ ρ
with Ψ such that Ψ (ρ) = p(ρ) . A weak solution u ∈ L ∞ ((0, T ) × Rd , U ) is called ρ2 an entropy solution of (1), (5) in DT if
T 0
Rd
η(u)φt + q(u) · φ dV dt ≥ −
Rd
η(u0 )φ(0, x) dV
(6)
holds for all φ ∈ C0∞ ([0, T ) × Rd , R), φ ≥ 0. It is well known (cf. [1]) that single-phase boundaries do not only have to satisfy the Rankine–Hugoniot conditions but also an additional so-called kinetic relation. Following [12], we require for some function K : P− × P+ × R → R that all phase boundaries connecting u− with u+ with velocity r satisfy the kinetic relation K (u− , u+ , r ) = 0
(7)
where K (u− , u+ , r ) = μ(ρ− ) + 0.5 (v− · n − r )2 − μ(ρ+ ) − 0.5 (v+ · n − r )2 + k∗ j, with the Gibb’s free energy μ given through μ = p /ρ, relative mass flux j = ρ− (v− · n − r ) and mobility k∗ > 0. In this paper, we are interested in entropy solutions u that split Rd for each time t ∈ [0, T ) in two disjunct ±-phase domains D− (t), D+ (t) and a hypersurface Γ (t) such that for almost all x ∈ Rd π u(t, x) = ± ⇒ x ∈ D± (t)
(8)
and Γ (t) = D− (t) ∩ D+ (t) hold. We call D± (t) the ±-phase domain and Γ (t) the sharp interface. For x ∈ Γ (t), let n(t, x) = (n 1 (t, x), . . . , n d (t, x))T ∈ S 1 denote the normal vector of Γ (t) that points into D− (t). Let the function u : DT → U be regular enough such that for (t, x) ∈ DT the traces u± (t, x) :=
lim u(t, x ± εn(t, x))
ε→0,ε>0
exist. Then, we define the interfacial jump for x ∈ Γ (t) by [[u(t, x)]] = u+ (t, x) − u− (t, x). We denote by r (t, x) the speed of Γ (t) in direction n(t, x). Necessary conditions for the function u to be a weak solution of (1), (5) are the Rankine–Hugoniot conditions − r (t, ·) [[u(t, ·)]] + [[n · f(u(t, ·))]] = 0.
(9)
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In the given setting a function u ∈ C 0 ([0, T ); L ∞ (Rd )), ±-phase domain families {D± (t)}t∈[0,T ) and a sharp-interface family {Γ (t)}t∈[0,T ) are called an entropycompatible sharp-interface solution of (1), (5) if the following conditions hold. (i) For t = 0, we have D± (0) = D±,0 , Γ (0) = Γ0 , and for each t ∈ (0, T ) we have Rd = D− (t) ∪ D+ (t) ∪ Γ (t) with disjunct d-dimensional ±-phase domains D± (t) and the hypersurface Γ (t). (ii) The condition (8) holds for almost all (t, x) ∈ DT . (iii) The function u is an entropy solution of (1), (5) in DT . (iv) For each t ∈ (0, T ), the function u satisfies the trace conditions (7), (9). For our numerical approach, the Riemann problem for the planar situation of (1) will be important. Fix some n ∈ S d−1 and define F(u) = n 1 f1 (u) + . . . + n d fd (u), u ∈ U .
(10)
Then, for states U± ∈ P± , the Riemann problem is the special initial value problem wt + (F(w))x = 0 in (0, ∞) × R ,
w(0, x) =
U− if x < 0, U+ if x > 0,
(11)
with unknown w = w(t, x) ∈ U . It is a reasonable assumption that the exact entropy solution w of (11) is a self-similar function that connects the left state U− and the right state U+ by at most 3 (for m-dimensional systems at most m) elementary waves (i.e., shock waves, contact waves, rarefaction waves, and attached shock-rarefaction waves, each of them within either P− or P+ ) and exactly one phase transition wave. The phase boundary wave is a shock wave that connects a state u− in P− with a state u+ in P+ . Across this wave, the conditions (9), (7) have to hold (see [13] for a general theory and [7, 9, 12, 14, 15] for specific cases). The range of the function w is in P− left to the phase transition and in P+ otherwise (see Fig. 2 for some illustration). In the following, we do not need to know the exact Riemann problem solution but only the speed of the phase transition, as well as the two adjacent values. This might be even given by an approximate solver [4, 16]. To combine both cases, we introduce an interface solver. For some kinetic relation (7), the mapping RF : P− × P+ → R × P− × P+ with RF (U− , U+ ) := (σ, u− , u+ )
(12)
is called an interface solver for (1) if the following conditions are satisfied. (i) RF is a continuous mapping. (ii) The states u± ∈ P± and σ satisfy − σ (u− − u+ ) + F(u− ) − F(u+ ) = 0, K (u− , u+ , σ ) = 0.
(13)
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t
Fig. 2 Sketch of a Riemann pattern that contains a phase transition wave with speed σ . The adjacent states are u∓ ∈ P∓
u−
u+
U−
U+
x
0
(iii) If there is a r ∈ R such that U− , U+ fulfill (13) with σ = r then RF (U− , U+ ) = (r, U− , U+ ). We call σ the speed of the interface.
3 An FV Moving Mesh Method with Interface Tracking In this section, we shortly summarize the used numerical method from [6] for multiple space dimensions. The method is a finite-volume scheme on moving meshes combined with an interface tracking and a post-processing step where the mesh is improved with regard to the volume/perimeter ratio. Assume that a (fixed) mesh is given as set of (d+1)-polygons K i (i ∈ I index set), i.e., τ = {K i |K i ∈ Pd+1 , i ∈ I } fulfilling the standard requirements of a mesh in Rd . Then, we call the pair T = (τ, {Φi }i∈I ) a moving mesh with continuous functions Φi : [t1 , t2 ] → La (K i , Rd ), t → Φit , Φi (t1 ) = id, mapping from an interval to the space of affine mappings La (K i , Rd ) from K i to Rd , if for all t ∈ [t1 , t2 ] the set {Φit (K i )}i∈I is a (fixed) mesh with index set I . This enables us to define the time-dependent elements and the time-dependent edges as K i (t) := Φit (K i ), Si, j (t) := Φit (K i ∩ K j ). A finite-volume scheme on a moving mesh then reads
K i (t n+1 ) un+1 = K i (t n ) un − Δt n
Si, j t n+1/2 gn (un , un ) + hn (un , un ) , i i, j i j i, j i j i j∈N (i)
with t n+1/2 = t n + 0.5Δt. The numerical flux function gi,n j and the geometrical flux function hi,n j are L-continuous functions, which obey the consistency conditions gi,n j (u, u) = f(u) · ni, j (t n+1/2 ) and hi,n j (u, u) = f(u) · ni, j (t n+1/2 )si, j u
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where we denote by si, j the speed of the midpoint of the edge Si, j , as well as the conservation property gi,n j (u, v) + hi,n j (u, v) = − gnj,i (v, u) + hnj,i (v, u) . The idea of the scheme in [6] is to choose the mappings Φi (and thus the timedependent elements K i (t)) with the aid of the interface solver RF such that the position of the phase transition is tracked. This has two advantages: First, we can treat interface edges separately via making a special choice for numerical and geometrical fluxes gi,n j (u, v) and hi,n j (u, v) when π(u) = π(v). Second, we do not have any smearing across the interface hypersurface due to averaging, since the phases are sharply separated. In an additional post-processing step, we define a new mesh in order to maintain a decent volume to perimeter ratio of the mesh triangles that might become either very small or very big due to the interface tracking. In our implementation, the mesh is chosen as a constrained Delaunay triangulation, which provides methods for insertion and removal of points. The complete description of the remeshing algorithm is out of scope. Let us mention that the remeshing, realized by point insertion and removal, leads only to small changes of the mesh, e.g., the insertion causes six new triangles in expectation [2]. In the numerical Example 2 of Sect. 4.2, we verify this by comparing the meshes before and after remeshing. The two most important properties of the complete scheme are the following. • The scheme fulfills the conservation property Rd
uh (t n , ·) − u0 dV = 0,
which does not hold, e.g., for the ghost fluid method for two-phase problems. • Assume that an entropy-compatible sharp-interface solution of (1), (5) is given as u(t, x) =
u L ∈ P− if x · ν + ct < 0 u R ∈ P+ if x · ν + ct > 0,
D± (t) = {x ∈ Rd | x · ν + ct ≶ 0}.
Then, the algorithm is able to resolve u exactly independent of the coarsity of the mesh when the numerical and geometrical fluxes are chosen as
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⎧ ⎪ ⎨f(U(u, v)) · ni, j n gi, j (u, v) = f(U(v, u)) · ni, j ⎪ ⎩ n g˜ i, j (u, v) ⎧ ⎪ ⎨−σ (u, v)U(u, v) hi,n j (u, v) = +σ (v, u)U(v, u) ⎪ ⎩˜n hi, j (u, v)
if u ∈ P− , v ∈ P+ , if u ∈ P+ , v ∈ P− , otherwise, if u ∈ P− , v ∈ P+ , if u ∈ P+ , v ∈ P− , otherwise,
where the values σ (u, v), U(u, v), and V(u, v) are obtained from the interface solver Rf·ni, j (see (12)) (σ (u, v), U(u, v), V(u, v)) = Rf·ni, j (u, v) and g˜ i,n j , h˜ i,n j are arbitrary numerical and geometrical fluxes, respectively. For a detailed explanation and proofs in two space dimensions, we refer to [6].
4 Numerical Results In this section, we will present numerical results for the two-dimensional isothermal Euler equations (1) and van der Waals pressure (2) with constants A = 3, B = 13 , θ = 0.85 and R = 83 . This gives us a state space that is separated by an interval, the unstable spinodal region, into two sets, cf. (3). The construction of an interface solver that allows for entropy-compatible sharp-interface solutions has just recently been established in full generality (see [18], and [9] for the framework). Since the details of the construction are not important in the sequel, we skip them referring to [18]. In the following examples, we apply two kinds of interface solvers, namely an exact and an approximate Riemann solver.
4.1 Numerical Results for the One-Dimensional Euler Equations As a first example, we show numerical results, where we apply the scheme from Sect. 3 reduced to the one-dimensional case. In one space dimension, the phase transitions boil down to a single point and the interface tracking consists of tracking the position of a moving cell edge in the computational mesh. The one-dimensional Euler equations are given by
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2
Fig. 3 Numerical solution of the 1-dimensional Euler equations for the initial data (15) at time t = 0.5. We used a mesh with 1000 cells
1.5 1 h h
0.5 0 −0.1
0
0.1
ρt + (ρv)x = 0
0.2
(14)
(ρv)t + (ρv 2 + p(ρ))x = 0,
with fluid density ρ(t, x), scalar velocity v(t, x) (in x-direction), and given pressure function p(ρ). For the interface solver, we take in this example the relaxation Riemann solver from [16]. The relaxation Riemann solver is an approximate Riemann solver and does in general not return the exact Riemann solution but satisfies the properties of an interface solver, see (i),(ii),(iii) on p. 5. Example 1. We check the quality of the scheme by a problem with known exact solution u. It consists of four constant states connect via two shock waves and one phase transition ⎧ (1.7, 1.0992) ⎪ ⎪ ⎪ ⎨(1.8074, 1) u(t, x) = ⎪ (0.3197, 1) ⎪ ⎪ ⎩ (0.2, 0.4903)
: x < s1 t, : s1 t < x < s2 t, : s2 t < x < s3 t, : s3 t < x,
u0 = u(0, ·),
(15)
with wave speeds s1 = −0.574, s2 = 1, and s3 = 1.8515 (all numbers rounded to four digits). Figure 3 depicts the numerical approximation. The experimental orders of convergence (EOC) are presented in Table 1 and show that the EOC tends to 1. This convergence rate might be expected for a single shock wave. Thus, one can conclude that the overall approach works, also with an inexact solver.
4.2 Numerical Results for the Two-Dimensional Euler Equations We will conclude this work with numerical examples for the two-dimensional isothermal Euler equations. We perform the computations on the bounded domain Ω := (−1, 1)2 in all cases. Appropriate boundary conditions are given for each test case.
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Table 1 L 1 errors and EOCs for the numerical scheme for solving the Riemann problem (14), (15). By h(τ ), we denote the mesh width |τ | (uh − u)(t, ·) L 1 h(τ ) EOC 80 160 320 640 1280 2560 5120 10240 20480
2.50 ·10−2 1.25 ·10−2 6.25 ·10−3 3.13 ·10−3 1.56 ·10−3 7.81 ·10−4 3.91 ·10−4 1.95 ·10−4 9.77 ·10−5
0.01895 0.01215 0.00736 0.00430 0.00243 0.00130 0.00067 0.00039 0.00017
0.64 0.72 0.77 0.82 0.90 0.96 0.80 1.20
The realization of the interface tracking and the remeshing operator was done with the 2D triangulation package of the C++ library CGAL [3]. Its Constrained_ Delaunay_Triangulation_2 class in combination with the hierarchy structure was extended to implement the moving mesh. This class manages a triangulation that is almost Delaunay except for a set of given constraints (in our case, prescribed edges of the interface curve) and it provides methods for the insertion, removal, and motion of points. For the computation of the numerical and geometrical fluxes, we will consider the one-dimensional problem (11). It is easy to see that the problem with the three unknown (ρ, ρv1 , ρv2 ) can be rewritten as the one-dimensional Euler equations (14) with density and the projected momentum ρv p = ρv · n = ρ(v1 n 1 + v2 n 2 ) as unknown. In fact, the used interface solver from [18] applies to this system which readily can be used to design an interface solver. Example 2 (Riemann problem). We start with a validation example and take initial conditions that correspond to a one-dimensional Riemann problem, where the entropy-compatible solution is known. At the boundary, we apply absorbing boundary conditions. The entropy solution under investigation consists out of two shock waves with velocities s1 = −1.2960, s3 = 1.5928 and one phase transition with velocity s2 = 0.2185 which is given as ⎧ (0.2, 0, 0) ⎪ ⎪ ⎪ ⎨(0.2646, −0.0837, 0) (ρ(t, x), ρv(t, x)) = ⎪ (1.8094, 0.2538, 0) ⎪ ⎪ ⎩ (1.65, 0, 0)
if x1 < s1 t, if s1 t < x1 < s2 t, if s2 t < x1 < s3 t, if s3 t < x1 .
Figure 4 shows the numerical solution. One can clearly see that the phase transition in the middle is resolved as a sharp vertical line while the two neighboring shock waves are slightly smeared out. With the exact solution, we compute L 1 errors and
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Fig. 4 Numerical solution with two shock waves and one phase transition
Table 2 Experimental orders of convergence (EOC) at time t = 0.1 for the Riemann problem |T (t)| (uh − u)(t, ·) L 1 h(T (t)) EOC(t) 760 3120 12757 50884 204158
2.0645 ·10−1 9.4325 ·10−2 5.3309 ·10−2 2.6078 ·10−2 1.1835 ·10−2
7.0457 ·10−2 3.6295 ·10−2 1.9390 ·10−2 1.1000 ·10−2 6.1603 ·10−3
0.8468 1.0986 0.7928 0.7338
the (experimental) orders of convergence, see Table 2. The errors and the orders listed in Table 2 show that we obtain again convergence with order close to 1. This means that the treatment of the phase transition does not influence the overall order of convergence, since we would already expect a convergence order of 1 for shock waves within one phase. We also check the remeshing routine of the algorithm. We compute the relative change in the number of triangles given as r (t n ) = | Iˆ(n) \ I (n) |/| Iˆ(n) |, where I (n) is the index set of the mesh T (t n ) and Iˆ(n) is the index set of the mesh Tˆ (t n ) that results from the remeshing of T (t n ) such that K i = Kˆ i is valid for all i ∈ I ∩ Iˆ. From Fig. 5, it can be seen that the change is decreasing for finer meshes and, for the three moving meshes depicted in the figure, limited by 1.56%, only. Example 3 (Oscillating Droplet). Next, we continue with an example, which involves more intricate geometric dynamics of the interface. This time, the initial condition consists of a liquid droplet surrounded by the fluid in vapor phase.
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(ρ0 (x), ρ0 v0 (x)) =
(1.7, 0, 0) if x2 < 0.2, (0.3, 0, 0) if x2 > 0.2
At the boundary, we choose reflecting boundary conditions. Figure 6 shows the numerical results that show how the initial bubble emits a wave to the inside and one wave to the outside of the bubble with a lower jump height (see color scale). Example 4 (Vapor bubbles). In the next example, we will consider a couple of vapor bubbles that start emitting waves and oscillate. The emitted waves interact thereby with the sharp phase boundaries, see Fig. 7. For this example, we again choose reflecting boundary conditions. Example 5 (With surface tension). In all previous examples, surface tension was neglected. Surface tension can be modeled via a modified Rankine–Hugoniot condition at the phase boundary given as [[ρ(v · n − σ )]] = 0, [[ρ(v · n − σ )v · n − p(ρ)]] = ζ ∗ κ where σ is the velocity of the phase boundary, ζ ∗ ≥ 0 is a constant surface tension coefficient, and κ = κ(x, t) is the mean curvature of the interface curve. In this
r(t n )
0.0156
T1 , |T1 (0.1)| = 3120 T2 , |T2 (0.1)| = 12757 T3 , |T3 (0.1)| = 50884
0.0073 0.0034 0
5 · 10−2
0.1
0.15
0.2
tn
0.25
Fig. 5 Relative change in the number of triangles of the remeshing routine performed after every time step of the algorithm. The three moving meshes correspond to the meshes 2, 3, and 4 of Table 2
Fig. 6 Numerical solutions for a single droplet surrounded by gas at t = 0, 0.5, and 1
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Fig. 7 Numerical solutions for three vapor bubbles in liquid at t = 0, 0.5, and 1
Fig. 8 Numerical solutions for a deformed droplet at times t = 0, 0.8, and 6
last example, we show numerical results where surface tension acts on a deformed droplet. We use again reflecting boundary conditions. Figure 8 shows how the droplet evolves to an equilibrium circular shape.
References 1. R. Abeyaratne, J.K. Knowles, Kinetic relations and the propagation of phase boundaries in solids. Arch. Ration. Mech. Anal. 114(2), 119–154 (1991) 2. M. de Berg, O. Cheong, M. van Kreveld, M. Overmars, Computational Geometry: Algorithms and Applications, 3rd edn. (Springer, USA, 2008) 3. CGAL Computational Geometry Algorithms Library, http://www.cgal.org 4. C. Chalons, F. Coquel, P. Engel, C. Rohde, Fast relaxation solvers for hyperbolic-elliptic phase transition problems. SIAM J. Sci. Comput. 34(3), A1753–A1776 (2012) 5. C. Chalons, P. Engel, C. Rohde, A conservative and convergent scheme for undercompressive shock waves. SIAM J. Numer. Anal. 52(1), 554–579 (2014) 6. C. Chalons, C. Rohde, M. Wiebe, A finite volume method for undercompressive shock waves in two space dimensions. ESAIM Math. Model. Numer. Anal , 51, 1987–2015 (2017) 7. C. Chen, H. Hattori, Exact Riemann solvers for conservation laws with phase change. Appl. Numer. Math. 94, 222–240 (2015) 8. A. Chertock, S. Karni, A. Kurganov, Interface tracking method for compressible multifluids. M2AN. Math. Model. Numer. Anal. 42(6), 991–1019 (2008) 9. R.M. Colombo, F.S. Priuli, Characterization of Riemann solvers for the two phase p-system. Commun. Partial Differ. Equ. 28(7–8), 1371–1389 (2003)
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10. A. Dressel, C. Rohde, A finite-volume approach to liquid-vapour fluids with phase transition, Finite volumes for complex applications V (ISTE, London, 2008), pp. 53–68 11. S. Fechter, C.D. Munz, C. Rohde, C. Zeiler, A sharp interface method for compressible liquidvapor flow with phase transition and surface tension. J. Comput. Phys. 336, 347–374 (2017) 12. M. Hantke, W. Dreyer, G. Warnecke, Exact solutions to the Riemann problem for compressible isothermal Euler equations for two-phase flows with and without phase transition. Q. Appl. Math. 71(3), 509–540 (2013) 13. P.G. LeFloch, Hyperbolic systems of conservation laws. Lectures in Mathematics ETH Zürich (Birkhäuser, Basel, 2002) 14. P.G. LeFloch, M.D. Thanh, Non-classical Riemann solvers and kinetic relations. II. An hyperbolic-elliptic model of phase-transition dynamics. Proc. R. Soc. Edinb. Sect. A 132(1), 181–219 (2002) 15. C. Merkle, C. Rohde, Computation of dynamical phase transitions in solids. Appl. Numer. Math. 56(10–11), 1450–1463 (2006) 16. C. Rohde, C. Zeiler, A relaxation Riemann solver for compressible two-phase flow with phase transition and surface tension. Appl. Numer. Math. 95, 267–279 (2015) 17. L. Truskinovsky, Kinks versus shocks, in Shock Induced Transitions and Phase Structures in General Media. The IMA Volumes in Mathematics and its Applications, ed. by J. Dunn (Springer, Berlin, 1993), pp. 185–229 18. C. Zeiler, Liquid Vapor Phase Transitions: Modeling, Riemann Solvers and Computation, Ph.D. thesis, Universität Stuttgart (2015)
Arbitrary Lagrangian–Eulerian Discontinuous Galerkin Method for 1D Euler Equations Jayesh Badwaik and Praveen Chandrashekar
Abstract We propose an explicit in time- discontinuous Galerkin scheme on moving grids using the arbitrary Lagrangian–Eulerian approach for one-dimensional Euler equations. The grid is moved with a velocity that is close to the local fluid velocity, which considerably reduces the numerical dissipation in the Riemann solvers. Local grid refinement and coarsening are performed to maintain the mesh quality and avoid very small or large cells. Second-, third-, and fourth-order methods are developed and several test cases are provided to demonstrate the accuracy of the proposed scheme. Keywords Discontinuous Galerkin methods Arbitrary Lagrangian–Eulerian methods · Euler equations · Numerical methods MSC 65M12 · 65M60
1 Introduction Finite volume schemes based on exact or approximate Riemann solvers are able to compute discontinuous solutions in a stable manner, since they have implicit dissipation built into them due to the upwind nature of the schemes. Higher order schemes are constructed following a reconstruction approach combined with a high-order time integration scheme. While formally high- order methods can converge at high rates for smooth solutions, they can still introduce too much numerical dissipation on coarse meshes. Springel [5] gives the example of a Kelvin–Helmholtz instability in which adding a large constant velocity to both states leads to suppression of the instability due to excessive numerical dissipation. This behavior is attributed to the J. Badwaik Department of Mathematics, University of Würzburg, Würzburg, Germany e-mail: [email protected] P. Chandrashekar (B) TIFR Center for Applicable Mathematics, Bangalore, India e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_26
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fact that fixed grid methods based on upwind schemes are not Galilean invariant. Upwind schemes, even when they are formally high- order accurate, are found to be too dissipative when applied to turbulent flows [2], since the numerical viscosity can overwhelm the physical viscosity. The inherent numerical dissipation in upwind schemes can be reduced if the grid moves along with the flow as in Lagrangian methods or arbitrary Lagrangian– Eulerian approach [1], where the mesh velocity can be chosen to be close to the local fluid velocity but maybe regularized to maintain the mesh quality. In [5], the mesh is regenerated after every time step based on a Delaunay triangulation, which allows it to maintain good mesh quality even when the fluid undergoes large shear deformation. However, these methods have been restricted to second- order accuracy. Traditionally, ALE methods have been used for problems involving moving boundaries as in wing flutter, store separation, and other problems involving fluid– structure interaction. Another class of methods solves the PDE on moving meshes where the mesh motion is determined based on a monitor function which is designed to detect regions of large gradients in the solution, see [6] and the references therein. These methods achieve automatic clustering of grid points in regions of large gradients. ALE schemes have been used to compute multi-material flows as in [3], since they are useful to accurately track the material interface. In the present work, we consider the one- dimensional problem and propose an explicit discontinuous Galerkin scheme that is conservative on moving meshes and automatically satisfies the geometric conservation law. The scheme is a single step method, which is achieved by using a predictor. Numerical results show the dramatic improvement in resolving discontinuities, especially contact waves. Apart from the geometric complexity, the proposed scheme can be extended to multiple dimensions.
2 Euler Equations The Euler equations are a hyperbolic system of conservation laws for mass, momentum, and energy, and can be written as ∂ u ∂ f (u) + =0 ∂t ∂x
(1)
where u is called the vector of conserved variables and f (u) are the corresponding fluxes given by ⎡ ⎤ ⎡ ⎤ ρ ρv u = ⎣ρv ⎦ , f (u) = ⎣ p + ρv2 ⎦ E ρHv In the above expressions, ρ is the density, v is the velocity, p is the pressure, and E is the total energy per unit volume, which for an ideal gas is given by E = p/(γ −
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t
Fig. 1 Space–time cell
xn+1 j− 1
xn+1 j+ 1
Cjn+1
2
dx dt
2
n = wj−1/2
xnj− 1
dx dt
Cjn
2
xnj+ 1 2
tn+1
n = wj+1/2
tn x
1) + ρv2 /2, with γ > 1 being the ratio of specific heats at constant pressure and volume, and H = (E + p)/ρ is the enthalpy.
3 Mesh and Solution Space Consider a partition of the domain into disjoint cells with the jth cell being denoted by C j (t) = (x j− 21 (t), x j+ 21 (t)). As the notation shows, the cell boundaries are time dependent which means that the cell is moving in some specified manner. The time levels are denoted by tn with the time step Δtn = tn+1 − tn . The boundaries of the cells move with a constant velocity in the time interval (tn , tn+1 ) given by w j+ 21 (t) = wnj+ 1 , 2
tn < t < tn+1
which defines a cell in space–time as shown in Fig. 1. The algorithm to choose the mesh velocity wnj+ 1 is explained in a later section. Let w(x, t) be the continuous linear 2 interpolation of the mesh velocity. We approximate the solution of the conservation law by piecewise polynomials, which are allowed to be discontinuous across the cell boundaries. For a given degree k ≥ 0, the solution in the jth cell is given by uh (x, t) =
k
u j,m (t)ϕm (x, t),
x ∈ C j (t)
m=0
where {u j,m ∈ R3 , 0 ≤ m ≤ k} are the degrees of freedom associated with the jth cell. The basis functions ϕm are defined in terms of Legendre polynomials by mapping to a reference cell.
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4 Discontinuous Galerkin Method Define the ALE flux as g(u, w) = f (u) − wu. The weak formulation after performing an integration by parts in the x variable, leads to x 1 (t) d x j+ 21 (t) ∂ j+ 2 uh (x, t)ϕl (x, t)dx = g(uh , w) ϕl (x, t)dx dt x 1 (t) ∂ x x j− 1 (t) j− 2
2
+ gˆ j− 1 (uh (t))ϕl (x +
j− 21
2
, t) − gˆ j+ 1 (uh (t))ϕl (x − 1 , t) j+ 2 2
where we have introduced the numerical flux gˆ j+ 21 (uh (t)) = gˆ (u−j+ 1 (t), u+j+ 1 (t), w j+ 21 (t)) 2
2
which provides an approximation to the ALE flux. The above scheme has an implicit nature since the unknown solution uh appears on the right-hand side integrals, whereas we only know the solution at time tn . In order to obtain an explicit scheme, we assume that we have available with us a predicted solution U h in the time interval (tn , tn+1 ), which is used in the time integrals to obtain an explicit scheme. Moreover, the integrals are computed using quadrature in space and time leading to the fully discrete scheme n+1 n n h n+1 j u j,l = h j u j,l + Δtn
+Δtn
r
θr h j (τr )
ηq g(U h (xq , τr ), w(xq , τr ))
q
∂ ϕl (xq , τr ) ∂x
θr [ gˆ j− 21 (U h (τr ))ϕl (x +j− 1 , τr ) − gˆ j+ 21 (U h (τr ))ϕl (x −j+ 1 , τr )]
r
2
2
where θr are weights for time quadrature and ηq are weights for spatial quadrature. In practice, the integrals are computed by mapping the cell to the reference cell, and the basis functions and its derivatives are also evaluated on the reference cell.
4.1 Mesh Velocity The mesh velocity must be close to the local fluid velocity in order to have a Lagrangian character to the scheme. Since the solution is discontinuous, there is no unique fluid velocity at the mesh boundaries. Some researchers, especially in the context of Lagrangian methods, solve a Riemann problem at the cell face to determine the face velocity. Since we use an ALE formulation, we do not require the exact velocity and in our work, we make a simple choice which is to take an average of the two velocities at every cell face
Arbitrary Lagrangian–Eulerian Discontinuous Galerkin Method …
w˜ nj+ 1 = 2
327
1 [v(x −j+ 1 , tn ) + v(x +j+ 1 , tn )] 2 2 2
We will also perform some smoothing of the mesh velocity, e.g., the actual face velocity is computed from wnj+ 1 = 2
1 n (w˜ 1 + w˜ nj+ 1 + w˜ nj+ 3 ) 2 2 3 j− 2
Note that our algorithm to choose the mesh velocity is very local and hence easy and efficient to implement as it does not require the solution of any global problems.
5 Computing the Predictor The predicted solution is used to approximate the flux integrals over the time interval (tn , tn+1 ) and the method to compute this must be local, i.e., it must not require solution from neighboring cells. For a second-order scheme, a Taylor expansion retaining only linear terms in t and x is sufficient. For higher order schemes, we adopt the approach of continuous explicit Runge–Kutta (CERK) schemes [4] to approximate the predictor. Let us choose a set of (k + 1) distinct nodes, e.g., Gauss– Legendre or Gauss–Lobatto nodes, which uniquely define the polynomial of degree k. These nodes are moving with velocity w(x, t), so that the time evolution of the solution at node xm is governed by ∂ dU m = −[A(U m (t)) − wm (t)I ] U h (xm , t) =: K m (t) dt ∂x with initial condition U m (tn ) = uh (xm , tn ). Using a Runge–Kutta scheme of sufficient order, we will approximate the solution at these nodes as U m (t) = uh (xm , tn ) +
ns
bs ((t − tn )/Δtn )K m,s , t ∈ [tn , tn+1 ), m = 0, 1, . . . , k
s=1
where K m,s = K m (tn + θs Δtn ), θs Δtn is the stage time and bs are certain polynomials related to the CERK scheme.
6 Positivity Property The solutions of Euler equations are well defined only if the density and pressure are positive quantities. This is not a priori guaranteed by the DG scheme even when the TVD limiter is applied. In the case of Runge–Kutta DG schemes, a positivity limiter
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has been developed in [7] which preserves accuracy in smooth regions. This scheme is built on a positive first- order finite volume scheme. For the first- order ALE-DG scheme using Rusanov flux, we can show positivity property provided the time step satisfies ⎧ ⎫ n ⎨ (1 − 1 β)h n ⎬ βh j j Δtn ≤ Δtn(1) := min 1 n 2 n , n (2) n j ⎩ (λ 1 + λ 1 ) |w 1 − w 1 | ⎭ 2 j+ j− j− 2
j+ 2
2
2
Here, β ∈ (0, 1) is the maximum allowed change in cell size during one time step relative to the previous size. Theorem 1. The scheme first-order ALE-DG scheme with Rusanov flux is positivity preserving if the time step condition (2) is satisfied. Remark 1. In the computations, we use the positivity preserving limiter of [7], which leads to robust schemes which preserve the positivity of the cell average value in all the test cases. Remark 2. An important property of schemes on moving meshes is their ability to preserve constant states for any mesh motion. This is related to the conservation of cell volumes in relation to the mesh motion. In our scheme, if we start with a constant state unh = c, then we can prove that the solution remains constant.
7 Grid Coarsening and Refinement The size of the cells can change considerably during the time evolution process due to the near Lagrangian movement of the cell boundaries. Near shocks, the cells will be compressed to smaller sizes which will reduce the allowable time step since a CFL condition has to be satisfied. In some regions, e.g., inside expansion fans, the cell size can increase considerably which may lead to loss of accuracy. In order to avoid too small or too large cells from occurring in the grid, we implement cell merging and refinement. If a cell becomes smaller than some specified size h min , then it is merged with one of its neighboring cells and the solution is transferred from the two cells to the new cell by performing an L 2 projection. If a cell size becomes larger than some specified size h max , then this cell is refined into two cells by division and the solution is again transferred by L 2 projection.
8 Numerical Results The numerical tests are performed with polynomials of degree one, two, and three, together with the linear Taylor expansion, two-stage CERK and four-stage CERK, respectively, for the predictor. For the quadrature in time, we use the midpoint rule,
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Table 1 Order of accuracy study on static mesh using Rusanov flux N k=1 k=2 k=3 Error Rate Error Rate Error 100 200 400 800 1600
4.370E-02 6.611E-03 1.332E-03 3.151E-04 7.846E-05
2.725 2.518 2.372 2.280
3.498E-03 4.766E-04 6.415E-05 8.246E-06 1.031E-06
2.876 2.885 2.910 2.932
3.883E-04 1.620E-05 9.376E-07 5.763E-08 3.595E-09
Rate 4.583 4.347 4.239 4.180
two- and three-point Gauss–Legendre quadrature, respectively. High-order schemes for hyperbolic equations suffer from spurious numerical oscillations when discontinuities or large gradients are present in the solution which cannot be accurately resolved on the mesh. To control these oscillations, we use standard TVD and TVB limiters applied to characteristic variable and account for nonuniform meshes. The time step is chosen based on Eq. (2), Δtn =
CFL Δt (1) 2k + 1 n
where the factor (2k + 1) comes from linear stability analysis, and in most of the computations we use CFL = 0.9. In all the solutions’ plots given below, symbols denote the cell average value.
8.1 Order of Accuracy We study the convergence rate of the schemes by applying them to a problem with a known smooth solution. The initial condition is taken as ρ(x, 0) = 1 + exp(−10x 2 ),
u(x, 0) = 1,
p(x, 0) = 1
whose exact solution is ρ(x, t) = ρ(x − t, 0), u(x, t) = 1, p(x, t) = 1. The initial domain is [−5, +5] and the final time is t = 1 units. The L 2 norm of the error in density are shown in Table 1 for the static mesh and in Table 2 for the moving mesh. In each case, we see that the error behaves as O(h k+1 ), which is the optimal rate for smooth solutions.
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Table 2 Order of accuracy study on moving mesh using Rusanov flux N k=1 k=2 k=3 Error Rate Error Rate Error 100 200 400 800 1600
2.331E-02 6.139E-03 1.406E-03 3.375E-04 8.278E-05
1.925 2.0258 2.0366 2.0344
3.979E-03 4.058E-04 5.250E-05 6.626E-06 8.304E-07
(a)
3.294 3.122 3.077 3.057
8.633E-04 1.185E-05 7.079E-07 4.340E-08 2.689E-09
6.186 5.126 4.760 4.573
(b)
1.0
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Rate
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Fig. 2 Sod problem using Roe flux, 100 cells, and TVD limiter: a static mesh b moving mesh
8.2 Sod Problem The initial condition for the Sod test case is given by (ρ, v, p) =
(1.0, 0.0, 1.0) (0.125, 0.0, 0.1)
if x < 0.5 if x > 0.5
and the solution is computed upto a final time of T = 0.2, where the domain is [0,1]. Since the fluid velocity is zero at the boundary, the computational domain does not change with time for the chosen final time. In Fig. 2, we show the results obtained using Roe flux with 100 cells and TVD limiter on static and moving mesh. The contact wave is considerably well resolved on the moving mesh as compared to the static mesh. To study the Galilean invariance or the dependence of the solution on the choice of coordinate frame, we add a boost velocity of V = 10 or V = 100 to the coordinate frame, while implies the initial fluid velocity if v(x, 0) = V and the other quantities remain as before. The results given in Fig. 3b clearly show the independence of the results on the moving mesh. Figure 3a shows that the accuracy of the static mesh results degrades with an increase in velocity of the coordinate frame, particularly the contact discontinuity is highly smeared.
Arbitrary Lagrangian–Eulerian Discontinuous Galerkin Method …
(a) 1.4
(b) 1.4
1.2
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331
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1.2
Fig. 3 Effect of coordinate frame motion on Sod problem using Roe flux, 100 cells, and TVD limiter: a static mesh b moving mesh
8.3 Shu–Osher Problem The initial condition is given by (ρ, v, p) =
(3.857143, 2.629369, 10.333333) if x < −4 (1 + 0.2 sin(5x), 0.0, 1.0) if x > −4
which involves a smooth sinusoidal density wave which interacts with a shock. The domain is [−5, +5] and the solution is computed upto a final time of T = 1.8. The solutions are shown in Fig. 4a, b on static and moving meshes using 200 cells and TVD limiter. The moving mesh scheme is considerably more accurate in resolving the sinusoidal wave structure that arises after interaction with the shock. In Fig. 4c, we compute the solution on static mesh with TVB limiter and the parameter M = 100 is used. In this case, the solutions on static mesh are more accurate compared to the case TVD limiter but still not as good as the moving mesh results. The moving mesh results have more than 200 cells in the interval [−5, +5] at the final time since cells enter the domain from the left side. Hence, in Fig. 4d, we show the static mesh results with 300 cells and using TVB limiter. The results are further improved but still not as accurate as the moving mesh case. The choice of parameters in the TVB limiter is very heuristic and hence it is still advantageous to use the moving mesh scheme which gives improved solutions even with TVD limiter.
8.4 Low-Density Problem The initial condition is given by
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(a) 5.0
(b) 5.0
4.5
4.5
4.0
4.0
3.5
3.5
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−3
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0
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5
(c) 5.0
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−2
−1
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1
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−3
−3
−2
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0
1
2
3
4
5
0.5 −4
Fig. 4 Shu–Osher problem using Roe flux: a static mesh, 200 cells, M = 0 b moving mesh, 200 cells, M = 0 c static mesh, 200 cells, M = 100 d static mesh, 300 cells, M = 100
(ρ, v, p) =
(1.0, −2.0, 0.4) x < 0.5 (1.0, +2.0, 0.4) x > 0.5
The computational domain is [0,1] and the final time is T = 0.15. The density using 100 cells is shown in Fig. 5 with static and moving meshes. The mesh motion does not significantly improve the solution. On the contrary, the mesh becomes rather coarse in the expansion region, though the solution is still accurate. By enabling grid refinement when h > 0.05, we obtain the result shown in Fig. 6, where better resolution of the low-density region is obtained.
8.5 Blast Problem The initial condition is given by
Arbitrary Lagrangian–Eulerian Discontinuous Galerkin Method …
(a)
(b)
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Fig. 5 Low-density problem using HLLC flux: a static mesh, b moving mesh
(a)
(b)
1.0
1.0 0.8
0.8
0.6
0.6
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0.0 0.0
0.2
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moving mesh
0.8
1.0
0.0 0.0
0.2
0.4
moving mesh + adaptation 108 cells at final time
Fig. 6 Low-density problem using HLLC flux with 100 cells and moving mesh a no adaptation b with adaptation
⎧ ⎪ ⎨(1.0, 0.0, 1000.0) x < 0.1 (ρ, v, p) = (1.0, 0.0, 0.01) 0.1 < x < 0.9 ⎪ ⎩ (1.0, 0.0, 100.0) x > 0.9 and the final time is T = 0.038. As shown in Fig. 7, static mesh results suffer from too much dissipation, especially, in the contact wave, while the moving mesh is able to resolve this more accurately. Since some cells can become very small in this problem, we have enabled mesh coarsening whenever h < 0.001 for any cell.
9 Summary We have developed an explicit DG scheme on moving meshes using ALE framework and space–time expansion of the solutions within each cell. The near Lagrangian nature of the mesh motion dramatically reduces the numerical dissipation especially
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7
6
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0 0.4
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0.8
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moving mesh 303 cells at final time
Fig. 7 Blast problem using HLLC flux with 400 cells on static and moving meshes
for contact waves. The scheme is shown to yield superior results in the presence of large boost velocity of the coordinate system indicating its Galilean invariance property. The mesh motion provides automatic grid adaptation near shocks, but may lead to very coarse cells inside expansion waves. A grid adaptation strategy is developed to handle the problem of very small or very large cells. The proposed methodology is general enough to be applicable to other systems of conservation laws modeling fluid flows. Acknowledgements The authors gratefully acknowledge the financial support received from the Airbus Foundation Chair on Mathematics of Complex Systems established in TIFR-CAM, Bangalore, for carrying out this work.
References 1. J. Donea, A. Huerta, J.-P. Ponthot, A. Rodríguez-Ferran, Arbitrary Lagrangian–Eulerian Methods (Wiley, New York, 2004) 2. E. Johnsen, J. Larsson, A.V. Bhagatwala, W.H. Cabot, P. Moin, B.J. Olson, P.S. Rawat, S.K. Shankar, B. Sjögreen, H.C. Yee, X. Zhong, S.K. Lele, Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves. J. Comput. Phys. 229(4), 1213–1237 (2010) 3. H. Luo, J.D. Baum, R. Löhner, On the computation of multi-material flows using ALE formulation. J. Comput. Phys. 194(1), 304–328 (2004) 4. B. Owren, M. Zennaro, Derivation of efficient, continuous, explicit runge-kutta methods. SIAM J. Sci. Stat. Comput. 13(6), 1488–1501 (1992) 5. V. Springel, E pur si muove: Galilean-invariant cosmological hydrodynamical simulations on a moving mesh. Mon. Not. R. Astron. Soc. 401(2), 791–851 (2010) 6. H. Weizhang, R.D. Russell, Adaptive Moving Mesh Methods, Applied Mathematical Sciences (Springer, Berlin, 2011) 7. X. Zhang, C.-W. Shu, On positivity-preserving high order discontinuous galerkin schemes for compressible euler equations on rectangular meshes. J. Comput. Phys. 229(23), 8918–8934 (2010)
A Runge–Kutta Discontinuous Galerkin Scheme for the Ideal Magnetohydrodynamical Model Praveen Chandrashekar, Juan Pablo Gallego-Valencia and Christian Klingenberg
Abstract A numerical scheme for solving the system of ideal Magnetohydrodynamics (MHD) model, using an explicit high-order Runge–Kutta Discontinuous Galerkin method (RKDG) is proposed. An entropy stable numerical flux introduced in the context of Finite Volume (FV) method in Chandrashekar and Klingenberg (SIAM J Numer Anal, 2016, [4]) is used in the RKDG scheme. To illustrate the usefulness of the implementation, some specific test cases for the ideal Magnetohydrodynamics model (MHD equations) are shown. Keywords Ideal MHD · Discontinuous Galerkin · Entropy stability
1 Introduction As a part of the EXAMAG project of the SPPEXA priority project funded by the DFG, a Runge–Kutta Discontinuous Galerkin (RKDG) scheme is proposed in order to solve the system of ideal Magnetohydrodynamics (MHD) equations, as an extension to the previous results published in [7] which showed a discontinuous Galerkin scheme to the compressible Euler equations of gas dynamics. Some techniques used to control oscillations near discontinuities are also available in this work, for example the limiting procedure used in [7] and the shock indicator criteria introduced in [6, 10]. In order to ensure the divergence-free condition on the magnetic field, the Godunov P. Chandrashekar Tata Institute for Fundamental Research, Sharada Nagar, Chikkabommsandra, Bangalore 560065, Karnataka, India e-mail: [email protected] J. P. Gallego-Valencia · C. Klingenberg (B) University of Würzburg, Campus Hubland Nord, Emil-Fischer-Strae, 40 97074 Würzburg, Germany e-mail: [email protected] J. P. Gallego-Valencia e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_27
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symmetrization procedure is followed [2], modifying the MHD system by adding the so-called Powell terms [12]. Two different numerical fluxes are used in the DG scheme, a local Lax–Friedrichs flux and an entropy stable flux introduced in [4]. Numerical test cases are presented to show the performance of the scheme through convergence rates and a direct comparison to the Athena Code [8].
2 Ideal MHD Equations The ideal magnetohydrodynamics model, or ideal MHD equations, is a system of equations which describes the conservation of mass, momentum, energy and magnetic field of a particular fluid. This system can be written in two dimensions as the conservation law system (1) qt + ∇x · F(q) = 0, with
q = [ρ, m, B, e]T , F (q) = f1 (q), f2 (q) ,
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ f1 = ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
⎡ ⎤ ⎤ m2 m1 m m 2 1 ⎢ ⎥ − B2 B1 ⎥ ρ + p − B12 ⎢ ⎥ ⎥ ρ ⎢ ⎥ 2 m1m2 ⎥ m2 2 ⎢ ⎥ − B1 B2 + p − B2 ⎥ ρ ρ ⎢ ⎥ ⎥ m1m3 m m ⎢ ⎥ 2 3 − B1 B3 ⎥ − B B 2 3 ρ ⎢ ⎥ ρ ⎥ f2 = ⎢ ⎥ m2 m1 ⎥ 0 B − B ⎢ ⎥ 1 2 ρ ρ ⎥ m2 m1 ⎢ ⎥ ⎥ B2 − B1 ρ ⎢ ⎥ 0 ρ ⎥ ⎢ ⎥ m3 m1 m3 m2 ⎦ B3 − B1 ρ B − B ⎣ ⎦ 3 2 ρ ρ ρ B2 m2 (E + p) − Bρ1 (m · B) + p) − · B) . (E (m ρ ρ
(2)
m 21
m1 ρ
(3)
where ρ is the density, m = [m 1 , m 2 , m 3 ]T is the momentum vector, p is the pressure, E is the total energy and B = [B1 , B2 , B3 ]T is the vector of magnetic components in each space dimension. Since an additional relation is needed to close the system, the following equation of state is used, where the total energy is a function of the thermal pressure, the kinetic energy and the magnetic pressure E=
1 1 p |m|2 + |B|2 . + γ − 1 2ρ 2
Here, γ is the adiabatic constant dependent on the type of gas. Finally, the magnetic field of this system of equations should satisfy the divergence free condition ∇ · B = 0,
(4)
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By calculating the Jacobian matrices A1 and A2 from f1 and f2 respectively, one realizes that they are singular, which means that the system is weakly hyperbolic. The set of eigenvalues for a Jacobiam matrix Aκ , are found to be • a zero eigenvalue λ0 = 0, • one entropy wave with speed λe = u κ , • two Alfvèn waves with speeds λa± = u κ ± • four magneto acoustic waves with speeds
Bκ √ , and ρ ± λ f,s = u κ
± c f,s ,
where u = [u 1 , u 2 , u 3 ]T is the vector of speeds which can be calculated by dividing each momentum component m κ by the density ρ. Finally, c f and cs stand for the fast and slow speeds given by c2f,s = Here, a = b2 =
1 ρ
γp ρ
1
2 1 2 a + b2 ± a 2 + b2 − 4a 2 (b · n)2 . 2 2
is the sound speed in the fluid and b :=
√1 B, ρ
which means that
|B| and n is a normal vector. 2
Divergence-Free Condition ∇ · B = 0 In order to deal with the divergence-free condition, a method based on the Godunov symmetrization of the MHD system shown in [2] will be used. This method arrives to the same modified system of equations published by Powell [12]. It modifies the eigensystem in primitive variables by replacing the zero eigenvalue (the κth-magnetic component) in each of the Jacobian matrices Aκ by a second advection wave with speed λ = u κ . The system becomes hyperbolic and the Jacobian matrices are no longer singular. This is done by adding the following source term ϕ (q) ∇ · B to the conservation form of the MHD equations as follows: ∂t q + ∇x · F(q) + ϕ (q) ∇x · B = 0,
(5)
where ϕ (qh ) = [0, B, u, u · B]T . The resulting system is non-conservative, but the additional terms are multiples of ∇ · B, so when its initial condition is zero it will be zero for any time for the exact solution of the PDE. Numerically, it is expected that these terms should remain very small, at least for smooth solutions.
3 Discontinuous Galerkin Semi-discrete Scheme for the MHD System The domain Ω is discretized by a tessellation T over the set Ω. The tessellation used here is a Cartesian grid with the following characteristics
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N • Ω = =1 τ , • each τ ∈ T is a closed set and the interior of τ o is non-empty, • τio ∩ τ jo = ∅, ∀i = j, τi , τ j ∈ T . As a result, and assuming that q is the solution of the Initial Value Problem (IVP)
∂t q + ∇x · F (q) + ϕ (q) ∇x · B = 0, q(0, x) = q0 (x),
(6)
in the cell τ ⊂ Ω, the solution q(t, x) to the PDE in the whole domain Ω can be written as the sum of the local solutions from each cell τ of the tessellation T . That is N 1 if x ∈ τ q (t, x)χ (x), where χ (x) = q(t, x) = 0 otherwise =0 Let Vhk be a test function space, defined as Vhk = {φ ∈ L p (D) : φ|τ ∈ Qk (τ ), ∀ = 1, ..., N }, where Qk (τ ) is the space of tensor product Legendre polynomials of degree k in the cell τ . By writing the approximated solution qh as a linear combination of functions φ(x, y) ∈ Vhk . 2 (k+1) qh = q˜ j (t)φ ( j) (x, y). j=1
Then, in order to solve the IVP in Eq. (6), the following discontinuous Galerkin (DG) semi-discrete scheme is proposed ∀τ ∈ T
∂t qh vh − ∇vh · (f(qh ) + g(qh )) + vh ϕ (qh ) ∇ · B dx τ
− + 1 − + − H qh , qh , n + ϕ qh (B − B ) · n vh− dσ = 0, 2 ∂τ
(7)
where H qh− , qh+ , n is a consistent numerical flux, n is a vector normal to the interface in ∂τ , where the numerical flux is computed, q− and q+ are, respectively, the values of qh from the inside and outside of the cell τ at each interface ∂τ . Two different numerical fluxes were used in this work, the first one is a local Lax–Friedrichs-type (LXF) numerical flux of the form. H (q+ , q−, n) =
λmax,τ + 1
F(q+ ) + F(q− ) − q − q− 2 2
(8)
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where λmax,τ is the maximum eigenvalue of the Jacobian matrix of system at the cell τ . The second numerical flux used was introduced in [4] and it was initially designed for a Finite Volume entropy stable scheme, and its general form is given by H (q− , q+ , n) = H ∗ (q− , q+ , n) −
1 D(q− , q+ )(q+ − q− ), D = D T ≥ 0 2
(9)
where H ∗ (q− , q+ , n) components are defined as ˆ n H1∗ = ρu H2∗ =P ∗ n 1 + u 1 H1∗ − B n B 1 ,
P∗ =
ρ 2β
1 + |B|2 2
H3∗ =P ∗ n 2 + u 1 H1∗ − B n B 2 , H4∗ =P ∗ n 3 + u 1 H1∗ − B n B 3 , 1
H5∗ = βu n B 1 − βu 1 B n β 1
H6∗ = βu n B 2 − βu 2 B n β 1
H7∗ = βu n B 3 − βu 3 B n β 1 1 − |u|2 H1∗ + u 1 H2∗ + u 2 H3∗ + u 3 H4∗ H8∗ = 2 (γ − 1)βˆ
1 + B1 H5∗ + B2 H6∗ + B3 H7∗ − u n |B|2 + u 1 B1 + u 2 B2 + u 3 B3 Bn 2 Here, u n = u · n and Bn = B · n denote the normal component of the vectors u and B to the interface for which the numerical flux is being computed, the values n i are the components of the normal vector n, the operator (·) denotes the arithmetic ˆ denotes average between the values at both sides of the interface and the operator (·) the logarithmic average denoted respectively by η=
1 − (η + η+ ), 2
ηˆ =
η+ − η− . ln(η+ ) − ln(η− )
The dissipation matrix D in (9) is a semi-definite positive matrix similar to the dissipation used in a Roe-type scheme. It is calculated using the scaled right eigenvectors matrix R˜ of symmetrization procedure to transform from conserved to entropy variables shown in [2, 4], as follows ˜ R˜ −1 D = R
(10)
These quantities were introduced by Chandrashekar and Klingenberg to ensure the entropy stability of the finite volume scheme presented in [4].
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This work will investigate the results using an explicit strong stability preserving (SSP) third-order Runge–Kutta (RK) time integration (as the one used in [5] and [7]) when it is applied to an entropy stable scheme designed at the semi-discrete level. Limiting Procedure For discontinuous solutions, it is necessary to use some limiting procedure to ensure non-oscillatory solutions. TVD-type limiters have been implemented inside the code. It has been pointed out in [7] the advantage in limiting over the characteristic variables instead of the conserved variables. It is basically a procedure in which the Jacobian matrices are diagonalized, then the limiting procedure is applied over the characteristic variables of the system and finally the system is transformed back to the conserved variables. The diagonalization procedure used in this work, was introduced by Bristo and Wu in [3], and it was also used in [9]. With other eigensystems, it may happen that the eigenvectors develop singularities in points where the eigenvalues degenerate. On the other hand, the eigensystem introduced in [3] has a proper choice of normalization, which avoids singularities in the eigenvectors and guarantees a complete set of eigenvectors. Additional to the limiting procedures implemented in the code, a shock indicator introduced in [6, 10] was also implemented and used in order to detect the places where the limiting procedure has to be applied. This shock indicator is based on the fact that for smooth regions the DG solution shows super-convergent approximation at the outflow boundaries of the cells. As the super-convergence is lost one can determine that the cell contains a discontinuous solution. The indicator criteria is defined as | ∂τ − (rh− − rh+ )dσ | > 1, (11) k+1 h 2 |∂τ − | rh τ were r is either a determined variable (for example the density or the energy) or a function of them (like the entropy), ∂τ − represent the set of outflow interfaces of the element and h is a characteristic length of the element.
4 Numerical Implementation The results presented in this work are an extension of the work in [7], which simulations were performed with the dflo code. The dflo code is an application based in the deal.II C++ libraries [1], and was initially developed to solve the compressible Euler equations of gas dynamics, and now it can solve the MHD equations with the following characteristics
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• • • •
works on a Cartesian grid, uses explicit time integration with third-order accuracy (SSP Shu scheme), uses tensor product Legendre polynomial basis, can use average gradient limiter (on the characteristic or the conserved variables), described in [7], • can use the KXRCF shock indicator based on super-convergence, introduced in [6, 10], • for the MHD equations, either a local Lax–Friedrichs (LXF) or an entropy stable (ES) numerical flux described in [4] can be used.
4.1 Polarized Alfvèn Wave Test Case The first MHD test case presented in this work is the Polarized Alfvèn wave, which is described by Tóth in [13], however, the setup of the implementation was taken from [8] with a different orientation. √ √ On a rectangular domain 0, 25 × 0, 5 , a grid of n × 2n square cells is used with periodic boundary conditions. The Alfvèn wave is fixed to propagate in the direction tan−1 ( 21 ) ≈ 26.6◦ with respect to the x axis. The density and pressure are constants fixed to ρ = 1 and p = 0.1. Then, the values of the magnetic field are given by B = 1, B⊥ = 0.1 sin(2π x ) and B3 = 0.1 cos(x ) and the velocities are given by u = 0.1, u ⊥ = 0.1 sin(2π x ) and u 3 = 0.1 cos(2π x⊥ ). In Cartesian coordinates, the expressions for the velocity components are u 1 = u cos(θ) − u ⊥ sin(θ), u 2 = u sin(θ) + u ⊥ cos(θ), u 3 = 0.1 cos(θ0 )
(12)
and for the magnetic field components are B1 = B cos(θ ) − B⊥ sin(θ ), B2 = B sin(θ ) + B⊥ cos(θ ), B3 = 0.1 cos(θ0 ) (13) where θ = tan−1 ( 21 ) and θ0 = 2π(x1 cos(θ ) + x2 sin(θ )). Due to the periodicity of the test case, the solution will return to the initial state every t = 1.0. This test case was simulated with two different numerical fluxes, and four different grid sizes. Since the solution is smooth and periodic, convergence rates where computed for t = 5 (Table 1). It can be seen that for Q1 polynomial basis the results are very similar for both numerical fluxes, however for Q2 the errors produced by the ES numerical flux are considerably smaller than those from the LXF flux for both time levels. Finally, Table 2 shows the L ∞ -errors and convergence rates for the ∇ · B = 0 condition at t = 5.0. It can be seen that for k = 1 there are not significant differences between the use of the LXF flux and the ES flux. However, for Q2 the L ∞ -errors are reduced considerably.
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Table 1 L 2 - errors of the polarized Alfvèn test case at t = 5.0 using four different grid sizes, Q1 and Q2 polynomial basis, and both the LXF and the ES numerical fluxes n LXF ES L2 -error Order L2 -error Order Q1
Q2
32 64 128 256 32 64 128 256
7.83E-04 1.60E-04 5.71E-05 7.59E-06 1.58E-04 1.36E-05 9.56-07 6.66E-08
– 2.29 2.10 2.03 – 3.54 3.83 3.84
9.30E-04 1.90E-04 4.47E-05 1.10E-05 2.04E-05 1.49E-06 1.56E-07 1.79E-08
– 2.29 2.09 2.02 – 3.78 3.26 3.12
Table 2 L ∞ -errors for the ∇ · B = 0 condition of the Polarized Alfvèn wave test case at t = 5.0, using Q1 and Q2 polynomial basis, the LXF and the ES flux, and four different grid sizes n LXF ES L∞ -error Order L∞ -error Order Q1
Q2
32 64 128 256 32 64 128 256
3.76E-02 1.88E-02 9.41E-03 4.71E-03 1.72E-03 3.89E-04 1.83E-04 7.88E-05
– 1.00 1.00 1.00 – 2.14 1.09 1.22
5.12E-02 2.49E-02 1.23E-02 6.16E-03 1.40E-03 3.95E-04 9.68E-05 2.41E-05
– 1.04 1.02 1.00 – 1.83 2.03 2.01
4.2 Orszag–Tang Vortex Test Case This test case was first introduced by Orszag and Tang in [11], and it has become a reference test case to validate numerical solutions for the MHD system. The setup of the test case is the following, in a square domain of [0, 1] × [0, 1] with periodic 5 and density ρ = boundary conditions a gas (γ = 53 ) has constant pressure p = 12π 25 . The initial speeds are u 1 = sin(2π x2 ) and u 2 = sin(2π x1 ), and the initial values 36π of the magnetic field are given by B1 = −B0 sin(2π x2 ) and B2 = B0 sin(4π x1 ), where B0 = √14π . The ES numerical flux was used for this test case, the limiting procedure used was the Qk polynomial basis limiter introduced in [7], and also the KXRCF shock indicator will be activated only using the total energy (E) as indicator variable. The indicator is seen to be activated in regions of discontinuous solutions (Fig. 1). The numerical solution is contrasted with the results of the Athena code in Figs. 2 and 3, which shows good agreement between the two schemes.
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Fig. 1 (Left) Density values of the Orszag–Tang test case at t = 0.5. (Right) Trouble cells flagged 1 by the shock indicator. ES numerical flux, Q2 polynomial basis, grid size x1 = x2 = 512
Fig. 2 Cut comparison (y = 0.3125) of the third-order test case solution computed using the dflo code for two different grid sizes (h = {1/256, 1/512}) using the ES numerical flux, and the results from the Athena code for h = 1/512
Fig. 3 Cut comparison (y = 0.4277) of the third-order test case solution computed using the dflo code for two different grid sizes (h = {1/256, 1/512}) using the ES numerical flux, and the results from the Athena code for h = 1/512
The results using the ES numerical flux are quite close to the solution of the Athena code, but sometimes a bit more dissipative. The implementation presented here uses the Powell terms, which are known to add some dissipation to the scheme. An improvement can be done by using locally divergence-free basis, in which case the volume Powell terms are absent and this will reduce the dissipation.
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5 Conclusions An explicit numerical scheme was proposed to solve the system of MHD equations. An entropy stable numerical flux designed initially for finite volumes was used in the context of DG. The numerical result shows the expected order of approximation for smooth solutions and a performance comparable to well-known methods as the Athena Code. Acknowledgements J.P. G.-V. thanks the GRK 1147 and the DAAD STIBET fellowship at the University of Würzburg for their support.
References 1. W. Bangerth, D. Davydov, T. Heister, L. Heltai, G. Kanschat, M. Kronbichler, M. Maier, B. Turcksin, D. Wells, The deal.II library, version 8.4. J. Numer. Math. 24 (2016) 2. T.J. Barth, Numerical methods for gasdynamic systems on unstructured meshes, An Introduction to Recent Developments in Theory and Numerics for Conservation Laws (Springer, Berlin, 1999), pp. 195–285 3. M. Brio, C.C. Wu, An upwind differencing scheme for the equations of ideal magnetohydrodynamics. J. Comput. Phys. 75(2), 400–422 (1988) 4. P. Chandrashekar, C. Klingenberg, Entropy stable finite volume scheme for ideal compressible mhd on 2-d cartesian meshes. SIAM J. Numer. Anal. (2016) 5. B. Cockburn, C.W. Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws v: multidimensional systems. J. Comput. Phys. 141(2), 199–224 (1998) 6. J.E. Flaherty, L. Krivodonova, J.F. Remacle, M.S. Shephard, Aspects of discontinuous galerkin methods for hyperbolic conservation laws. Finite Elem. Anal. Des. 38(10), 889–908 (2002): Robert J. Melosh Medal Compet. (2001) 7. J.P. Gallego-Valencia, C. Klingenberg, P. Chandrashekar, On limiting for higher order discontinuous galerkin method for 2d euler equations. Bull. Braz. Math. Soc. New Ser. 47(1), 335–345 (2016) 8. T.A. Gardiner, J.M. Stone, An unsplit godunov method for ideal mhd via constrained transport. J. Comput. Phys. 205(2), 509–539 (2005) 9. G.S. Jiang, C.c. Wu, A high-order weno finite difference scheme for the equations of ideal magnetohydrodynamics. J. Comput. Phys. 150(2), 561–594 (1999) 10. L. Krivodonova, J. Xin, J.F. Remacle, N. Chevaugeon, J. Flaherty, Shock detection and limiting with discontinuous galerkin methods for hyperbolic conservation laws. Appl. Numer. Math. 48(3–4), 323–338 (2004). (Workshop on Innovative Time Integrators for PDEs) 11. S.A. Orszag, C.M. Tang, Small-scale structure of two-dimensional magnetohydrodynamic turbulence. J. Fluid Mech. 90(01), 129–143 (1979) 12. K.G. Powell, An approximate riemann solver for magnetohydrodynamics (that works in more than one dimension). Technical report (1994) 13. G. Tóth, The ∇ · b = 0 constraint in shock-capturing magnetohydrodynamics codes. J. Comput. Phys. 161(2), 605–652 (2000)
Well-Balanced Central-Upwind Schemes for 2 × 2 Systems of Balance Laws Alina Chertock, Michael Herty and Seyma ¸ Nur Özcan
Abstract In this study, we have developed a well-balanced second-order centralupwind scheme for 2 × 2 systems of balance laws, in particular, the models of isothermal gas dynamics with source and traffic flow with relaxation to equilibrium velocities. The new scheme is based on modifications in the reconstruction and evolution steps of a Godunov-type central-upwind method. The first step of this modification is to introduce an equilibrium variable obtained from incorporating the source term into the flux. By reconstructing equilibrium variables and using them in the wellbalanced evolution process, we have illustrated that the proposed scheme being well balanced, namely, it preserves steady states of the system. Keywords Well-balanced schemes · 2 × 2 system of balance laws
1 Introduction We consider a 2 × 2 system of equations of the following type:
ρt + f 1 (ρ, q)x = 0, qt + f 2 (ρ, q)x = −s(ρ, q),
(1)
¸ N. Özcan A. Chertock (B) · S. Department of Mathematics, North Carolina State University, 2311 Stinson Drive, 27695 Raleigh, NC, USA e-mail: [email protected] S. ¸ N. Özcan e-mail: [email protected] M. Herty Department of Mathematics, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_28
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which can be rewritten in the vector form as Ut + F(U)x = S(U),
(2)
where ρ U := , q
F(U) :=
f 1 (ρ, q) , f 2 (ρ, q)
0 S(U) := , −s(ρ, q)
(3)
are the vectors of the conservative variables, flux, and source terms, respectively, and x ∈ R and t ∈ R+ are the spatial and time variables. Systems of type (1) are called balance laws and appear as mathematical models in many applications, see, e.g., [6, 13, 15, 17, 38]. System (1) is also a common model for gas flow in highpressure transmission pipelines [3, 35] and traffic flow [2, 12], both will be our primal motivation for designing of a numerical method and validating computational results. One of the main difficulties one may encounter when numerically solving systems (1) is the loss of smoothness of the solution. Typically, the solutions of these systems possess complicated nonlinear waves such as shocks and rarefaction waves. Capturing such solutions numerically requires the use of high-resolution shock-capturing techniques, see e.g., [37, 39]. In addition to capturing the discontinuities, preserving certain steady states is an essential part of such problems. In many relevant examples, small perturbations of the steady states are also obtained as a solution of balance laws and these perturbations may not be accurately handled on a coarse mesh. Thus, one needs to implement the so-called well-balanced (WB) schemes which are capable of balancing flux and source terms exactly and maintain the small perturbations of the steady states in an accurate and stable way. These methods were first established in [20] and then developed and used widely in many hyperbolic systems, such as shallow water equations [1, 4, 8, 10, 11, 16, 21, 30, 32, 34], Euler equations with gravitation [5, 9, 22, 41–43], etc. In this paper, we propose a WB Godunov-type finite volume scheme, which conserves the steady-state solutions of (1) exactly. These schemes consist of reconstruction-evolution- projection procedure, in which cell averages of the variables are employed. In particular, we consider a second-order central-upwind (CU) scheme which was first introduced in [27] in the context of the hyperbolic conservation laws and further developed in [24, 25, 28]. The steady states, ρt = qt = 0, of (1) satisfy the following time-independent system: f 1 (ρ, q)x = 0, (4) f 2 (ρ, q)x = −s(ρ, q), as well as f 1 (ρ, q)t = f 2 (ρ, q)t = s(ρ, q)t = 0, which yields x f 1 (ρ, q) ≡ Const,
f 2 (ρ, q) +
s(ρ, q)dξ ≡ Const, ∀x, t.
(5)
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Our method is based on incorporating the source term into the flux in the second equation of the system (1) and introducing a new reconstruction-evolution process to guarantee that all steady states of (1) are captured exactly. Following [9, 10], we introduce new equilibrium variables, which are preserved during the reconstruction and propagate in time according to a modified evolution step. The rest of the paper is organized as follows. In Sect. 2, we outline a secondorder CU scheme and its WB modification with the reconstruction of the equilibrium variables in place of the conservative ones and revised numerical fluxes. In Sect. 3, we apply the WB scheme to the examples of the gas transmission systems, particularly the isothermal Euler equations with source term depends on friction or bottom profile. We also apply the method to the Aw–Rascle–Zhang traffic flow model with relaxation terms [2].
2 Numerical Method In this section, we describe a second-order semi-discrete CU scheme originally introduced in [27] and show that it does not balance the source and flux terms exactly at the discrete level. We, then, present a modification for the reconstruction procedure as well as the numerical flux used in the CU scheme to guarantee the exact preservation of the steady states.
2.1 Second-Order Central-Upwind Scheme We discretize the computational domain Ω into finite volume cells, C j = [x j− 21 , x j+ 21 ] of size Δx centered at x j = ( j − 1/2)Δx, j = 1, . . . , N , where N is the total number of grid cells in Ω. We assume that the approximated cell averages of the computed solution at fixed time level t, 1 U j (t) := U(x, t)d x, (6) Δx Cj
are known. Considering the system (1), we write the semi-discrete CU scheme as described in, e.g., [23, 26, 36]: F j+ 21 − F j− 21 d + Sj , Uj = − dt Δx
(7)
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where, F j+ 21 are the numerical fluxes: F j+ 21 =
a +j+ 1 F(UEj ) − a −j+ 1 F(UW j+1 ) 2
2
a +j+ 1 − a −j+ 1 2
α j+ 21 :=
a +j+ 1 a −j+ 1 2
2
a +j+ 1 − a −j+ 1 2
E + α j+ 21 (UW j+1 − U j ),
2
(8)
,
2
and S j is the vector of the cell averages of the source term: S j = (0, −s( ρ j , q
j ))
T
.
(9)
In (8), UEj and UW j+1 are the left and right point values of the solution at the cell interfaces x = x j+ 21 computed by using the piecewise linear reconstruction: U(x) = U j + (Ux ) j (x − x j ), x ∈ C j , that is,
(10)
Δx (Ux ) j , 2 Δx (Ux ) j+1 . + 0) = U j+1 − 2
U(x j+ 21 − 0) = U j + UEj := UW j+1
:= U(x j+ 21
(11)
The reconstruction (10) yields a second-order accurate scheme-provided slopes (Ux ) j give at least first-order approximations to the derivative Ux (x j , t). To avoid oscillations, the slopes (Ux ) j are to be computed using a nonlinear limiter applied to the cell averages U j . In our experiments reported below, we have used a generalized minmod limiter (see e.g., [29, 33, 40]):
Uj+1 − Uj Uj+1 − Uj−1 Uj − Uj−1 (Ux ) j = minmod θ , ,θ Δx 2Δx Δx
where
, θ ∈ [1, 2],
⎧ ⎨ min(z 1 , z 2 , . . .), if z i > 0 ∀i, minmod(z 1 , z 2 , . . .) := max(z 1 , z 2 , . . .), if z i < 0 ∀i, ⎩ 0, otherwise,
(12)
(13)
and the parameter θ is used to control the amount of the numerical dissipation—the larger θ results in less dissipative, but more oscillatory scheme. The one-sided local speeds of propagation, a ±j+ 1 , in (8) are obtained from the 2 largest and smallest eigenvalues λ(U) of the Jacobian matrix ∂ F(U)/∂U:
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− E W a +j+ 1 = max λ(UEj ), λ(UW j+1 ), 0 , a j+ 1 = min λ(U j ), λ(U j+1 ), 0 . 2
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(14)
2
Finally, the semi-discrete ODE system, (7) should be integrated in time by an appropriate accurate and stable ODE solver for which the CFL condition satisfies (see e.g., [26]) 1 Δx (15) , κ≤ . Δt ≤ κ ± 2 max |a j+ 1 | j
2
It is instructive to note that the described scheme does not necessarily preserve the steady-state solutions (5). To cite an example, we consider the case where f 1 (ρ, q) = q and, therefore, q = Const and ρ = ρ(x) satisfies the steady state (5). Implementing the CU scheme (7)–(14) for, say, the first component of the solution will result in the following semi-discrete approximation: ⎡ + − E W d ρj 1 ⎣ a j+ 21 q j − a j+ 21 q j+1 E + α j+ 21 (ρ W =− j+1 − ρ j ) dt Δx a +j+ 1 − a −j+ 1 2 2 ⎤ + − E W a j− 1 q j−1 − a j− 1 q j E 2 2 ⎦ − + α j− 21 (ρ W j − ρ j−1 ) . a +j− 1 − a −j− 1 2
2
The last equation reduces to E W E α j+ 21 (ρ W d ρj j+1 − ρ j ) − α j− 21 (ρ j − ρ j−1 ) =− , dt Δx
(16)
E W since q Ej = q W j+1 = q j−1 = q j = Const. However, in general, the piecewise linear approximation, (10), forms discontinuities at the cell interfaces, so that the point E W E values ρ W j+1 and ρ j (ρ j and ρ j−1 ) are not necessarily equal. Thus, right-hand side of the ODE (16) does not vanish and the scheme fails to preserve the steady state.
2.2 Well-Balanced Modification In this section, we present a WB modification of the CU scheme described in the previous section. To this end, we first define new variables x K := f 1 (ρ, q), and L := f 2 (ρ, q) + R, and rewrite the system as
ρt + K x = 0, qt + L x = 0,
R :=
s(ξ, ρ, q)dξ,
(17)
(18)
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which can be put into the vector form (2) with the different flux and zero source term: where U = (ρ, q)T , F(U) = (K , L)T and S(U) ≡ (0, 0)T . Obviously, the steady state of (18) will be in the following form: K ≡ Const,
2.2.1
L ≡ Const.
(19)
Reconstruction
We start by describing a special reconstruction procedure, which is implemented to obtain the point values, UEj and UW j+1 , used in (8), and is based on reconstructing equilibrium variables, K and L, instead of conservative ones, ρ and q. To this end, we first compute the values K j and L j from the cell averages, ρ j and q j , i.e., K j = f 1 ( ρ j , q j ),
L j = f2 ( ρ j , q j ) + R j ,
(20)
where the values of R j are evaluated by applying the midpoint quadrature rule to the integral in (17) and using the following recursive relation: Rj =
1 (R 1 + R j+ 21 ), 2 j− 2
R j+ 21 = R(x j+ 21 ) = R j− 21 + Δx s(x j , ρ j , q j ), (21)
starting from R1/2 ≡ 0. The point values of K and L at the cell interfaces x = x j± 21 are then obtained from (10)–(13): Δx Δx (K x ) j , L Ej = L j + (L x ) j , 2 2 Δx Δx (K x ) j , L W (L x ) j . = Kj − j = Lj − 2 2
K Ej = K j + KW j
(22)
, L E,W , and R j± 21 , we compute the corFinally, equipped with the values of K E,W j j responding point values of ρ and q by solving the following four nonlinear equations E W in terms of ρ Ej , ρ W j , q j , and q j , respectively: K Ej = f 1 (ρ Ej , q Ej ),
L Ej = f 2 (ρ Ej , q Ej ) + R j+ 21 ,
W W KW j = f 1 (ρ j , q j ),
W W LW j = f 2 (ρ j , q j ) + R j− 21 .
Clearly, the procedure would significantly simplify when one of the conservative variables is also an equilibrium one, say, K = f 1 (ρ, q) = q. In such case, the point can be obtained directly from (10)–(13) and thus only two nonlinear values q E,W j equations should be solved to obtain ρ E,W for each j. In all of our examples presented j below, the set of nonlinear equations (22) was solved analytically.
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2.2.2
351
Evolution
We then evolve the cell averages, U j = ( ρ j , q j )T , in time by using the following system of ODEs: F j+ 21 − F j− 21 d , (23) Uj = − dt Δx where F j± 21 are the numerical fluxes whose two components are given as follows: (1) F j+ 1 2
=
a +j+ 1 K Ej − a −j+ 1 K W j+1 2
2
a +j+ 1 − a −j+ 1 2
+ (2) F j+ 1 =
2
α j+ 21 (ρ W j+1
− ρ Ej ) H
|K j+1 − K j | |Ω| · , Δx max j {K j , K j+1 }
a +j+ 1 L Ej − a −j+ 1 L W j+1
2
2
2
a +j+ 1 − a −j+ 1 2
+
(24)
2
α j+ 21 (q W j+1
−
q Ej ) H
|L j+1 − L j | |Ω| · , Δx max j {L j , L j+1 }
and α j+ 21 is defined in (8). The second components in the numerical flux functions (24) are modified (compared to (8)) to accommodate to preserve the steady states. Namely, a smooth function H , satisfying H (φ) =
(Cφ)m , H (0) = 0, 1 + (Cφ)m
(25)
is introduced for some constants C > 0 and m > 0. When the solution is a steady E W state, e.g., both K Ej = K W j+1 = K j ≡ Const and L j = L j+1 = L j ≡ Const, H vanishes, so is each component of the numerical flux in (24). Otherwise, H is very close to 1 and then the scheme reduces to the classical semi-discrete central-upwind scheme |Ω| |Ω| and , where |Ω| (8). The normalization factors, max j {K j , K j+1 } max j {L j , L j+1 } is the size of the computational domain, are introduced in order to make the function H nondimensional and independent of the choice of C and m. We summarize this observation in the following theorem. Theorem 1 The semi-discrete CU scheme, (23)–(25), with the reconstruction described in Sect. 2.2.1 gives an absolute balance between the source and flux terms and thus preserves the steady state, (5), exactly, i.e., the scheme is WB.
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Proof Let us start with assuming that at a certain time level t the solution reaches its steady state and ∗ ∗ and L Ej = L W K Ej = K W j+1 = K j ≡ K j+1 = L j ≡ L ,
∀ j,
(26)
where K ∗ and L ∗ are constants. We show that U j = ( ρ j , q j )T remains constant in time, which means the right-hand side of the ODE system (23) diminishes with given conditions (26). Indeed, identities in (26) imply H = 0, which in turns results (1) (2) (1) (1) (2) ∗ ∗ and F j+ − F j− − in F j+ 1 = K 1 = L . Therefore, both F 1 = 0 and F j+ 1 j+ 1 2
(2) F j− 1 2
2
2
d Uj = 0, ∀ j. = 0 and thus from (23) we obtain dt
2
2
3 Computational Results In this section, we test the performance of the developed WB method and show that it preserves steady-state solutions exactly for several 2 × 2 systems. In particular, the system of isothermal Euler equations of gas dynamics with friction and with the bottom profile and the model for traffic flow with relaxation are studied. In all of the experiments reported below, we implemented the second-order WB CU scheme (23)–(25) and compared the obtained results with those computed by the non well-balanced (NWB) CU scheme (7)–(9). The scheme parameters were taken as θ = 1.3 in Examples 1, 2 and θ = 1 in Example 3; C = 200 in Examples 1, 3 and C = 400 in Example 2 and m = 1 in (25) in all of the examples. For the time evolution, we used the third-order strong stability preserving Runge–Kutta method (see, e.g., [18, 19]) to solve the semi-discrete ODE system (23) with the CFL constant in (15) taken as κ = 0.4 in Examples 1, 3 and κ = 0.1 in Example 2. Example 1 – Gas dynamics with pipe-wall friction. In this example, we solve the isothermal Euler equations of gas dynamics with pipe-wall friction, which is used for the simulation of high-pressure gas transmission systems [7, 35]. The model is governed by the following system of hyperbolic balance laws: ⎧ ⎪ ⎨ ρt + qx = 0, q2 q 2 ⎪ q + c ρ + = −μ |q|, ⎩ t ρ x ρ
(27)
where ρ(x, t) is the density of the fluid with the velocity u(x, t), q(x, t) is the momentum, μ > 0 is the friction coefficient (divided by the pipe cross section) and c > 0 is the speed of sound.
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We first check the WB property of the developed scheme by considering (27) with c = μ = 1 and subject to the following initial data (given in terms of equilibrium variables): K (x, 0) = q(x, 0) = K ∗ = 0.15 and L(x, 0) = L ∗ = 0.4,
(28)
in a single pipe x ∈ [0, 1]. Here, q2 2 (x, t) + R(x, t), K (x, t) = q(x, t) and L(x, t) = c ρ + ρ
x
(29)
q(ξ, t) |q(ξ, t)|dξ . ρ(ξ, t) To run the computations, we divide the interval Ω = [0, 1] into N uniform grid cells and apply the WB second-order CU scheme (23)–(25) to the system (27) with zero-order extrapolations for both K and L at the boundaries of the domain. We compute the solution until the final time T = 1 with N = 100, 200, 400 and 800 and report L 1 -errors, measured as K (·, T ) − K ∗ 1 and L(·, T ) − L ∗ 1 , in Table 1 (left). As one can see, on all of these grids, the initial data are preserved within the machine accuracy. For comparison, we run the same computations using the NWB CU scheme (7)–(9), in which case the initial equilibria are preserved within the accuracy of the scheme only, as can be seen in Table 1 (right). Next, we solve the system (27) with the perturbed initial data as follows: are the steady states and R(x, t) =
μ
K (x, 0) = K ∗ + ηe−100(x−0.5) = 0.5 + ηe−100(x−0.5) ,
L(x, 0) = L ∗ = 0.4, (30) with the perturbation constant η > 0. In Fig. 1, we plot the obtained momentum perturbations computed using both WB and NWB schemes with two different perturbation constants, η = 10−3 and η = 10−6 at time T = 0.2 on N = 100 uniform grid cells. We also calculate a solution using the NWB method on finer grids, i.e., N = 1600 for η = 10−3 and N = 3200 for η = 10−6 . We observe that for the larger value of the constant η = 10−3 , both the WB and NWB schemes can capture the perturbation even on a coarse mesh. However, when the perturbation is relatively small, η = 10−6 , the WB scheme still can resolve the perturbation on a coarse grid 2
2
Table 1 Example 1: L 1 -errors of the results from the WB (left) and NWB (right) computations at time T = 1 N
q
K
N
q
Rate
K
Rate
100 200 400 800
1.94E-18 9.71E-19 1.66E-18 2.18E-18
7.77E-18 9.71E-18 9.57E-18 1.18E-17
100 200 400 800
1.29E-06 3.30E-07 8.34E-08 2.09E-08
– 1.9668 1.9843 1.9965
8.81E-07 2.25E-07 5.69E-08 1.43E-08
– 1.9692 1.9834 1.9924
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10
-4
perturbation of q,
= 10 -3
10
initial state WB, N=100 NWB, N=100 NWB, N=1600
12 10
10 8
6
6
4
4
2
2
0
0.2
0.4
0.6
0.8
1
0
perturbation of q,
= 10 -6
initial state WB, N=100 NWB, N=100 NWB, N=3200
12
8
0
-7
0
0.2
x
0.4
0.6
0.8
1
x
Fig. 1 Example 1: Momentum perturbation computed by the WB and NWB schemes at time T = 0.2 for η = 10−3 (left) and η = 10−6 (right)
(N = 100), while the NWB method is not capable of catching it unless it is employed on a very fine mesh, say N = 3200. Example 2 – Gas dynamics with the bottom profile. In the second example, we consider the 2 × 2 system of gas dynamics with bottom profile where the governing equations are given by ⎧ ⎪ ⎨ ρt + qx = 0, q2 2 ⎪ = −gρh x (x), ⎩ qt + c ρ + ρ x
(31)
with h(x) being the bottom profile. This case is relevant to the practical applications when gas pipes are not horizontal. In particular, the gravitational force needs to be considered in mountainous regions with high-pressure gas transmission. Here, we consider the system (31) with c = 1, g = 9.81 and an exponential function 2 (32) h(x) = e−(x−0.5) . We solve the system on the computational domain x ∈ [0, 1] and subject to the following initial data (again given in terms of equilibrium variables): K (x, 0) = q(x, 0) = K ∗ = 1 and L(x, 0) = L ∗ = 20,
(33)
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Table 2 Example 2: L 1 -errors of the results from the WB (left) and NWB (right) computations at time T = 1 N
q
K
N
q
Rate
K
Rate
100 200 400 800
3.19E-16 3.90E-16 1.99E-16 1.41E-16
1.17E-15 9.76E-16 8.70E-16 9.45E-16
100 200 400 800
8.97E-03 2.25E-03 5.64E-04 1.41E-04
– 1.9951 1.9961 2.0000
0.117 2.98E-02 7.54E-03 1.89E-03
– 1.9731 1.9826 1.9972
where q2 (x, t) + R(x, t), K (x, t) = q(x, t) and L(x, t) = c2 ρ + ρ
(34)
x and R =
gρ(ξ, t)h x (ξ )dξ . Since (33) is a steady-state solution of (31), we adopt
it to illustrate that the CU scheme (23)–(25) is WB. Similarly to the first example, we obtain the solutions of the system (31) by implementing both the WB and NWB CU schemes on a uniform grid with N = 100, 200, 400 and 800 cells. Table 2 indicates the L 1 -errors as estimated in the previous example in measuring the equilibrium states K and L computed by both the WB (left) and NWB (right) schemes. One can clearly see that while the WB scheme gives errors within machine accuracy, the NWB method requires very fine grid, to preserve steady- state solution. We, then, introduce an initial perturbation on momentum as follows: K (x, 0) = K ∗ + ηe−100(x−0.5) = 1 + ηe−100(x−0.5) , 2
2
L(x, 0) = L ∗ = 20, (35)
where η > 0 is the perturbation constant. We first run the computations with η = 10−1 and plot the results in Fig. 2 (left) obtained at time T = 0.25 by both the WB and NWB methods with N = 100 uniform grid cells. In both cases, zero-order extrapolations are implemented at the boundaries of the computational interval Ω = [0, 1]. For comparison, we also plot a solution obtained by the NWB scheme with N = 1600. We observe that, while the WB scheme is capable of resolving the perturbation on a coarse mesh, the NWB method requires a finer mesh, e.g., N = 1600. In Fig. 2 (right), we illustrate the momentum perturbation at time T = 0.25 obtained by both the WB and NWB schemes for a smaller value of the perturbation constant η = 10−3 . We note that our WB scheme can capture smaller perturbations of the steady states on a coarse mesh, N = 100, while to obtain corresponding results with the NWB method, one needs to use a very refined mesh, N = 6400, which would be costly in most of the cases.
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10
-2
perturbation of q,
= 10 -1
10
initial state WB, N=100 NWB, N=100 NWB, N=1600
12 10
10 8
6
6
4
4
2
2
0
0.2
0.4
0.6
0.8
1
0
perturbation of q,
= 10 -3
initial state WB, N=100 NWB, N=100 NWB, N=6400
12
8
0
-4
0
0.2
0.4
x
0.6
0.8
1
x
Fig. 2 Example 2: Momentum perturbations computed by the WB and NWB schemes for η = 10−1 (left) and η = 10−3 (right) at time T = 0.25
Example 3 – Traffic flow with relaxation to equilibrium velocities. In the last example, we study a second-order model for traffic flow, which has been introduced in [2] to model driver-dependent traffic conditions. The model has been investigated since then by many authors and we refer to [14] for a recent comparison and discussion. The governing equations are written in terms of the density of cars ρ(x, t) and the velocity u(x, t), as well as a driver property w(x, t). The latter can be viewed as distance towards an equilibrium velocity Veq (ρ). For simplicity, we chose as Veq (ρ) = 1 − ρ, where ρ = 1 represents maximum density and introduce a fixed relaxation time τ > 0 for all drivers, in which case the model reads as follows: ⎧ ρt + (ρu)x = 0, ⎪ ⎪ ⎨ ρ (ρw)t + (ρuw)x = ((1 − ρ) − u) , ⎪ τ ⎪ ⎩ w = u + ρ. We substitute u = w − ρ, introduce a new variable q = ρu = ρ(w − 1) and rewrite the above system in the conservative form as follows: ⎧ ⎪ ⎨ ρt + (q + ρ(1 − ρ))x = 0, 2 q 1 ⎪ q + q(1 − ρ) + = − q. ⎩ t ρ τ x
(36)
We observe that in the limit of small relaxation times (τ → 0), the second equation in (36) formally ensures q → 0 and ρ ∈ [0, 1], and the model predictions of (36) are
Well-Balanced Central-Upwind Schemes for 2 × 2 Systems of Balance Laws
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expected to be close to those of the classical Lighthill–Whitham–Richards (LWR) model [31] given by ρt + (ρ(1 − ρ))x = 0. Clearly, q = 0 is a steady-state solution of the system (36) for any constant ρ. However, for fixed positive τ , the system has steady states deviating from the LWR model. In view of the previous discussion, we introduce the equilibrium variables K and L as q2 + q(1 − ρ) + R, (37) K = q + ρ(1 − ρ), L = ρ
x
1 q(ξ, t)dξ . Then, the steady states are K , L =Const. τ We consider the system (36) with τ = 1 and set the following initial data given with respect to the equilibrium variables:
where R(x, t) =
K (x, 0) = K ∗ = 0.375,
L(x, 0) = L ∗ = 0.5,
(38)
which also satisfy the steady-state solutions of (36). As before, we first verify that the developed WB CU scheme (23)–(25) is capable of preserving steady states of the system (36) exactly. To this end, we partition the computational domain Ω = [0, 1] into N uniform cells and assign zero-order extrapolations for K and L at the boundaries. We obtain the results at final time T = 1 by implementing the WB CU scheme with N = 100, 200, 400, and 800 grid cells. In Table 3 (left), we present the L 1 -errors computed as before, for equilibrium variables K and L, that is, K (·, T ) − K ∗ 1 and L(·, T ) − L ∗ 1 , and observe that the errors of machine accuracy for the WB scheme. However, we can conclude that NWB scheme can maintain the steady states only within the order of the scheme, as seen in Table 3 (right). We then investigate the performance of the WB scheme by capturing the perturbations of the steady states. Here, we add a small perturbation to the initial value of the variable q: 2 (39) q p (x, 0) = q(x, 0) + ηe−50(x−0.5) , Table 3 Example 3: L 1 -errors of the results from the WB (left) and NWB (right) computations at time T = 1 N
K
L
N
K
Rate
L
Rate
100 200 400 800
4.21E-17 5.57E-17 1.48E-15 5.50E-17
1.00E-16 8.74E-17 2.46E-15 1.17E-16
100 200 400 800
2.59E-06 6.47E-07 1.61E-07 4.04E-08
– 2.0011 2.0067 1.9946
8.10E-06 2.02E-06 5.04E-07 1.25E-07
– 2.0035 2.0028 2.0114
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10
-8
pertubation of K,
= 10 -7
10 35
initial state WB, N=100 NWB, N=100 NWB, N=6400
16
-8
= 10 -7
initial state WB, N=100 NWB, N=100 NWB, N=6400
30 25
12
pertubation of L,
20 8
15 10
4 5 0
0
0.2
0.4
0.6
0.8
1
0
0
0.2
0.4
x 10
-8
pertubation of K,
= 10 -7
10
initial state WB, N=200 NWB, N=200 NWB, N=6400
16
0.6
0.8
1
x -8
pertubation of L,
= 10 -7
initial state WB, N=200 NWB, N=200 NWB, N=6400
30 25
12 20 15
8
10 4 5 0
0
0.2
0.4
0.6
x
0.8
1
0
0
0.2
0.4
0.6
0.8
1
x
Fig. 3 Example 3: Perturbations on the equilibrium variable K (left column) and L (right column), computed by the WB and NWB schemes at time T = 0.1 for η = 10−7
where η = 10−7 is taken in this example. In Fig. 3, we plot the perturbations on the equilibrium variables K and L, respectively, obtained by both WB and NWB schemes with N = 100 and 200 uniform grid cells at time T = 0.1. For observation, we also plot the solutions computed by NWB method on a very fine mesh with N = 6400. We conclude that while the WB scheme is capable of capturing the perturbations on a relatively coarse grid, the NWB scheme needs to be implemented on a much finer grid.
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4 Conclusion We have generated a well-balanced second-order central-upwind scheme for 2 × 2 systems of balance laws. This is a first work in a series of well-balanced methods that preserve general steady states of the underlying system. Particularly, we consider gas flow in high-pressure transmission pipelines and traffic model with relaxation. We have presented the results of one-dimensional model, in which, steady states are captured exactly by the proposed well-balanced scheme. As an ongoing study, different physical models including the two-dimensional systems will be examined. Acknowledgements The work of A. Chertock was supported in part by the NSF Grants DMS1521051. The work of M. Herty was supported in part by DFG STE2063/1-1 DFG HE5386/1315, the cluster of excellence DFG EXC128 ‘Integrative Production Technology for High-Wage Countries’ and the BMBF project KinOpt. The work of S. ¸ N. Özcan was supported in part by the Turkish Ministry of National Education. The authors also acknowledge the support by NSF RNMS Grant DMS-1107444. The authors thank M. Lukáˇcová for her hospitality at the University of Mainz.
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On The Relative Entropy Method For Hyperbolic-Parabolic Systems Cleopatra Christoforou and Athanasios Tzavaras
Abstract The work of Christoforou and Tzavaras (Arch Rat Mech Anal 229(1):1– 52, 2018, [5]) on the extension of the relative entropy identity to the class of hyperbolic-parabolic systems whose hyperbolic part is symmetrizable is the context of this article. The general theory is presented and the derivation of the relative entropy identities for both hyperbolic and hyperbolic-parabolic systems is presented. The resulting identities are useful to provide measure valued weak versus strong uniqueness theorems as well as convergence results in the zero-viscosity limit. An application of this theory is given for the example of the system of thermoviscoelasticity. Keywords Relative entropy · Conservation laws · Hyperbolic-parabolic Convergence · Weak-strong uniqueness · Dissipative measure-valued Weak solutions
1 The Relative Entropy Method This manuscript serves as a review article of the relative entropy method as it has been recently extended in the work of Christoforou and Tzavaras in [5] for general systems of hyperbolic-parabolic conservation laws ∂t A(u) + ∂α Fα (u) = ε∂α (Bαβ (u)∂β u) .
(1)
C. Christoforou (B) Department of Mathematics and Statistics, University of Cyprus, 1678 Nicosia, Cyprus e-mail: [email protected] A. Tzavaras Computer, Electrical, Mathematical Sciences & Engineering Division, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia e-mail: [email protected] A. Tzavaras Institute of Applied and Computational Mathematics, FORTH, Heraklion, Greece © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_29
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Here u(t, x) takes values in Rn , t ∈ R+ , x ∈ Rd with n, d being integers representing the number of the conserved quantities and the space dimension. The functions A, Fα : Rn → Rn , Bαβ : Rn → Rn×n are smooth, α, β = 1, ..., d, and the summation convention over repeated indices is employed throughout the article. Also it is assumed that the associated hyperbolic problem ∂t A(u) + ∂α Fα (u) = 0
(2)
is symmetrizable in the sense of Friedrichs and Lax [17]. The idea of relative entropy, introduced by Dafermos [8, 9] and DiPerna [12], is quite powerful in comparing solutions of conservation laws (e.g. [3, 11, 12, 27]), or balance laws (e.g. [23, 30]), and has recently being applied to problems that are classified under the domain of hyperbolic-parabolic systems (e.g. [4, 14, 21, 22]). The aim is to extend the class of computations that go under the general term relative entropy to the broader class of systems (2) and (1) and to systematize the derivation of relative entropy identities in a unifying framework. Moreover, the connection of the relative entropy theory with its natural framework, the L 2 theory of hyperbolicparabolic systems of Kawashima [19] and the developments on Green functions by Liu-Zeng [31] is shown and in particular with the framework of thermodynamics. It should be emphasised that from the very early developments of the method [8, 9, 18] the theory appears in the context of thermodynamics. Therefore in the end of this article, this connection is revisited in the context of the general constitutive theory of thermoviscoelasticity, whose thermodynamical structure is specified in [6, 7, 29]. Hypotheses on the constitutive functions and the viscosity matrices that characterise the class of systems (1) and (2) for which the relative entropy method is extended are the following: (H1 ) (H2 )
A : Rn → Rn is a C 2 globally invertible map with ∇ A(u) nonsingular, existence of an entropy-entropy flux pair (η, q) , that is ∃ G : Rn → Rn , G = G(u) smooth such that ∇η = G · ∇ A ∇qα = G · ∇ Fα , α = 1, ..., d ,
(H3 ) (H4 )
the symmetric matrix ∇ 2 η(u) − G(u) · ∇ 2 A(u) is positive definite, and either the matrices ∇G(u)T Bαβ (u) induce entropy dissipation to (1) , namely d
ξα · ∇G(u)T Bαβ (u)ξβ ≥ 0
∀ξ = (ξ1 , . . . , ξd ) ∈ Rd×n ,
α,β=1
(H5 )
or that the stronger dissipative structure holds for (1): ∃ μ > 0 such that
On The Relative Entropy Method For Hyperbolic-Parabolic Systems
α,β
∇G(u)∂α u · Bαβ (u)∂β u ≥ μ
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| Bαβ (u)∂β u|2 . α
β
These hypotheses are set and guided by the goal of rendering the relative entropy identity useful and applying it to some standard problems. Let us add that Hypotheses (H1 )–(H3 ) are equivalent to the usual symmetrizability hypothesis in the sense of Friedrichs and Lax and therefore, they render system (2) hyperbolic. The additional hypotheses (H4 ) for the hyperbolic-parabolic systems (1) guarantees that the entropy dissipates along the evolution. Hypothesis (H5 ) appears in Dafermos [10, Ch IV], with the motivation that dissipation controls the diffusion in (1), and it allows for degenerate viscosity matrices. The hypotheses (H4 ) and (H5 ) are related to the Kawashima condition; we refer to Dafermos [10, Ch IV] and the articles by Serre [24–26] for further discussions. In [5], we prolong the relative entropy method to this broader class of systems and derive the relative entropy identities for systems (1) and (2). The derivation of these identities (19) and (20) is presented in Sect. 2. The structure of this manuscript is as follows: In Sect. 2 we state the main hypotheses that make the relative entropy into a workable quantity, and establish their connections to the theory of symmetrizable systems [17] and to the L 2 theory of hyperbolicparabolic systems [19]. We then derive the relative entropy identities in Sect. 3 for both hyperbolic systems (2) and hyperbolic-parabolic systems (1) and see that the usual dissipative structure for hyperbolic-parabolic systems suffices to control the various error terms that appear. We conclude in Sect. 4 by undertaking this issue in the context of a specific application. We take up the system of thermoviscoelasticity in several space dimensions under its constitutive theory. We derive the relative entropy identity that is pertinent to this theory and describe how the general theory for hyperbolic-parabolic systems takes particular shape when applied to the constitutive theory of thermoviscoelasticity. Related formulas in more special situations have been computed in [14, 15] for gases with Stokes viscosity and Fourier heat conduction and in [18] for the constitutive theory of thermoelasticity. It should be noted that for this example, the convexity of the entropy in the conserved variables translates into the usual thermodynamic stability conditions ψ F F (F, θ ) > 0 and ηθ (F, θ ) > 0 familiar from the work of Gibbs for a theory with thermal and elastic effects. The identities derived in the general theory in Sect. 2 or the example in thermoviscoelasticity in Sect. 3 are quite powerful in establishing weak versus strong uniqueness theorems, stability of bounded smooth solutions of hyperbolic-parabolic systems as well as convergence results in the zero viscosity limit to a smooth solution of inviscid system (2). Several propositions in each setting are proven in [5] using the relative entropy identities. However these are not presented in this review article. We refer the reader to [5] to realise how these identities are exploited in these theorems. Let us only add that an interesting feature of the analysis is how concentration measures are defined for a symmetrizable hyperbolic system (2) and the associated form of the averaged relative entropy identity. Last, systems of balance laws are also studied in [5] and the role of the source terms in the derivation of the relative entropy identity is investigated in the proof of the weak-strong uniqueness result.
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2 The General Framework Consider first the constituent system of conservation laws (2) and assume that A : Rn → Rn is a C 2 map which is one-to-one and satisfies ∇ A(u) is nonsingular ∀u ∈ Rn .
(H1 )
By the inverse function theorem the map v = A(u) is locally invertible with the inverse map u = A−1 (v) a C 2 map. Next, a structural hypothesis is added that is the system (2) is endowed with an additional conservation law ∂t η(u) + ∂α qα (u) = 0 .
(3)
Indeed this is expressed as follows: The functions η − q, q = (qα ), α = 1, ..., d, are called an entropy pair (η is called entropy and q = (qα ), the associated entropy-flux) if there exists a smooth function G : Rn → Rn , G = G(u), such that simultaneously ∇η = G · ∇ A ∇qα = G · ∇ Fα , α = 1, ..., d .
(H2 )
If (H2 ) is satisfied then smooth solutions of (2) satisfy the additional identity (3). One checks that (H2 ) is equivalent to requiring that G satisfies the simultaneous equations ∇G T ∇ A = ∇ A T ∇G
(4)
∇G ∇ Fα = ∇ Fα ∇G , α = 1, ..., d . T
T
(5)
That is, if there exists a multiplier G(u) satisfying (H2 ) (equivalently (4), (5)) then (2) is endowed with the additional conservation law (3) and it is well known that systems from mechanics naturally inherit the entropy pair structure from the second law of thermodynamics. Given two solutions u, u¯ of (2), the relative entropy is defined via η(u|u) ¯ = η(u) − η(u) ¯ − G(u) ¯ · (A(u) − A(u)) ¯
(6)
while the relative flux(es) by qα (u|u) ¯ = qα (u) − qα (u) ¯ − G(u) ¯ · (Fα (u) − Fα (u)) ¯ .
(7)
Formula (6) will be used to estimate the distance between two solutions u and u. ¯ To make it amenable to analysis, we note that ∇ 2 η(u) − G(u) · ∇ 2 A(u) is symmetric and require that it is positive definite, that is ξ · ∇ 2 η(u) − G(u) · ∇ 2 A(u) ξ > 0
for ξ ∈ Rn \ {0} .
(H3 )
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Under (H3 ), expression (6) is useful for comparing the distance between two solutions u(t, x) and u(t, ¯ x). The definition of relative entropy and flux(es) given by (6)–(7) extends to the case of system (2) a well known definition pursued in [8, 12] for the case A(u) = u with the same objective of calculating the distance between two solutions. Also in [19], one can find a special case of the quantity (6) for comparing a general solution u(t, x) to a constant state u, ¯ in connection to asymptotic behavior problems. Regarding the hyperbolic–parabolic system (1), in addition to hypotheses (H1 ), (H2 ), (H3 ) on the hyperbolic part, we assume that the viscosity matrices that induce a dissipative structure. Using the multiplier G(u) in (H2 ), we deduce that smooth solutions of (1) satisfy the identity ∂t η(u) + ∂α qα (u) = ε∂α (G(u) · Bαβ (u)∂β u) − ε∇G(u)∂α u · Bαβ (u)∂β u .
(8)
We will require that the entropy dissipates along the evolution, namely that the following positive semi–definite structure holds true: d
d n ij j ξα · ∇G(u)T Bαβ (u) ξβ = ξαi Dαβ ξβ ≥ 0
α,β=1
∀ξα , ξβ ∈ Rn ,
α,β=1 i, j=1
(H4 ) . where Dαβ = ∇G(u)T Bαβ (u), for α, β = 1, ..., d and note that (H4 ) is rewritten in extended coordinates for the convenience of the reader. Hypothesis (H4 ) is natural in the context of applications to mechanics as it is connected to entropy dissipation and the Clausius–Duhem inequality. When exploiting the identities, we often impose a strengthened version of (H4 ): d α,β=1
ξα · ∇G(u)T Bαβ (u) ξβ > 0 ∀ξ = (ξ1 , . . . , ξd ) ∈ Rd×n , ξ = 0 .
(H4s )
n For ξ ∈ Rd×n , we denote by |ξ | = ( dα=1 i=1 |ξα,i |2 )1/2 its euclidean norm. The minimum ν(u) and maximum N (u) eigenvalues of the associated quadratic form, for u ∈ Rn , may be used to express (H4s ) in an equivalent (more quantitative) format: 0 < ν(u)|ξ |2 ≤
d
ξα · ∇G(u)T Bαβ (u) ξβ ≤ N (u)|ξ |2 ,
(9)
α, β=1
∀ξ ∈ Rd×n \ {0}. Remark 1. We can compare the Hypotheses (H2 ) and (H3 ) with the familiar notion of symmetrizable first-order systems of Friedrichs and Lax [17] using the transformation v = A(u) since this is invertible by Hypothesis (H1 ). Then systems (2) and (3) can be expressed in terms of the conserved variables v,
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∂t v + ∂α (Fα ◦ A−1 )(v) = 0 −1
−1
∂t (η ◦ A )(v) + ∂α (qα ◦ A )(v) = 0 .
(10) (11)
Setting f α (v) = Fα ◦ A−1 (v) ,
H (v) = η ◦ A−1 (v) ,
Q α (v) = qα ◦ A−1 (v) ,
(12)
we obtain the formulas η(u) = H (A(u)) , qα (u) = Q α (A(u)) .
(13)
One can easily check that (H2 ) is equivalent to ∇v Q α (v) = ∇v H (v) · ∇v f α (v) ,
(h2 )
G(u) = (∇v H )(A(u)).
(14)
and
Hence η − q is an entropy pair for (2) satisfying (H2 ) if and only if the pair H − Q is an entropy pair for (10) satisfying (h2 ). Also Hypothesis (H3 ) translates to the requirement that the entropy H (v) is convex, ζ · ∇v2 H (v)ζ > 0 for ζ ∈ Rn , ζ = 0 .
(h3 )
In summary, (H1 ), (H2 ) and (H3 ) are equivalent to the usual symmetrizability hypothesis of [17]. Regarding next the hyperbolic-parabolic system (1), it is instructive to compare the structural hypotheses pursued here to the L 2 theory of hyperbolic-parabolic systems. As already mentioned, hypothesis (H4 ) on the diffusion coefficients render the last term of (8) as semi-positive definite. We refer the reader to Kawashima [19, 20] for the early developments and to Liu–Zeng [31] for the connection to Green’s functions for hyperbolic-parabolic systems as well as to Kawashima [19, Ch II] and Dafermos [10] for results concerning well-posedness of hyperbolic-parabolic systems. An exposition on the structure of dissipative viscous systems is also given by Serre in [24–26]. Note here that the effective diffusion matrices Dαβ should at least satisfy the well-known Kawashima condition, which guarantees that waves of all characteristic families are properly damped. This can be deduced from hypothesis (H4 ) by setting ξα = να Ri with Ri standing for the right eigenvector associated to the i-characteristic speed of system (1) and ν ∈ S d−1 . Remark 2. The analysis by Christoforou and Tzavaras [5] is not a direct application of the change of variable v = A(u) to the existing theory for A(u) = u that one might at a first glance believe. Indeed the theory for hyperbolic-parabolic systems cannot be studied only based on the change of variables due to the viscous part of the system
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as one can see from the computation of the previous remark. This raised the need to extend the relative entropy method to a more general framework for systems (1) and (2).
3 The Relative Entropy Identity In this section we extend a well known calculation developed in [8, 12] for the case A(u) = u to the hyperbolic system (2), subject to hypotheses (H1 ), (H2 ) and (H3 ). Let u be an entropy weak solution of (2), that is u is a weak solution of (2) that satisfies in the sense of distributions the inequality ∂t η(u) + ∂α qα (u) ≤ 0 .
(15)
Let u¯ be a strong (conservative) solution of (2) that is satisfying the entropy identity ∂t η(u) ¯ + ∂α qα (u) ¯ = 0.
(16)
We proceed to compute the relative entropy identity for the quantities relative entropy (6) and relative flux (7). Observe first that u, u¯ satisfy the chain of identities ¯ · (A(u) − A(u)) ¯ + ∂α G(u) ¯ · (Fα (u) − Fα (u)) ∂t G(u) ¯ ¯ + ∇G(u) ¯ u¯ xα · (Fα (u) − Fα (u)) ¯ = ∇G(u)∂ ¯ t u¯ · (A(u) − A(u)) ¯ T ∇ A(u) ¯ −T ∇G(u) ¯ T (A(u) − A(u)) ¯ = −u¯ xα · ∇ Fα (u) + ∇G(u) ¯ u¯ xα · (Fα (u) − Fα (u)) ¯ ¯ A(u) ¯ −1 (A(u) − A(u)) ¯ = −∇G(u) ¯ u¯ xα · ∇ Fα (u)∇ + ∇G(u) ¯ u¯ xα · (Fα (u) − Fα (u)) ¯ =: ∇G(u) ¯ u¯ xα · Fα (u|u) ¯ ,
(17)
where ¯ := Fα (u) − Fα (u) ¯ − ∇ Fα (u)∇ ¯ A(u) ¯ −1 (A(u) − A(u)) ¯ . Fα (u|u)
(18)
Combining (15), (16) and (17), we obtain ¯ + ∂α qα (u|u) ¯ ≤ −∂α G(u) ¯ · Fα (u|u) ¯ . ∂t η(u|u)
(19)
The above calculation is formal, it can however be made rigorous following ideas that are well developed (see e.g. [10, Ch V]) and provides a way of comparing a weak entropic to a strong solution of (2). There exist variants of this calculation that compare entropic measure valued solutions to strong solutions of hyperbolic conservation laws (see [3, 5, 11]).
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An analogous but more complicated analysis leads us to the relative entropy identity ∂t η(u|u) ¯ + ∂α qα (u|u) ¯ + ε∇G(u)∂α (u − u) ¯ · Bαβ (u) ∂β (u − u) ¯ = −∇G(u)(∂ ¯ α u) ¯ · Fα (u|u) ¯ + ε∂xα (Jα + jα ) + ε
6
Qi
(20)
i=1
among two solutions u and u¯ of (1) under the framework (H1 )–(H4 ). Here the viscous fluxes Jα and jα are Jα = (G(u) − G(u)) ¯ · (Bαβ (u)u xβ − Bαβ (u) ¯ u¯ xβ ) + Bαβ (u) ¯ u¯ xβ · G(u|u) ¯ ,
with
(21)
jα := Bαβ (u) ¯ u¯ xβ · ∇G(u)φ(u| ¯ u) ¯ ,
(22)
¯ . G(u|u) ¯ := G(u) − G(u) ¯ − ∇G(u)∇ ¯ A(u) ¯ −1 (A(u) − A(u))
(23)
while Q i represent quadratic “error” terms. The details of this derivation can be found in Sect. 2.3 of [5].
4 Application in Thermoviscoelasticity Here we present the relative entropy calculation for the system of thermoviscoelasticity in several space dimensions. In [5] this calculation is also performed for the system of thermoviscoelasticity when restricted to one-space dimension (when restricted to the particular case of Stokes viscosity and Fourier heat conduction). The requirements imposed from thermodynamics on the constitutive theory of thermoviscoelasticity were developed in [6, 7] and a summary can be found in [10, Sect 3.2]. The constitutive theory takes the form ψ = ψ(F, θ ) , ∂ψ (F, θ ) , Σ= ∂F (24) ∂ψ η=− (F, θ ) , ∂θ e = ψ + θη . The total stress is decomposed into an elastic part Σ and a viscoelastic part ˙¯ where Σ and Z are both symmetric tensor valued functions, Z = Z (F, θ, g, F) Z (F, θ, 0, 0) = 0 so that
On The Relative Entropy Method For Hyperbolic-Parabolic Systems
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˙¯ S = Σ(F, θ ) + Z (F, θ, g, F) ∂ψ ˙¯ , = (F, θ ) + Z (F, θ, g, F) ∂F ˙¯ . Q = Q(F, θ, g, F) Moreover, the heat flux Q and the viscoelastic contribution to the stress Z have to satisfy 1 ˙¯ ≥ 0 ∀(F, θ, g, F) ˙¯ , g · Q(F, θ, g) + F˙¯ : Z (F, θ, g, F) θ
(H)
which along with (24) guarantee consistency for smooth processes with the Clausius– Duhem inequality [6, 7]. ˙¯ and Q = Q(F, θ, g) i.e. Z is For simplicity we assume that Z = Z (F, θ, F) ˙¯ Hence condition (H) implies taken independent of g = ∇θ and Q independent of F. Q(F, θ, 0) = 0, Z (F, θ, 0) = 0, and accordingly (H) decomposes into two distinct inequalities 1 ˙¯ ≥ 0 . g · Q(F, θ, g) ≥ 0 and F˙¯ : Z (F, θ, F) θ
(H )
The system of thermoviscoelasticity then takes the form Ft = ∇v vt = div(Σ + Z ) + f ∂t ( 21 |v|2
(25)
+ e) = div(v · Σ + v · Z ) + divQ + v · f + r .
Recall that x stands for the Lagrangean variable, div is the usual divergence operator ˙¯ One (in spatial coordinates), while ∂t is here the material derivative, hence Ft = F. checks that smooth solutions of (25) satisfy the energy dissipation identity ∂t e = ∇v : (Σ + Z ) + divQ + r and the entropy production identity ∂t η − div
1 r Q 1 = 2 ∇θ · Q + ∇v : Z + . θ θ θ θ
To relate now to the general theory of the previous sections, we set ⎛ ⎞ F 2 U = ⎝ v ⎠ ∈ Rd +d+1 , θ
⎛
⎞ F ⎠ v A(U ) = ⎝ 1 2 v + e(F, θ ) 2
(26)
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and impose eθ (F, θ ) > 0 so that A(U ) is invertible. Also we set η(U ˆ ) := −η(F, θ ) where η(U ˆ ) is the mathematical entropy and η(u, θ ) the thermodynamic one. Here the relative entropy identity takes the form ¯ θ¯ ) η(U ˆ |U¯ ) = − η(F, θ ) + η( F, 1 ¯ v − v¯ , e(F, θ ) + 1 |v|2 − e( F, ¯ θ¯ ) − 1 |¯v|2 ¯ v¯ , −1) · F − F, − (Σ, 2 2 θ¯
(24) 1 ¯ θ¯ ) + 1 |v − v¯ |2 + (η(F, θ ) − η( F, ¯ θ¯ ))(θ − θ¯ ) . = ψ(F, θ | F, 2 θ¯ We use the notation ∂ψ ¯ ¯ − ∂ψ ( F, ¯ θ¯ )(θ − θ¯ ) ( F, θ¯ ) : (F − F) ∂F ∂θ ¯ + η(θ = ψ − ψ¯ − Σ¯ : (F − F) ¯ − θ¯ ) (27) ¯ θ¯ ), η¯ = η( F, ¯ θ¯ ) and so on. We see that and set ψ¯ = ψ( F, ¯ θ¯ ) = ψ(F, θ ) − ψ( F, ¯ θ¯ ) − ψ(F, θ | F,
⎛1
⎞ ψF F 0 0 ∇ 2 η(U ˆ ) − G(U ) · ∇ 2 A(U ) = ⎝ 0 θ1 0 ⎠ 0 0 θ1 ηθ (24)
θ
and the positivity for the matrix ∇ 2 η(U ˆ ) − G(U ) · ∇ 2 A(U ) is equivalent to the usual Gibbs thermodynamic stability conditions ψ F F > 0 and ηθ > 0. A careful analysis in a similar fashion as in Sect. 3 leads to the relative entropy identity or the system of thermoviscoelasticity (25) ¯ θ¯ ) + (η − η)(θ ¯ + 1 |v − v¯ |2 ¯ − θ) ∂t ψ(F, θ | F, 2 Q Q¯ − div (v − v¯ ) · (Σ + Z − Σ¯ − Z¯ ) + (θ − θ¯ ) − θ θ¯ ¯ ¯ ¯ ¯ ¯ ¯ = −θt η(F, θ | F, θ ) + Ft : Σ(F, θ | F, θ ) ∇v ∇ v¯ Z Z¯ Q Q¯ ∇θ ∇ θ¯ − θ θ¯ − − − : − θ¯ − θ · θ θ θ θ θ¯ θ¯ θ¯ θ¯ r r ¯ . + (v − v¯ ) · ( f − f¯) + (θ − θ¯ ) − θ θ¯ Here we have set
(28)
On The Relative Entropy Method For Hyperbolic-Parabolic Systems
373
∂Σiα ¯ ∂η ¯ ¯ (θ − θ) ¯ , ( F, θ¯ )(Fiα − F¯iα ) − ( F, θ) ∂θ ∂θ ∂Σ jβ ∂Σiα ¯ ¯ θ¯ ) (θ − θ¯ ) . ¯ θ) ¯ θ) ¯ := Σ jβ (F, θ) − Σ jβ ( F, ¯ − ¯ (Fiα − F¯iα ) − Σ jβ (F, θ| F, ( F, ( F, θ) ∂ F jβ ∂θ ¯ θ) ¯ θ¯ ) + ¯ := η(F, θ) − η( F, η(F, θ| F,
Let us just comment that in (28), the effect of viscous dissipation and heat conduction is captured respectively by the terms ∇ v¯ Z Z¯ − : θ θ θ¯ θ¯ Q ¯ ¯ ∇θ Q ∇θ · . − Dq := θ¯ − θ θ θ θ¯ θ¯ Dv := θ θ¯
∇v
−
Moreover it can be easily checked that the same relative entropy formula can be derived for the case that we compare two different constitutive theories that have the same thermoelastic part but different viscoelastic and heat conduction formulas. Acknowledgements Christoforou would like to thank the organizers of XVI International Conference on Hyperbolic Problems Theory, Numerics, Applications (Hyp2016) that took place in Aachen from August 1st until 5th of 2016 for the invitation and the warm hospitality.
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12. R.J. DiPerna, Uniqueness of solutions to hyperbolic conservation laws. Indiana Univ. Mah. J. 28, 137–187 (1979) 13. R.J. DiPerna, A.J. Majda, Oscillations and concentrations in weak solutions of the incompressible Euler equations. Commun. Math. Phys. 108, 667–689 (1987) 14. E. Feireisl, A. Novotny, Weak-strong uniqueness property for the full Navier–Stokes–Fourier system. Arch. Ration. Mech. Anal. 204, 683–706 (2012) 15. E. Feireisl, Asymptotic analysis of compressible, viscous and heat conducting fluids, in Nonlinear dynamics in partial differential equations. Advanced Studies in Pure Mathematics, vol. 64 (Mathematical Society of Japan, Tokyo, 2015), p. 133 16. U.S. Fjordholm, R. Käppeli, S. Mishra, E. Tadmor, Construction of approximate entropy measure-valued solutions for hyperbolic systems of conservation laws. Preprint (2014) 17. K.O. Friedrichs, P.D. Lax, Systems of conservation equations with a convex extension. Proc. Nat. Acad. Sci. USA 68, 1686–1688 (1971) 18. D. Iesan, On the stability of motions of thermoelastic fluids. J. Therm. Stress. 17, 409–418 (1994) 19. S. Kawashima, Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, Doctoral thesis, Kyoto University, 1984 20. S. Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications. Proc. R. Soc. Edinb. Sect. A 106, 169–194 (1987) 21. C. Lattanzio, A.E. Tzavaras, Structural properties of stress relaxation and convergence from viscoelasticity to polyconvex elastodynamics. Arch. Ration. Mech. Anal. 180, 449–492 (2006) 22. C. Lattanzio, A.E. Tzavaras, Relative entropy in diffusive relaxation SIAM. J. Math. Anal. 45, 1563–1584 (2013) 23. A. Miroshnikov, K. Trivisa, Relative entropy in hyperbolic relaxation for balance laws. Commun. Math. Sci. 12, 1017–1043 (2014) 24. D. Serre, The structure of dissipative viscous systems of conservation laws. Phys. D 239(15), 1381–1386 (2010) 25. Local existence for viscous system of conservation laws: H s data with s>1 + d/2, Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena. Contemporary Mathematics (American Mathematical Society, Providence, RI, 2010), pp. 339–358 26. D. Serre, Viscous system of conservation laws: singular limits, Nonlinear Conservation Laws and Applications. The IMA Volumes in Mathematics and its Applications (Springer, New York, 2011), pp. 433–445 27. D. Serre, A. Vasseur, L2-type contraction for systems of conservation laws. J. École Polytech. Math. 1, 1–28 (2014) 28. L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics. Heriot–Watt Symposium, ed. by R.J. Knops (Pitman Research Notes in Mathematics, Pitman, Boston, 1979), pp. 136–192 29. C. Truesdell, W. Noll, The non-linear field theories of mechanics. Handbuch der Physik, Bd. III/3, (Springer, Berlin 1965), pp. 1–602 30. A.E. Tzavaras, Relative entropy in hyperbolic relaxation. Commun. Math. Sci. 3, 119–132 (2005) 31. T.-P. Liu, Y. Zeng, Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws. Mem. Amer. Math. Soc. 125, 1–120 (1997)
A Multispecies Traffic Model Based on the Lighthill-Whitham and Richards Model Rinaldo M. Colombo, Christian Klingenberg and Marie-Christine Meltzer
Abstract We consider an extension of the macroscopic traffic model of Lighthill– Whitham and Richards to a multispecies traffic model. As has already been observed by Benzoni-Gavage and Colombo (Eur J Appl Math 14:587–612, 2003, [1]), the system of PDEs lacks strict hyperbolicity. We study the Riemann problem for the two species extension with focus on the values around the umbilic point, where the eigenvalues coalesce. For this purpose, we examine the behavior of the solutions around the critical point like it is done by Meltzer (Multispecies traffic models based on the Lighthill–Whitham and the Aw-Rascle model, 2016, [9]). Due to the difficulty of the umbilic point, we are not able to prove well-posedness analytically. But we provide an understanding of the systems properties via numerical experiments which leads to the conjecture that the Riemann problem has a unique solution not only near the umbilic point but also away from it in the set where it is defined. Keywords Macroscopic traffic model · Multispecies vehicular traffic Conservation laws
1 Introduction In this paper, we consider a macroscopic traffic model which means that we assume a large number of vehicles on the road and describe them through the density. A famous and well-studied model describing the traffic density and its evolution is the model of Lighthill–Whitham and Richards (LWR). It is deduced assuming the conservation of vehicles and that their speed depends solely on their density. The wish R. M. Colombo INdAM Unit, Brescia University, Via Valotti 9, 25133 Brescia, Italy e-mail: [email protected] C. Klingenberg (B) · M.-C. Meltzer Department of Mathematics at Würzburg University, Emil Fischer Str. 40, 97074 Würzburg, Germany e-mail: [email protected] M.-C. Meltzer e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_30
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to discriminate different types of vehicles, like, for example, cars and trucks, leads to the consideration of a multispecies extension of this model that has already been proposed in literature. This extension leads to difficulties in proving well-posedness already for two species, since it contains an umbilic point where the system is not hyperbolic, and hence, we are not able to use the standard theory for hyperbolic systems. In this paper, the solution to the Riemann problem is studied around the critical point to see how the lack of hyperbolicity affects the solutions. First of all, we introduce the LWR Model, its variables, and the Greenshields velocity function which was first studied by [7] and [10]. From the macroscopic point of view, there are three fundamental variables describing car traffic. As a first step, we consider a single-lane road where all traffic participants are of one species and overtaking is forbidden. Then, we describe the velocity, the density, and the traffic flow as functions of the space coordinate x ∈ R and of time t ∈ R+ . The velocity of cars is described by the velocity field u, and the traffic density ρ measures the number of vehicles per unit length of the road. Alternatively, ρ can be viewed as the occupancy, i.e., as the fraction of road length occupied by vehicles. Finally, the traffic flow f is the third macroscopic variable, defined as the number of cars passing a fixed point of the road in a given amount of time or, equivalently, as the product of the density by the velocity: f = ρ u. The LWR model is obtained from the postulate of conservation of vehicles, which yields to the nonlinear partial differential equation ∂t ρ + ∂x f = 0,
(1)
together with the assumption that the speed is a function of the density, namely, u(ρ) = V ψ(ρ)
(2)
where the positive constant V is the vehicular maximal speed, while the function ψ describes how the attitude of drivers depends on the local traffic speed, so that ψ is a monotone (weakly) decreasing C 1 -function, i.e., ψ < 0, normalized so that ψ(0) = 1 and ψ(1) = 0. Below, we use the usual Greenshields speed–density relation, namely, ψ(ρ) = 1 −
ρ ρmax
(3)
where V > 0 is the maximal velocity and the maximal density ρmax is normalized to 1, coherently with its interpretation as occupancy.
2 Multispecies Extension of the LWR Model For the purpose of distinguishing different traffic participants, we extend the LWR model introduced above obtaining a many-species model, see [1].
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The extension to i populations, with i ∈ N+ , of the conservation law (1) consists of the following system of PDEs: ∂t ρ j + ∂x ρ j u j (ρ) = 0
j = 1, . . . , i ,
(4)
where ρ = (ρ j )ij=1 is the vector of the populations’ densities. A natural extension of the choice (2) leads to ⎞ ⎛ i ρj⎠ u j (ρ) = V j ψ ⎝
j = 1, . . . , i ,
(5)
j=1
where we assume that the speeds are indexed so that V j > V j for j < j. With the choice (3), the system is naturally defined on the simplex i
S = (ρ j )ij=1 | ρ j ≥ 0 ∀ j = 1, . . . , i and ρj ≤ 1 . i=1
In this paper, we only focus on the two-species extension of the LWR model, its peculiarity being that it leads to one umbilic point and to a variety of unexpected features. Nevertheless, we mention that for i > 2 there exist more difficulties than umbilic points. For example, in a three-species case, we obtain umbilic lines and planes on the boundary of the set of definition. However, the most interesting properties of the multispecies model (1)–(2)–(3) can already be seen in the two-species case. Setting i = 2 in (4)–(5), we obtain the two species model
∂t ρ1 + ∂x (ρ1 V1 ψ(r )) = 0 ∂t ρ2 + ∂x (ρ2 V2 ψ(r )) = 0
r = ρ1 + ρ2 .
(6)
First of all, we investigate the hyperbolicity of (6). The Jacobian J (ρ) of the system is given by
V1 ρ1 ψ (r ) V1 ψ(r ) + ρ1 ψ (r ) J (ρ) = V2 ρ2 ψ (r ) V2 ψ(r ) + ρ2 ψ (r )
(7)
with characteristical polynomial πρ (λ) = (β1 − λ) (β2 − λ) − α1 α2
where
αi = Vi ρi ψ (r ) βi = Vi ψ(r ) + ρi ψ (r )
Then, the eigenvalues λ1/2 (ρ) are found as the roots of πρ λ1 =
1 (β1 + β2 ) − (β1 − β2 )2 + 4α1 α2 . 2
and
λ2 =
and r = ρ1 + ρ2 .
(8)
1 (β1 + β2 ) + (β1 − β2 )2 + 4α1 α2 . 2
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Note that, with the above choice, λ1 ≤ λ2 . Moreover, we see that there is no way to find a velocity function ψ for which the eigenvalues of system (6) are distinct on the whole set S . The LWR model for two species provides one umbilic point where the eigenvalues are the same, and hence at this point, the system is not strictly hyperbolic. In the case of the Greenshield’s relation (3), we obtain the following theorem about the system’s hyperbolicity. Theorem 1. System (3)–(6) is strictly hyperbolic in S \{ρ u }. At the umbilic point ρ u , where V1 − V2 , (9) ρ u = (ρ1u , 0) with ρ1u = 2V1 − V2 the eigenvalues λ1 and λ2 coalesce. We now study the consequences of the existence of this point ρ u on the wellposedness of (6). Another system, where hyperbolicity is not strictly given, is studied by [5] and [8]. Yet, the model discussed here differs from the Keyfitz–Kranzer model because there the umbilic point lies in the interior of the set of definition. In our case, the umbilic point lies on the boundary of the simplex S . Before we study the Riemann problem of (6), we state some of its global features. Mention that for ρ2 = 0 the eigenvalues are linear functions in ρ1 and coincide at ρ1u . Indeed, λ1 (ρ1 , 0) =
V2 (1 − ρ1 )
for ρ1 < ρ1u
V1 (1 − 2ρ1 )
for ρ1 > ρ1u ,
λ2 (ρ1 , 0) =
V1 (1 − 2ρ1 )
for ρ1 < ρ1u
V2 (1 − ρ1 )
for ρ1 > ρ1u .
(10)
With the above abbreviations (8), a choice of corresponding eigenvectors is given by
λ1 − β2 − α1 , v1 = λ1 − β1 − α2
−λ2 + β2 − α1 v2 = . λ2 − β1 + α2
(11)
Hence, we see that in the umbilic point the following holds. Lemma 1. At the umbilic point ρ u given in (9), the Jacobian matrix (7) is not diagonalizable and, in addition to its eigenvalues, its eigenvectors (11) also coalesce. Hence, for this value there exists no basis of eigenvectors for system (6). The next proposition describes the corresponding characteristic fields. For the basic terminology, we refer to [11]. Proposition 1. In S \{ρ u }, with reference to system (3)–(6), the first characteristic field is genuinely nonlinear; the second characteristic field is linearly degenerate for ρ1 + ρ2 = 1 and genuinely nonlinear elsewhere. For the proof of this statement, we refer to [1]. Note that the standard technique relying on the direct computation of dλi · vi is not immediately of use, here. We recall the following classical result ensuring the invariance of subsets of R2 , see [4] for more details.
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Proposition 2. A set with regular boundary is locally invariant under a strictly hyperbolic and genuinely nonlinear system of conservation laws if the domain is convex and the normal to the boundary is a left eigenvector of the system. We already know that (6) is not strictly hyperbolic and that along the line ρ1 + ρ2 = 1 it is linearly degenerate, so that Proposition 2 cannot be applied to the simplex S . Nevertheless, by (7), (1, 1) J (ρ1 , ρ2 ) = −(ρ1 V1 + ρ2 V2 ) (1, 1) (1, 0) J (0, ρ2 ) = V1 (1 − ρ2 ) (1, 0) (0, 1) J (ρ1 , 0) = V2 (1 − ρ1 ) (0, 1)
ρ1 + ρ2 = 1; ρ1 , ρ2 ∈ [0, 1]; ρ2 ∈ [0, 1]; ρ1 ∈ [0, 1]. (12) Therefore, the normals to the convex set S are indeed left eigenvectors of the Jacobian J of (6). Thus, we conjecture that S is invariant, which implies that for initial data inside S , the solution lies inside S , too. With the above discussion, one can now study the Riemann problem.
2.1 The Riemann Problem When a Species Is Absent After the examination of general features of the LWR model for two species for the Greenshields velocity function, we investigate its well-posedness. As we have already seen, the existence of the umbilic point hinders us from using general existence, uniqueness, and invariance theorems about hyperbolic conservation laws. Hence, we start with the discussion of existence for particular Riemann problems (RPs). In the case of i = 2 populations, the general (RP) consists of system (6) with initial datum ρ(x, 0) =
ρ L = (ρ1L , ρ2L )
for x < 0
ρ =
for x > 0
R
(ρ1R , ρ2R )
(13)
Since we know that (6) is not strictly hyperbolic, the question is whether the existence of the umbilic point influences the solution to the RP (6)–(13) or not. Note that if we define system (6) on S \{ρ u }, then it will indeed be strictly hyperbolic, and thus, the solution to the RP (6)–(13) can be found by following the standard Lax theory [6]. Moreover, (6) is well-posed on all initial data with small variation for all times [2]. But if we take the point ρ u into account, it will not be clear whether the solution to the RP is well defined. Nevertheless, we can use the Lax theory to discover the Lax curves and construct the solution to the RP taking care of what happens near the umbilic point. The algebraic expressions are hard to handle and, to our knowledge, there exists no direct proof of well-posedness. But one can examine how the umbilic point affects the solution to the RP for a special case. Hence, consider the situation where the slower species is absent at the beginning, i.e., ρ2L , ρ2R = 0. This yields the RP on the ρ1 -axis, which is (6) together with (13) which now reads
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ρ(x, 0) =
ρ L = (ρ1L , 0) for x < 0 , ρ R = (ρ1R , 0) for x > 0 .
(14)
Then, we have to include ρ u into the discussion since the initial data lies on the same axis as the umbilic point. Given any point ρ in S , the Lax curves L i (ρ) are determined by the integral rarefaction curve of the ith eigenvector along increasing λi together with the Hugoniot shock curves. As a next step, we compute the solutions to the RP (6)–(14). Since the Greenshield’s function is concave in each entry, the solution to the RP for one species consists of a shock if ρ1L < ρ1R and of a rarefaction wave if ρ1L > ρ1R . First, we turn to the discontinuities. The shock curves are described with the help of the Rankine– Hugoniot (RH) condition. Proposition 3. If the Riemann problem (6)–(13) is solved by two shocks with middle sate ρ, then the Rankine Hugoniot condition
σ (ρ L − ρ) = f(ρ L ) − f(ρ) γ (ρ − ρ R ) = f(ρ) − f(ρ R )
(15)
has to be fulfilled, where σ, γ ∈ R are the shock speeds and ρ ∈ S . If ρ belongs to the ρ1 -axis, i.e., ρ = (ρ1 , 0), we are able to describe the curves exiting ρ completely. Recall that we call Hugoniot set through the point ρ 0 ∈ S the set H (ρ 0 ) = {(ρ1 , ρ2 ) ∈ R2 , σ ∈ R| (16) holds} where
ρ1 (1 − ρ1 − ρ2 )V1 − σ = ρ10 (1 − ρ10 − ρ20 )V1 − σ ρ2 (1 − ρ1 − ρ2 )V2 − σ = ρ20 (1 − ρ10 − ρ20 )V2 − σ
(16)
and, when necessary, we distinguish between shocks of the first family (H 1 ) and of the second family (H 2 )., see [2, 3] for more details. Whenever ρ 0 lies along the 1 axis, that is we have ρ20 = 0, Eq. (16) become
ρ1 (1 − ρ1 − ρ2 )V1 − σ = ρ10 (1 − ρ10 )V1 − σ ρ2 (1 − ρ1 − ρ2 )V2 − σ = 0 .
(17)
Proposition 4. For ρ 0 = (ρ10 , 0), the Hugoniot curves exiting ρ 0 can be described depending on where ρ10 lies along the ρ1 -axis. 1. If ρ10 = 0, then H 2 (ρ 0 ) = {ρ2 = 0} and H 1 (ρ 0 ) = {ρ1 = 0}. 2. If ρ10 < ρ1u , then H 2 (ρ 0 ) = {ρ2 = 0} and H 1 (ρ 0 ) is monotone in ρ1 and exits S at a point with ρ2 = 0 and ρ1 > ρ1u .
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Fig. 1 A sample of H 1 and H 2 -curves where ρ20 = 0 with V1 = 1 and V2 = 0.75. (The umbilic point is ρ u = (0.2, 0).)
3. If ρ1∗ > ρ10 > ρ1u , then H 1 (ρ 0 ) = {ρ2 = 0} and H 2 (ρ 0 ) is monotone in ρ1 and exits S at a point with ρ2 = 0 and 0 < ρ1 < ρ1u . 4. If ρ10 > ρ1∗ , then H 1 (ρ 0 ) = {ρ2 = 0} and H 2 (ρ 0 ) is monotone in ρ1 and exits S at a point with ρ1 = 0 and 0 < ρ2 < 1. 5. If ρ10 = 1, then H 1 (ρ 0 ) = {ρ2 = 0} and H 2 (ρ 0 ) = {ρ1 + ρ2 = 1}. where ρ1∗ = 1 − V2 /V1 . One can see this by studying the Hugoniot set (17). The hyperbolæ are given by ρ2 (ρ1 ) =
(ρ1 − ρ10 )((V1 − V2 )(1 − ρ1 ) − ρ10 V1 ) . (V1 − V2 )ρ1 + ρ10 V2
In Fig. 1a, one sees a sample of the Hugoniot curves for V1 = 1 and V2 = 0.75. The next proposition confirms that the Hugoniot curves are tangent to the congestion axis ρ1 + ρ2 = 1, except at the vertexes (1, 0) and (0, 1). Proposition 5. Let ρ 0 ∈ S with ρ10 + ρ20 < 1. Then, the Hugoniot curves exiting ρ 0 intersect the {ρ1 + ρ2 = 1}-axis only for ρ10 = 0 or ρ10 = 1. For the proof, we refer to [9]. As a next step, we want to find solutions to the RH (16) condition and see whether there is a solution to the RP (6)–(14) consisting of shocks. From the first line in (15), we get the equations
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⇔
σ (ρ1L − ρ1 ) = f 1 (ρ1L , ρ2L ) − f 1 (ρ1 , ρ2 ) σ (ρ2L − ρ2 ) = f 2 (ρ1L , ρ2L ) − f 2 (ρ1 , ρ2 ) σ (ρ1L − ρ1 ) = V1 (ρ1L − ρ1 )(1 − ρ1L − ρ1 ) + V1 ρ1 ρ2 σρ2 = ρ2 V2 (1 − ρ1 − ρ2 )
where a first solution is given by ρ = (ρ1R , 0) with the shock speed σ = V1 (1 − ρ1R − ρ1L ). The second line of (15) yields ⇔
γ (ρ1 − ρ1R ) = f 1 (ρ1 , ρ2 ) − f 1 (ρ1R , ρ2R ) γ (ρ2 − ρ2R ) = f 2 (ρ2 , ρ2 ) − f 2 (ρ1R , ρ2R ) γ (ρ1R − ρ1 ) = V1 (ρ1R − ρ1 )(1 − ρ1R − ρ1 ) + V1 ρ1 ρ2 γρ2 = ρ2 V2 (1 − ρ1 − ρ2 )
A first solution is ρ = (ρ1L , 0) with γ = V1 (1 − ρ1R − ρ1L ). One obtains that σ = γ . Thus, the solution of only one shock connecting ρ L and ρ R may be possible. From now on, ρ2 = 0 in (15). Hence, the solution to the Riemann problem could also consist of two shock curves. The one-shock curve from ρ L going to an intermediate state ρ m = (ρ1m , ρ2m ) ∈ S and the two-shock curve connecting ρ m with ρ R . It is necessary that ρ m is an element of S . If it lies outside of S , the solution cannot be constructed by two shock curves. Both shocks have to fulfill the RH condition
σ (ρ1L − ρ1m ) = f 1 (ρ1L , ρ2L ) − f 1 (ρ1m , ρ2m )
σ (ρ2L − ρ2m ) = f 2 (ρ1L , ρ2L ) − f 2 (ρ1m , ρ2m ) ⎧ m m ⎪ ⎨ σ = V1 (1 − ρ L − ρ m ) + V1 ρ1 ρ2 1 1 ρ1L − ρ1m ⇔ ⎪ ⎩ σ = V (1 − ρ m − ρ m ) 2
1
2
where ρ1L = ρ1m and
γ (ρ1m − ρ1R ) = f 1 (ρ1m , ρ2m ) − f 1 (ρ1R , ρ2R ) γ (ρ2m − ρ2R ) = f 2 (ρ1m , ρ2m ) − f 2 (ρ1R , ρ2R )
⎧ m m ⎪ ⎨ γ = V1 (1 − ρ R − ρ m ) + V1 ρ1 ρ2 1 1 R ρ1 − ρ1m ⇔ ⎪ ⎩ γ = V (1 − ρ m − ρ m ) 2
1
2
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where ρ1R = ρ1m . Again, σ = γ and solving for ρ m = (ρ1m , ρ2m ) yields the intermediate state V2 ρ1L ρ1R , (V1 − V2 )(1 − ρ1L − ρ1R ) (18) V1 − V2 V2 ρ1L ρ1R m L R L R (1 − ρ1 − ρ1 ) − ρ1 − ρ1 + ρ2 = − V2 (V1 − V2 )(1 − ρ1L − ρ1R )
ρ1m =
with
σ = V2 (1 − ρ1m − ρ2m ) = V1 (1 − ρ1L − ρ1R )
(19)
and V1 = V2 , ρ1L + ρ1R = 1. The solution is given by two shock curves only if the middle state lies inside of S . We have to check the condition under which this holds. Corollary 1. For ρ1L < ρ1u < ρ1R , the intermediate state ρ m with coordinates (18) lies in the interior of the simplex S and hence, the solution to the RP with data (ρ1L , 0) and (ρ1R , 0) consists of two intersecting shock curves, if V1 ρ L + ρ1R ≤ 1, V1 − V2 1 V1 ρ1L + ρ R ≥ 1. V1 − V2 1
(20)
for V1 = V2 . Proof. It is clear that ρ m lies inside S if ρ1m + ρ2m ≤ 1 and if ρ1m , ρ2m > 0. Hence, by checking these conditions one obtains (20). 2 The second condition is needed because for ρ1R < V1V−V the curves of the second 1 family exit the simplex S on the ρ1 -axis and not the ρ2 -axis and hence may intersect with the curves of the first family on the axis, i.e., on the boundary and not in the interior of S . Then, the middle state (18) does not lie in the interior, either. The set of values where (20) holds can be seen in Fig. 2 for different maximal velocities. The blue and green lines denote the values of ρ1L and ρ1L where equality holds in (20). The shaded region is the set of values where the inequalities are fulfilled. Altogether, we can state the following.
Proposition 6. For a RP with ρ1L < ρ1u < ρ1R and ρ2L = ρ2R = 0 the Rankine Hugoniot condition (15) yields two solutions. The first consists of one shock with speed σ = V1 (1 − ρ1L − ρ1R ) connecting ρ L to ρ R . The second solution contains one shock going from ρ L to an intermediate state ρ m and one from ρ m to ρ R with the same speed σ . The coordinates of the middle state are
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Fig. 2 Values of initial data ρ1L and ρ1R (ρ2L = ρ2R = 0) for which condition (20) holds with different maximal velocities. The red lines denote the umbilic point and hence restrict the possible values of the initial data. The blue and green lines are the values where equality holds in (20). Altogether, corollary 1 is fulfilled in the blue shaded region. Note that, different from the other plots, we are in the ρ1L -ρ1L -plane
ρ1m = ρ2m
V2 ρ1L ρ1R (V1 − V2 )(1 − ρ1L − ρ1R )
V1 − V2 V2 ρ1L ρ1R =− (1 − ρ1L − ρ1R ) + (ρ1L + ρ1R ) − . V2 (V1 − V2 )(1 − ρ1L − ρ1R )
The second solution is restricted to initial data where ρ m fulfills (20).
(21)
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Depending on the position of ρ1L on the ρ1 -axis, the Hugoniot curves yield 1. ρ1L < ρ1u : then S2 (ρ1L , 0) = {ρ2 = 0} and S1 (ρ1L , 0) is monotone in ρ1 and 2. ρ1L > ρ1u : then S1 (ρ1L , 0) = {ρ2 = 0} and S2 (ρ1L , 0) is monotone in ρ1 . Since ρ R = (ρ1R , 0), it follows that 1. for ρ1L < ρ1R < ρ1u the solution consists only of S2 = {ρ2 = 0}, 2. for ρ1u < ρ1L < ρ1R the solution consists of S1 = {ρ2 = 0}, and 3. for ρ1L < ρ1u < ρ1R the solution consists either only of S1 or of S1 and S2 , intersecting at (21). Note that due to the behavior of the Hugoniot curves, the solution consists either of one or two intersecting shock curves, depending on the initial data. If both values of the RP lie left or right of the umbilic point, we immediately get the same result as for the one species model. Here, the consistency with the LWR model for one species is given. As a next step, it is interesting to see how the intermediate state computed from the RH behaves depending on the initial data. For this, we consider some special cases of RP. We start with values close to the umbilic point and examine ρ m . Corollary 2. If the initial data is given by ρ1L = ρ1u − ε and ρ1R = ρ1u + ε with ε > 0, the intermediate state has the coordinates ρ1m = ρ1u − ρ2m
ε2 ρ1u
ε2 = u. ρ1
(22)
Then, one immediately sees that ε → 0 yields ρ m = ρ u . This is convenient since then both initial data equal ρ u . Now, if ε becomes bigger, the initial data lies further apart from the umbilic point and ρ m wanders to smaller ρ1 and bigger ρ2 . This goes on until ε = ρ1u and thus ρ1L = 0. At this point, we have ρ1L = 0 ρ1R = 2ρ1u
(23)
and ρ m = (0, ρ1u ). Here, the shock speed does not depend on ρ1L and ρ1R σ = V1 (1 − 2ρ1u ) = V2 (1 − ρ1u )
(24)
and is always positive since ρ1u < 1/2. Moreover, here the intermediate state lies in the interior of the simplex S which is consistent with proposition 1 because the initial data fulfills (20). The described behavior can be seen in Fig. 3 where we also consider RP with initial data changing in a different relation to each other.
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Fig. 3 Sample of intersecting Hugoniot curves for V1 = 1 and V2 = 0.75 and different initial data on the ρ1 -axis. In the first three pictures, the solution consists of two shock waves intersecting in the interior of S while in the last one, the solution is given by one curve connecting ρ L to ρ R
In Fig. 3b, we see the intersecting Hugoniot curves for initial data with ρ1L = − ε and ρ1R = ρ1u + 2ε. This means that the distance between the umbilic point and the right initial datum ρ1R grows two times faster than the one between ρ1u and ρ1L . In the other two plots in Fig. 3c, d, we see the same with 3ε and 4ε. In (d), we only see one curve because here the equality of condition (20) holds. This implies that for these data, there is one-shock curve connecting ρ1L to ρ1R . For data which do not fulfill (20), we have already seen that the solution consists of only one shock curve. We conclude the continuous dependence of the intermediate state from the initial data. Another special case is to start with initial data with nearly maximal distance in S . This means that ρ1L is nearly 0 or ρ1R is nearly 1. ρ1u
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Corollary 3. Consider the RP with ρ1L = 0 and ρ1R = 1 − ε with ε > 0. The coordinates of the middle state are ρ1m = 0 ρ2m
V1 − V2 . =1−ε 1− V2
(25)
Hence, ρ m lies on the ρ2 -axis and moves downward for bigger ε. The shock speed is given by (26) σ = εV1 . We observe that for ε → 0 the middle state equals ρ m = (0, 1). Altogether, we described the intermediate state for all values of ρ1L and ρ1R , respectively. There is continuity between Corollary 2 and 3. Mention that the solution consists of two shocks with speed σ = V1 (1 − ρ1L − ρ1R ) for both shocks. Hence, the solution does not attain the value ρ m . For all initial data with ρ2L = ρ2R = 0, the solution to the two-species LWR model is given by ρ(x, t) = (ρ1 (x, t), 0) with ρ1 (x, 0) =
ρ1L for x < σ t ρ1R for x > σ t
.
(27)
But before, one must check which of the solutions of Proposition 6 is admissible and to which family the Hugoniot curves belong. For this purpose, the Lax condition must be checked. Lemma 2. A shock of the ith family, connecting ρ L to ρ R with speed σ , is admissible in the sense of [6], if (28) λi (ρ R ) ≤ σ ≤ λi (ρ L ) holds. If both initial data lie either left or right of the umbilic point, the solution will consist of only one shock. Checking the Lax inequality leads to 1. for ρ1L < ρ1R < ρ1u the condition λ2 (ρ1R , 0) < σ < λ2 (ρ1L , 0) holds 2. for ρ1u < ρ1L < ρ1R the condition λ1 (ρ1R , 0) < σ < λ1 (ρ1L , 0) holds. For ρ1L < ρ1u < ρ1R , we have the same shock speed for both shocks, and thus, the Lax inequality (28) is checked for only one shock from ρ L to ρ R . The Lax admissibility conditions are λ1 (ρ1R , 0) < σ < λ1 (ρ1L , 0), (29) λ2 (ρ1R , 0) < σ < λ2 (ρ1L , 0).
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with shock speed σ . The eigenvalues yield λ1 (ρ1L , 0) = V2 (1 − ρ1L ), λ1 (ρ1R , 0) = V1 (1 − 2ρ1R ), λ2 (ρ1L , 0) = V1 (1 − 2ρ1L ),
(30)
λ2 (ρ1R , 0) = V2 (1 − ρ1R ). Hence, we get the next proposition. Proposition 7. The Lax admissibility demands that for ρ1L < ρ1u < ρ1R the shock connecting ρ L to ρ R with speed σ is • a one shock if ρ1R > φ(ρ1L ) and φ(ρ1R ) < ρ1L , • a two shock if ρ1R < φ(ρ1L ) and φ(ρ1R ) > ρ1L , and • an over compressive shock if ρ1R < φ(ρ1L ) and φ(ρ1R ) < ρ1L with the C 1 -function φ(ρ1 ) =
(1 − ρ1 )V1 − V2 . V1 − V2
(31)
with V1 = V2 . Proof. From the expressions (30), one sees that λ1 (ρ1R , 0) < σ λ2 (ρ1L , 0) > σ.
(32)
The other inequalities are obtained by computation σ < λ1 (ρ1L ) ⇔ φ(ρ1R ) < ρ1L σ > λ2 (ρ1R ) ⇔ φ(ρ1L ) > ρ1R .
(33)
We observe that the existence of the umbilic point leads to overcompressive shocks. This is also discussed in [1]. Since we closed the case ρ1L < ρ1R we now propose that ρ1L > ρ1R . By [3], the solution to a single PDE consists of a rarefaction wave. Since we are in the case of the second species absent, we can use this. The question is, whether the existence of the umbilic point affects the solution similarly to the previous discussed shock waves or not. The rarefaction waves are obtained by integration along the eigenvectors. A sample of eigenvectors can be seen in Fig. 4, oriented so that dλi · vi > 0 for i = 1, 2. Note that from Theorem 1 both characteristic fields are genuinely nonlinear for data on the ρ1 -axis. One has already stated in Lemma 1 that the eigenvectors coalesce in the umbilic point, too. We observe this in Fig. 4. On the ρ1 -axis, the first eigenvector is parallel to the axis for ρ1 > ρ1u while the second one is parallel for
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Fig. 4 Sample of eigenvectors with V1 = 1 and V2 = 0.75, oriented so that dλi · vi > 0
ρ1 < ρ1u . In the umbilic point (for the plots ρ1u = 0.2), they coincide. Moreover, the eigenvectors change continuously with the densities as can be seen in the zoom in Fig. 4c, d. We do not integrate the curves explicitly here, because of the complexity of the algebraic expressions of the eigenvectors. But we are able to compute the eigenvectors on the ρ1 -axis from equation (11) by using ρ2 = 0. Due to the behavior of the eigenvalues in (10) we also have to make a distinction for the eigenvectors.
⎧ −V1 ρ1 ⎪ ⎪ ⎨ V2 (1 − ρ1 ) − V1 (1 − 2ρ1 )
v1 (ρ1 , 0) = ⎪ ⎪ ⎩ V1 (1 − 3ρ1 ) − V2 (1 − ρ1 ) 0
⎧ −V1 (1 − 2ρ1 ) + V2 (1 − ρ1 ) ⎪ ⎪ ⎨ 0
v2 (ρ1 , 0) = ⎪ −V1 ρ1 ⎪ ⎩ −V1 (1 − 2ρ1 ) + V2 (1 − ρ1 )
for ρ1 < ρ1u (34) for ρ1 >
ρ1u
for ρ1 < ρ1u (35) for ρ1 >
ρ1u
We have already seen that in the umbilic point the eigenvectors coalesce.
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v1 (ρ1u , 0) = v2 (ρ1u , 0) =
−V1 ρ1u 0
(36)
For ρ1u < ρ1L , the rarefaction wave of the first family passing through ρ1L is given by R1 (ρ1L , 0) = {ρ2 = 0}. If ρ1R < ρ1u , the 2-rarefaction wave is also described by R2 (ρ1R , 0) = {ρ2 = 0}. Hence, for ρ1R < ρ1u < ρ1L and ρ2R = ρ2L = 0 the solution consists of two rarefaction waves with intermediate state ρ m = (ρ1u , 0). Thereby, R1 (ρ1L , 0) goes from ρ L to ρ m and R2 (ρ1u , 0) connects the middle state with ρ R . The solution consists of only one rarefaction curve in the case that both initial data lie left (second family) or right (first family) of the umbilic point. This is consistent with Fig. 4 and is the only admissible solution in the sense of [6]. The two rarefaction curves are equal to each other and so the solution to the RP for the fast species (ρ2L = ρ2R = 0) for the LWR model of two species can again be constructed with the help of the standard LWR model. In the same way as for the shock curves, we get ρ(x, t) = (ρ1 (x, t), 0), with ⎧ ⎪ ρ1L ⎪ ⎪ ⎪ ⎪ ⎨ x 1 ρ1 (x, t) = (1 − ) ⎪ 2 V t ⎪ ⎪ ⎪ ⎪ ⎩ ρ1R
x < V1 (1 − 2ρ1L ) t x for V1 (1 − 2ρ1L ) < < V (1 − 2ρ1R ). t x > V1 (1 − 2ρ1R ) for t for
(37)
Again, we do not see the middle state because the solution does not attain this value. The existence of the umbilic point does not affect the solution to the RP for rarefaction waves, either. Closing this subsection, one has to mention that the existence of the solution to the general RP (6)–(14) was not fully proved due to the fact that the expressions of eigenvalues, eigenvectors and shock curves are hard to handle.
2.2 Perturbation of the Riemann Problem We now want to see whether the solution depends continuously on the initial data or not. Therefore, one can look at a small perturbation ε > 0 of the Riemann problem (6)–(14) which is (6) together with ρ(x, 0) =
ρ L = (ρ1L , ε) for x < 0 ρ R = (ρ1R , ε) for x > 0
.
(38)
We assume that there are a small number of vehicles of the slower species on the road, too. Now, we want to examine how this small perturbation of the initial data affects the solution. We again examine the discontinuities and rarefaction waves. Two shocks with speeds σ and γ , connecting ρ L to ρ m and ρ m to ρ R , have to fulfill the RH condition
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σ (ρ1L − ρ1m ) = f 1 (ρ1L , ρ2L ) − f 1 (ρ1m , ρ2m )
σ (ρ2L − ρ2m ) = f 2 (ρ1L , ρ2L ) − f 2 (ρ1m , ρ2m ) ⎧ ρ1m ρ2m − ερ1L ⎪ L m ⎪ σ = V (1 − ρ − ρ ) + V ⎪ 1 1 1 1 ⎨ ρ L − ρm 1
(39)
1
⎪ ρ m ρ m − ερ L ⎪ ⎪ ⎩ σ = V2 (1 − ε − ρ2m ) + V2 1 2 m 1 ε − ρ2
γ (ρ1m − ρ1R ) = f 1 (ρ1m , ρ2m ) − f 1 (ρ1R , ρ2R ) γ (ρ2m − ρ2R ) = f 2 (ρ1m , ρ2m ) − f 2 (ρ1R , ρ2R )
⇔
⎧ ρ1m ρ2m − ερ1R ⎪ R m ⎪ ⎪ ⎨ γ = V1 (1 − ρ1 − ρ1 ) + V1 ρ R − ρ m ⎪ ⎪ ⎪ ⎩ γ = V2 (1 − ε − ρ2m ) + V2
(40)
1 ρ1m ρ2m
ε
1 − ερ1R − ρ2m
Then, the two shock speeds are not equal. They differ by γ − σ = V2 ε
ρ1R − ρ1L . ρ2m − ε
(41)
One observes that the speed of the two-shock is always greater than the speed of the one shock. For ε → 0, the speeds are the same again and equal to the shock speed of the unperturbed system. Here, different from the RP ρ1 -axis, the middle state appears in the solution. This is convenient because the second species is present in this case.
Fig. 5 Sample of Hugoniot curves exiting a point near the ρ1 -axis with the umbilic point (ρ u = (0.2, 0)) where V1 = 1 and V2 = 0.75
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Fig. 6 Sample of Hugoniot curves with V1 = 1 and V2 = 0.75 for initial data lying in the interior of S
Now, the intermediate state ρ m cannot be computed explicitly from (39) and (40), and thus, the exact shock speeds cannot be found either. We can only solve Eqs. (39) and (40) by inserting data for the RP. Moreover, the solution could consist of both shocks and rarefactions because we are not in a one-species case anymore. By plotting the Hugoniot curves in Fig. 5, one sees that the Hugoniot curves depend continuously on the initial data. For data close to the ρ1 -axis, the behavior of the curves changes only slightly. The rarefaction curves are again obtained by integrating along the eigenvectors (4). We get solutions different from the previous case, too, because the eigenvectors are not parallel to the {ρ2 = 0}-axis for data lying in the interior of S .
2.3 Conclusion In this contribution, we have considered the Riemann problem for the two-species extension of the Lighthill–Whitham traffic model given by [1]. We found a solution to the RP for data on the {ρ2 = 0}−axis. We have seen that the solution depends continuously on the initial data, because if we perturb the RP on the fast axis by ε > 0, small, we will observe a small variation in the Lax curves. The general RP is defined in (6)–(13) for the LWR model for two species. The complexity of the expressions hinders us from further general search for explicit expressions related to system (6). We know that for initial data with small variation different from the umbilic point, the RP is well-posed. But for a general RP with data lying far apart, it is difficult to prove well-posedness due to the complexity of the
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expressions. The idea is to find a middle state ρ m as the intersection of the two Lax curves L 1 (ρ L ) and L 2 (ρ R ). Mention that now the solution can also contain both shocks and rarefaction waves and on the {ρ1 + ρ2 }-axis contact discontinuities. In Fig. 6, one sees samples of shock curves belonging to the first and second family. The rarefaction waves are obtained by integrating along the eigenvectors, which have already been plotted in Fig. 4. These plots give no indication that the RP (6) should be ill-posed, as was conjectured in by [1]. For data different from the umbilic point, the system is indeed strictly hyperbolic and around ρ u we solved the RP. Moreover, the simplex S is convex and because of (12) we assume its invariance. But since we have no (global) proof of well-posedness for all initial data, we cannot be sure that the lack of global strict hyperbolicity does not lead to ill-posedness. To summarize, the extension of the LWR model to a two populations model proves to be difficult, since global hyperbolicity is not given. There exists an umbilic point on the boundary of the set where we define the system, meaning at the boundary the eigenvalues coalesce. To the best of our knowledge well-posedness of a similar model (w. umbilic point on the boundary) has not been discussed in the literature. Moreover, the model, although it is of simple structure, yields intricate expressions for the corresponding eigenvalues and vectors. Even though our studies seem to hint at well-posedness of this model, a proof of well-posedness of our model seems elusive.
References 1. S. Benzoni-Gavage, R.M. Colombo, An n-populations model for traffic flow. Eur. J. Appl. Math. 14(5), 587–612 (2003). https://doi.org/10.1017/S0956792503005266. ISSN 0956-7925 2. A. Bressan, Hyperbolic systems of conservation laws, in Oxford Lecture Series in Mathematics and its Applications, vol. 20. (Oxford University Press, Oxford, 2000), (The one-dimensional Cauchy problem). ISBN 0-19-850700-3 3. C.M. Dafermos, Hyperbolic conservation laws in continuum physics, in Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 4th edn., vol. 325. (Springer, Berlin, 2016), https://doi.org/10.1007/978-3-642-04048-1, https://doi.org/10. 1007/978-3-662-49451-6. ISBN 978-3-642-04047-4 4. D. Hoff, Invariant regions for systems of conservation laws. Trans. Amer. Math. Soc. 289(2), 591–610 (1985). https://doi.org/10.2307/2000254. ISSN 0002-9947 5. B.L. Keyfitz, H.C. Kranzer, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory. Arch. Rational Mech. Anal. 72(3), 219–241 (1979/80). https://doi.org/10. 1007/BF00281590. ISSN 0003-9527 6. P.D. Lax, Hyperbolic systems of conservation laws II. Commun. Pure Appl. Math. 10, 537–566 (1957). ISSN 0010-3640 7. M.J. Lighthill, G.B. Whitham, On kinematic waves II. a theory of traffic flow on long crowded roads. Proc. R. Soc. Lond. Ser. A 229, 317–345 (1955). ISSN 0962-8444 8. S. Liu, F. Chen, Existence of global L p solutions to a symmetric system of Keyfitz–Kranzer type. Appl. Math. Lett. 52, 96–101 (2016). https://doi.org/10.1016/j.aml.2015.08.011. ISSN 0893-9659 9. M.-C. Meltzer, Multispecies traffic models based on the Lighthill–Whitham and the Aw-Rascle model, Thesis (2016), https://www.mathematik.uni-wuerzburg.de/~klingen/Workgroup_files/ Thesis_Marie-Christine_Meltzer.pdf
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10. P.I. Richards, Shock waves on the highway. Oper. Res. 4, 42–51 (1956). ISSN 0030-364X 11. D. Serre, Systems of Conservation Laws 1 (Cambridge University Press, Cambridge, 1999). https://doi.org/10.1017/CBO9780511612374. Hyperbolicity, entropies, shock waves, Translated from the 1996 French original by I. N. Sneddon. ISBN 0-521-58233-4
Semi-Lagrangian Particle Methods for Hyperbolic Equations Georges-Henri Cottet
Abstract Particle methods with remeshing of particles at each time step can be seen as forward semi-Lagrangian conservative methods for advection-dominated problems, and must be analyzed as such. In this article, we investigate the links between these methods and finite-difference methods and present convergence results as well as techniques to control their oscillations. We emphasize the role of the size of the time step and show that large time steps, only limited by the flow strain, can lead to significant gains in both computational cost and accuracy. Our analysis is illustrated by numerical simulations in level set methods and in fluid mechanics for compressible and incompressible flows. Keywords Particle methods · Semi-Lagrangian methods Non-linear conservation laws
1 Particle Methods for Conservation Laws Particle methods have long been considered as natural tools for the discretization of conservation laws, written in general form as ∂U + div (a : U) + AU = F. ∂t
(1)
In the above equation, U is a vector in Rm , a = (a j ) is a vector field in Rn and ⎛ div (a : U) ≡ ⎝
n j=1
⎞ ∂(a j u i ) / ∂ x j ⎠
.
i∈[1,m]
G.-H. Cottet (B) University Grenoble Alpes and Institut Universitaire de France, Grenoble, France e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_31
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The term F represents both external forces and terms that are not related to advection, typically diffusion terms or pressure gradients. This PDE can be deduced from the following conservation property d dt
Ω(t)
U dx +
Ω(t)
AU dx =
F dx, Ω(t)
where Ω(t) is a domain of Rn moving with velocity a. System (1) must of course be supplemented with boundary conditions (at least inflow boundary condition in case F = 0). However, we will here focus on unbounded problems, or problems with periodic boundary conditions. Particle methods consist of concentrating the mass of U on points (particles), which means that the following approximation is considered: Uh (x, t) =
α p (t)δ(x − x p (t)).
(2)
p
Particle trajectories are along the velocity field dx p = a(x p , t) dt and particle strengths change to account for zero-order terms and right-hand side F. When smooth quantities must be recovered/plotted one can rely on mollified particles (or blobs) α p (t)ζε (x − x p (t)) (3) Uhε (x, t) = p
where ζε (x) = ε−n ζ (x/ε) with ζ a smooth function satisfying ζ (x) dx = 1. The strengths α p of the particles represent local masses. It may be convenient to write these masses in terms of local values of u and local volumes v p : α p (t) = U(x p (t), t)v p (t) with
dv p = diva(x p , t)v p . dt
In case of an incompressible flow (div a = 0), strengths, local values, and volumes of the particles are conserved along the flow. The velocity field, as well as the matrix A and the right-hand side F can either be given (linear problems) or be a function of the solution U (nonlinear problems). In the latter case, the coupling of the advected quantities with the flow field requires to consider the mollified particle distribution, which, as we will see below, has important consequences on the convergence properties of the method.
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1.1 Classical Examples In the linear case, passive scalars advected by a flow, like concentration or density in incompressible flows or level set functions to capture an interface, are natural examples. When scalars are also involved in the flow equations, like temperature or density in compressible flows, or interface capturing level set functions in presence of surface tension, one obtains nonlinear cases where particle methods can also be used. More elaborate examples include the Vlasov–Maxwell system, the Navier–Stokes equations in vorticity form, and the equations of gas dynamics. In the Vlasov–Maxwell system the advected quantity is the density probability f associated to each type of ions or electrons, moving under the action of the electric and magnetic fields. The phase-space variables are the positions and velocities (in other words n = 6 in the general case). The conservation law for f can be written as (1) with a(x, v) = (v, E(x, t) + v × B(x, t)), A = F = 0 (assuming an elementary electric charge equal to 1). The electric and magnetic fields are computed from the momentum of f with respect to v (charge density and electric current) through the Maxwell equations. We refer to [4, 14] for descriptions and numerical analysis of particle methods in this context. Inviscid incompressible flows can be described by the vorticity form of the incompressible Euler equations [5]. In this case, n = m = 3, U = ω = ∇ × u, a = u and A = −[∂u i /∂ x j ]. For the Navier–Stokes equations, viscosity and diffusion come in the right-hand side F. Note that in two dimensions, it is convenient to represent the vorticity as a vector along the axis perpendicular to the flow plane. In this case, the so-called stretching term AU vanishes and the mass of the particles coincide with the local circulation of the flow. In 3D, the stretching term is responsible for reorientation and amplification of the local vorticity, and ultimately of the onset of 3D turbulence. Finally, gas dynamics equations can be recast in terms of density, momentum, and energy: U = (ρ, ρu, ρ E), which in the most general case gives m = 5 and n = 3. The term F is made of pressure gradients terms to complete the momentum and energy equations. Its discretization by means of particles led to the Smooth Particle Hydrodynamics (SPH) methods [22].
1.2 Sketch of Numerical Analysis and Overlapping Condition To give an idea of the numerical analysis of these methods, let us assume for a sake of simplicity a linear equation where A = F = 0. One way to understand the convergence properties of particle methods [6] is to realize that particle approximation given by (2) are exact weak solutions to the advection equation (1). One can further show that, for smooth enough velocity fields, the advection equation is stable in distribution spaces of the form W −m, p (Rn ). As a result one can write, for any given time interval [0, T ],
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(U − Uh )(·, t)W −m, p ≤ exp (C T ) (U − Uh )(·, 0)W −m, p for t ∈ [0, T ], where C depends on the derivatives of a. If at t = 0, particles are initialized on a regular grid of grid size h and if the initial condition is of class C ∞ with compact support (or periodic) the right-hand side above can be bounded through classical quadrature rules by O(h m ) for arbitrary m, which gives (U − Uh )(·, t)W −m, p = 0(h m ) for t ∈ [0, T ]. Now, to obtain error bounds in L p norms for the mollified particles (3), one needs to add regularization errors and to “pay the price” for the mollification. This easily leads to the following estimate (U − Uhε )(·, t)W −m, p ≤ exp (C T ) (εr + h m /εm ) for t ∈ [0, T ],
where r is such that xγ ζ (x) dx = 0 for 1 ≤ |γ | ≤ r − 1. The above estimate exhibits two scales in the convergence process: ε, the mollifying range, which eventually dictates the overall order of the method, and h the particle spacing. It also shows that convergence requires h ε, or in other words that many particles lie in the mollifying range. In the case when the right-hand side F involves pressure gradients or diffusion term, one can easily predict that this overlapping condition will be even more demanding. A more precise analysis also shows that the flow strain (derivatives of the velocity) is responsible for the exponential term in the error estimate, which indicates that the overlapping requirement gets more difficult to fulfill in presence of strong shear in the flow.
2 From Grid-Free to Semi-Lagrangian Particles The overlapping condition just outlined has long been recognized as a major difficulty to perform accurate simulations, in particular for nonlinear problems. Figure 1 illustrates typical numerical artifacts appearing in the simulation of a smooth axisymmetric (and thus steady) vortex for the incompressible 2D Euler equations, when the blob size ε remains constant and of the order of the initial particle spacing. The situation is even more problematic when the nonlinearity does not involve any smoothing effect, unlike in the case of incompressible flows in vorticity formulation just mentioned or for the Vlasov–Poisson equations. In that case, the flow strain prohibits any convergence proof. This includes the simple 1D Burger’s equations. As a matter of fact, convergence analysis in the SPH literature (e.g., [1]) always assume that particles are carried by a smooth flow field and focus on the particle treatment of pressure gradient terms. An important effort has been made starting in the 80s to overcome this difficulty for flow simulations. In particular, a class of methods aims at adapting the particle weights or mollifying range to the local flow strain (see [5] and the references therein).
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Fig. 1 Numerical artifact due to a lack of overlapping in the simulation of a steady smooth axisymmetric vortex in the 2D incompressible Euler equation (courtesy of P. Koumoutsakos). Left picture: exact solution; right picture: numerical solution with ε h
Triangulated particle methods for vortex methods [27] or renormalization methods for gas dynamics [23] fall in this category. More recently [26], methods have been designed in particular to improve the treatment of diffusion with grid-free particles. However, these methods have been mostly validated for 2D flows and it is not clear that they offer viable tools (in terms of accuracy and cost) for complex 3D flows. An alternative approach consists of remeshing particles from time to time in order to maintain the initial overlapping. This strategy can be traced back to inviscid equations and the first vortex sheet calculations [17] or vortex filament calculations [21]. In these cases, particles where lying on curves and the overlapping could be maintained by simply inserting fresh particles when successive particles were too far apart. The weights of the fresh particles could easily be computed by interpolation along filaments (in 3D) or sheets (in 2D). For more general vortex topology and/or to handle the Navier–Stokes equations, a more systematic approach, together with appropriate treatment of no-slip boundary conditions, was introduced and implemented for flow past 2D cylinders at high Reynolds numbers in [15] and for the study of the axisymmetrization of elliptical vortices in [16], where it allowed to obtain reference results. These calculations were soon followed by 3D simulations [9, 24, 25]. Until recently, particle remeshing was considered as an ad hoc fix, to maintain the regularity of the particle distribution at the price of some truncation error every few time steps. In practice, the number of time steps between two successive remeshing steps never increases as the particle spacing tends to zero, which means that the remeshing error cannot be considered as an additional truncation error, but must be analyzed as part of the particle scheme. On the other hand, the time scale which governs the particle distortions is of order ∇u−1 L ∞ , similar to the time scale which is used for the time discretization of the particle motion. As a result, these time scales are routinely taken equal, which means that remeshing is performed at each time step. It is important to point out that this particular implementation offers several additional advantages. It allows particle methods to be combined with grid-based methods when appropriate, either in the same computational domain (for example, to rely on
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FFTs to compute velocity or electric/magnetic fields) or in domain decomposition approaches [24]. It also enables Adaptive Mesh Refinement [2] or wavelet-based multilevel [3] approaches. For advection equations with uniform grids, the resulting method becomes a forward, conservative semi-Lagrangian method and must be analyzed as such.
3 Semi-Lagrangian Particle Methods for Linear Hyperbolic Equations In the following, we denote by Δx the grid size and Δt the time step. Let us consider the following one-dimensional advection equation in conservation form ∂u + div (au) = 0, x ∈ R, t > 0, ∂t
(4)
where a is a given smooth velocity field. Assuming that particles are initialized and remeshed on a uniform grid with grid size Δx, we obtain u in+1 =
j
u nj Γ
x n+1 − xi j Δx
, i ∈ Z, n ≥ 0,
(5)
where Γ is the interpolation function used to remesh particles. In the above equation xin+1 = xi + a˜ in Δt,
(6)
where a˜ in is the velocity field used to push particle and depends on the chosen timestepping scheme. As we will see later, the construction of accurate interpolation kernels is done by requiring the conservation of successive moments of the particle distribution. This means that Γ satisfies
1 if α = 0 (x − k)α Γ (x − k) = , x ∈ R, (7) 0 if 1 ≤ α ≤ p k∈Z for some value of p ≥ 1. The simplest example of interpolation kernelis the piecewise xi u in ( p = 1). linear function, which conserves mass i u in and first momentum In the case of constant velocity this choice corresponds, for a CFL number less than 1, to the classical first-order upwind scheme. However, it is readily seen that this choice can lead to inconsistency if the local CFL number crosses an integer value, in particular when the velocity changes sign, whatever time step is chosen. For instance, for a(x) = x, a particle initialized and remeshed at x = 0 will keep its strength unchanged although the exact solution at this point is given by u 0 (x) exp (−t).
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The number of integers inside the support of Γ must increase with p. The choice p = 2 leads to a piecewise quadratic function, with support on three grid points, which yields the following weights (according to Fig. 2) α j = −λ(1 − λ)/2 , β j = 1 − λ2 , γ j = λ(1 + λ)/2,
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where λ is the algebraic distance, normalized by Δx, between the particle, after advection, and the nearest grid point. For a constant velocity and a CFL number below 1/2, this formula corresponds to the Lax–Wendroff finite-difference scheme [8]. However, for CFL numbers larger than 1/2, this kernel can lead to inconsistencies which can be corrected by first- order terms [20]. The consistency issues just mentioned are actually related to the lack of regularity of the kernels: the piecewise linear kernel is not differentiable, and the piecewise quadratic kernel is not even continuous. Indeed, requiring more regularity, together with moment properties and the interpolation property, which means that remeshing does not change the weights if particles do not move, allows to prove a general consistency theorem [10]. Under the following assumptions Γ Γ Γ Γ
is even and piecewise polynomial in intervals of the form [i, i + 1], is of class C r , for r ≥ 1 satisfies the moment properties (7) for p ≥ 1, satisfies the interpolation property Γ (i − j) = δi, j , i, j ∈ Z
and provided that the time step Δt satisfies the following condition (sometimes referred to as a Lagrangian CFL condition) Δt < |a |−1 L∞ ,
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one can prove that, for an Euler time-stepping scheme for the motion of particles, the consistency error of the semi-Lagrangian method is bounded by O(Δt + Δx β ) where β = min ( p, r ). Moreover, at least for kernels of order up to 4, under appropriate decay properties for the kernel Γ one can prove the stability of the method under the sole assumption (9). Higher order in time can be recovered by using classical Runge–Kutta methods to push particles. The kernel conditions listed above can be used to construct in a systematic fashion high order kernels, denoted Λ p,r (see in [10] examples for β up to 6).
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Fig. 3 Advection of a level set in a 2D off-center vorticity field [10]. Comparison of the exact solution (left) and fourth-order particle solution using N = 2562 particles and a CFL number equal to 30
Fig. 4 Advection of a level set in a 3D off-center vorticity field [10]. Comparison of the results obtained by a first-order and fourth-order kernels, using N = 2563 particles with a CFL number equal to 30
The above discussion concerns 1D equations but carries on to several dimensions using directional splitting for advection equations. In practice, it means that particles are transported and then remeshed in successive directions, with the possibility of using high order splitting methods. As a matter of fact, dimensional splitting allows to use high order (with large support) kernels at an affordable cost. Figure 3 presents a comparison between computed and exact solutions in the classical level set case of a disk undergoing filamentation in the velocity field resulting from a smooth off-center vortex. The particle method uses the fourth-order kernel Λ6,4 , 2562 points and a CFL number equal to 30. For this experiment, a secondorder directional splitting and a fourth- order Runge–Kutta method were used and the observed order of accuracy was 5.9 (we refer to [10] for more examples and refinement studies). Figure 4 shows a comparison of the efficiency of kernels Λ2,1 and Λ8,4 for a similar 3D experiment, with 2563 points and a CFL number equal to 30.
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A striking example where the combination of high-order accuracy and large time steps can lead to very efficient implementations of semi-Lagrangian particle methods in the study of scalar turbulence. In the passive advection of a scalar in a turbulent flow, when the diffusivity of the scalar is smaller than the viscosity of the flow, the spectrum of the scalar exhibits a k −1 decay range, where k is the wave number, beyond the classical k −5/3 range of the flow. This implies that, for Direct Numerical Simulations, it is necessary to use a finer resolution for the scalar than for the flow. The ratio of the flow viscosity to the scalar diffusivity is called the Schmidt number (Sc) and √ the ratio between the corresponding grid discretizations has to be of the order of Sc. Using semi-Lagrangian particles allows to increase the scalar resolution while keeping time steps defined by the flow strain and not by the fine particle discretization. In [18] systematic simulations coupling semi-Lagrangian particles for the scalar and spectral methods for the flow, allowed to exhibit the k −1 scalar spectrum decay for a large range of Schmidt numbers and to confirm the Kraichnan prediction for the dissipative scales of the scalar (see Fig. 5). In the case of a Schmidt number equal to 128, resolutions of up to 30003 were made possible because the time step could be larger than what would be necessary in a spectral method by a factor close to 100. Figure 6 shows scalar and vorticity contours in a periodic turbulent plane jet [13], for a Reynolds number equal to 103 and a Schmidt number equal to 64. This simulation used semi-Lagrangian particles both for the vorticity to solve the Navier–Stokes equations, and for the scalar, respectively, at grid resolutions of 1283 and 10243 . The flow and scalar particles were computed on different hardware, to account for the different parallel scalability of te different parts of the algorithm: 8 CPUs for the flow and 8 GPUs for the scalar for a computational cost of about 1.5 sec per iteration.
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Fig. 6 Periodic plane jet for Re = 103 and Sc = 64 [13]. Side (left) and top (right) views showing the different scales in the vorticity (bottom) and the scalar (top). Simulation with 1283 particles for the flow and 10243 particles for the scalar
4 Semi-Lagrangian Particles for Nonlinear Conservation Laws Let us first consider scalar equations of the form u t + (g(u)u)x = 0.
(10)
The situation is different from the linear case. If one considers the piecewise linear remeshing kernel together with an explicit first-order time discretization, it is easy to see that, for a CFL number smaller than 1, it translates into the following 3 points scheme: (11) = u nj − λ u j+1 g −j+1 + u j g +j − u j g −j − u j−1 g +j−1 , u n+1 j
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where λ = Δt/Δx and the superscript + and − denote the positive and negative parts. This scheme has a numerical flux given by F(u, v) = ug(u)+ + vg(v)− and is, therefore, consistent with the original equation. Let us now consider the three-point second-order formula (8) associated to the piecewise quadratic kernel corresponding to the Lax–Wendroff scheme in the constant coefficient case. To obtain a second-order scheme both in time and space, one can consider a Runge– Kutta scheme, where the particle velocities are predicted by advancing the equation for half a time step. That can be done using the following approximation for the particle velocity at time (tn + tn+1 )/2) [7] n+1/2
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= u nj 1 − Δt g(u)x (x j )/2 .
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In practice, g(u)x (x j ) in the above formula is evaluated by a centered finite difference (g(u nj+1 ) − g(u nj−1 ))/(2Δx). The finite-difference formula corresponding to this method for a CFL number below 1/2 can be then derived along the same lines as in the linear case and is given by Δt n n (g˜ u − g˜ nj−1 u nj−1 ) 2Δx j+1 j+1 Δt 2 n 2 n + (g˜ j+1 ) u j+1 − 2(g˜ nj )2 u nj + (g˜ nj−1 )2 u nj−1 2 2Δx
= u nj − u n+1 j
n+1/2
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where we have set g˜ nj = g(u j ). It can be checked [8, 28] that this is a second-order scheme. For systems of conservation laws, the above ideas extend to flux splitting methods, where advective fluxes are dealt with by semi-Lagrangian particles, whereas pressure gradient terms are handled by grid-based finite-volume methods. These methods bear some similarities with the Advection Upstream Splitting Method [19], with the difference that in our case the advection does not involve the pressure flux in the energy equation. It also can be seen as a variation of the Lagrange-projection methods [12], with the difference that in our case the advection step is performed in the conservation form. Note that, in the case of systems, the up-winding implicitely resulting from particle motions is based on material velocities and not on wave speeds. Figure 7 shows a comparison of a semi-Lagrangian particle method using the Λ2,1 kernel with a Mac–Cormack scheme and a third-order limiter derived in [11] for the calculation of the interaction of a shock wave with a boundary layer of a Reynolds number of 200, with the same grid resolution Δx = 10−3 .
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Fig. 7 Density levels in a shock / boundary layer interaction. Comparison of the results of a semi-Lagrangian particle method (top) and of the third-order Mc Cormack finite-difference scheme [11] (bottom). In both calculations, Δx = 10−3
5 Non-oscillatory Semi-Lagrangian Particles Like all high-order methods, semi-Lagrangian particle methods can produce spurious oscillations near sharp variations of the solution. To prevent this, limiters can be derived in a similar way to finite-difference schemes. The general idea is to start from the equivalent finite-difference scheme when the CFL number is below a certain value, to derive limiters for these schemes and to go back to remeshing formulas by interpreting the modified finite-difference coefficients.
5.1 The Linear Case In this section, we give the derivation presented in [20] for non-oscillatory remeshing schemes obtained by limiting second-order centered formulas with first-ordercentered formulas. More specifically, since we have seen in Sect. 3 that the piecewise linear remeshing is not consistent for nonconstant velocities, we choose a 3-point first-order-centered formula. For |λ = aΔt/Δx| ≤ 1/2, with the the notations of Fig. 8 the weights corresponding to this remeshing kernel are given by:
Semi-Lagrangian Particle Methods for Hyperbolic Equations Fig. 8 Comparison of a semi-Lagrangian particle method (green curve) and a fifth-order WENO scheme (blue curve), using the same grid size, with the exact solution (red curve), for the advection of a double top-hat function in a sinusoidal velocity field [20]. The CFL number for the particle method is 12
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Exact solution Particle method Weno scheme
α j = 3/4 − λ2 , β j = 1/2(1/2 − λ)2 , γ j = 1/2(1/2 + λ)2 . This is equivalent to a first-order scheme written in incremental form as (1) = u nj + C (1) u n+1 j j+1/2 Δv j+1/2 − D j−1/2 Δv j−1/2
where Δv j+1/2 = v j+1 − v j and (1) 2 2 C (1) j+1/2 = 1/2(λ − 1/2) , D j−1/2 = 1/2(λ + 1/2)
Similarly, the second-order remeshing scheme (8), which corresponds to the Lax– Wendroff scheme when |λ| ≤ 1/2, can be recast in a similar incremental form with (1) (2) (1) coefficients C (2) j+1/2 = C j+1/2 − 1/8 and D j−1/2 = D j−1/2 − 1/8. It is, therefore, natural to look for TDV scheme of the form 1 1 (1) (1) u n+1 = u nj + C j+1/2 Δv j+1/2 − D j−1/2 Δv j−1/2 − φ j+1/2 Δv j+1/2 + φ j−1/2 Δv j−1/2 j 8 8
(14) where, classically, φ is a function of the slopes with values in [0, 1]. To derive appropriate TVD conditions for φ let us first assume that λ ≥ 0. We then define φ j+1/2 = φ(r j+1/2 ), where r j+1/2 = Δv j+1/2 /Δv j−1/2 and rewrite (14) as
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u n+1 = u nj + j
1 1 1 1 φ j+1/2 (λ − 1/2)2 Δv j+1/2 − Δv j−1/2 [ (λ + 1/2)2 + φ j−1/2 − ]. 2 2 8 8 r j+1/2
Using Harten’s theorem, it is readily seen that we obtain a TVD scheme provided φ satisfies |φ(s) − φ(r )/r | ≤ 1 which allows to use the classical limiters. This scheme can be interpreted back into a remeshing formula with weights (according to Fig. 2) given by 1 1 1 (λ − )2 − φ j−1/2 , 2 2 8 3 1 2 β j = − λ + (φ j−1/2 + φ j+1/2 ), 4 8 1 2 1 1 γ j = (λ + ) − φ j+1/2 . 2 2 8
αj =
If λ ≤ 0 we rewrite (14) as 1 1 1 1 φ j−1/2 u n+1 = u nj + (λ + 1/2)2 Δv j−1/2 − Δv j+1/2 [ (λ − 1/2)2 − φ j+1/2 − ] j 2 2 8 8 r˜ j−1/2
where r˜ j−1/2 = Δv j+1/2 /Δv j−1/2 . Remeshing weights are then given by a formula similar to the ones above. The point is now that the resulting scheme can be used even for a CFL number larger than 1 (in this case, one has to use the corrected Λ2 formula derived in [7]). The method then of course does not reduce anymore to a finite-difference scheme. To illustrate the method we consider the case of a double top-hat function advected in a flow with a positive velocity with sinusoidal modulation [20]. This flow, although very smooth, results in compression and dilatation, associated to an increase or decrease of local values, which are delicate to capture. Figure 8 shows a comparison, with the same grid size Δx = 0.510−2 , of the non-oscillatory particle scheme at a CFL number equal to 12, with a fifth-order WENO scheme for a CFL number equal to 2. Strikingly the particle scheme performs better, although it is at most second order and locally only first order. The reason is that, since it uses large time steps, it “sticks” more to the exact condition.
5.2 The Nonlinear Case The above discussion extends to the nonlinear case [7, 28]. Reference [28], in particular, contains a detailed derivation of the TVD formulas associated to a large class of first and second-order remeshing kernels, for scalar conservation laws and for the Euler equations. It also gives a proof, for these TVD particle schemes, of convergence toward entropy solutions in the case of scalar equations. Note that, to
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Fig. 9 Semi-Lagrangian particle solution (red dots) for a shock tube problem, compared to the exact solution (green curve), for Δx = 10−2 . Left picture: density; right picture: velocity
our knowledge, this is the only convergence proof of particle methods for nonlinear scalar conservation laws. Unlike in the case of linear equations, these schemes are restricted to CFL conditions similar to the finite-difference methods. Indeed, due to the shocks, the Lagrangian CFL condition, where the time step is constrained by flow derivatives, reduce in this case to a classical CFL condition since the flow derivatives can reach values of the order of max |u|/Δx. In Fig. 9 we show an illustration of a TVD semiLagrangian particle scheme for a classical shock tube. In this example, the particle scheme uses the second-order formula (8), corresponding to the second- order scheme (13), limited by the upwind first-order piecewise linear formula (corresponding to (11)) and the pressure gradient terms are computed using the first-order method in the Euler–Lagrange method in [12]. This illustration shows that, in contrast with the linear case, the method does not avoid the numerical dissipation of first-order schemes, in particular in the contact discontinuity. Based on the preceding remarks concerning the linear case, one may expect that using local time steps, with CFL numbers larger than 1 away from shocks, could improve this part of the solution. This possibility has yet to be tested.
6 Conclusion Particle methods with particle remeshing at each time step can be analyzed as semiLagrangian conservative methods. High-order methods can be easily implemented and the analogy with finite differences for small time steps enables the derivation of non-oscillatory schemes. For linear problems, the possibility to use large times-steps, only constrained by the flow strain, can lead to significant savings. For nonlinear problems, the method can be seen as a particular case of advective flux splitting. Using local time steps or decoupling the scales of the conservative variables and of the velocity field are two possible directions to enhance the performance of the methods.
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References 1. B. Ben, Moussa, J.-P. Vila, Convergence of SPH method for scalar nonlinear conservation laws. SIAM J. Numer. Anal. 37, 863–887 (2000) 2. M. Bergdorf, G.-H. Cottet, P. Koumoutsakos, Multilevel adaptive particle methods for convection-diffusion equations. SIAM Multiscale Model. Simul. 4, 328–357 (2005) 3. M. Bergdorf, P. Koumoutsakos, A Lagrangian particle-wavelet method. SIAM Multiscale Model. Simul. 5(3), 980–995 (2006) 4. G.-H. Cottet, P.-A. Raviart, Particle methods for the one-dimensional Vlasov-Poisson equations. SIAM J. Numer. Anal. 21, 52–76 (1984) 5. G.-H. Cottet, P. Koumoutsakos, Vortex Methods (Cambridge University Press, 2000) 6. G.-H. Cottet, A new approach for the analysis of vortex methods in 2 and 3 dimensions. Ann. Inst. Henri Poincaré 5, 227–285 (1988) 7. G.-H. Cottet, A. Magni, TVD remeshing schemes for particle methods, C. R. Acad. Sci. Paris, Ser. I 347, 1367–1372 (2009) 8. G.-H. Cottet, L. Weynans, Particle methods revisited: a class of high-order finite-difference schemes, C. R. Acad. Sci. Paris, Ser. I 343, 51–56 (2006) 9. G.-H. Cottet, B. Michaux, S.Ossia, G. Vanderlinden, A comparison of spectral and vortex methods in three-dimensional incompressible flows, J. Comput. Phys. 175 (2002) 10. G.-H. Cottet, J.-M. Etancelin, F. Perignon, C. Picard, High order Semi-Lagrangian particles for transport equations: numerical analysis and implementation issues. ESAIM Math. Model. Numer. Anal. 48, 1029–1060 (2014) 11. V. Daru, C. Tenaud, Evaluation of TVD high resolution schemes for unsteady viscous shocked flows. Comput. Fluids 30, 89–113 (2001) 12. B. Despres, F. Lagoutiere, Contact discontinuity capturing schemes for linear advection and compressible gas dynamics. J. Sci. Comput. 16, 479–524 (2002) 13. J.-M. Etancelin, G.-H. Cottet, F. Perignon, C. Picard, Multi-CPU and multi-GPU hybrid computations of multi-scale scalar transport. 26th International conference on parallel computational fluid dynamics, Trondheim, 2014 14. R.W. Hockney, J.W. Eastwood, Computer Simulation Using Particles (McGraw-Hill Inc., 1981) 15. P. Koumoutsakos, A. Leonard, High resolution simulations of the flow around an impulsively started cylinder using vortex methods. J. Fluid Mech. 296, 1–38 (1995) 16. P. Koumoutsakos, Inviscid axisymmetrization of an elliptical vortex. J. Comput. Phys. 138, 821–857 (1997) 17. R. Krasny, Desingularization of periodic vortex sheet roll-up. J. Comput. Phys. 65, 292–313 (1986) 18. J.-B. Lagaert, G. Balarac, G.-H. Cottet, Hybrid spectral particle method for the turbulent transport of a passive scalar. J. Comput. Phys. 260, 127–142 (2014) 19. M.-S. Liou, C.J. Steffen Jr., A new flux splitting scheme. J. Comput. Phys. 107, 23–39 (1993) 20. A. Magni, G.-H. Cottet, Accurate, non-oscillatory remeshing schemes for particle methods. J. Comput. Phys. 231(1), 152–172 (2012) 21. J.E. Martin, E. Meiburg, Numerical investigation of three-dimensional evolving jets subject to axisymmetric and azimuthal perturbation. J. Fluid Mech. 230, 271 (1991) 22. J.J. Monaghan, Particle methods for hydrodynamics. Comput. Phys. Rep. 3, 71–124 (1985) 23. G. Oger, M. Doring, B. Alessandrini, P. Ferrant, An improved SPH method: towards higher order convergence. J. Comput. Phys. 225, 1472–1492 (2007) 24. M.L. Ould-Salihi, G.-H. Cottet, M. El Hamraoui, Blending finite-differences and vortex methods for incompressible flow computations. SIAM J. Sci. Comput. 22, 1655–1674 (2000) 25. P. Ploumhans, G.S. Winckelmans, Vortex methods for high-resolution simulations of viscous flow past bluff bodies of general geometry. J. Comput. Phys. 165, 354–406 (2000)
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26. S. Reboux, B. Schrader, I. Sbalzarini, A self-organizing Lagrangian particle method for adaptive-resolution advectiondiffusion simulations, J. Comput. Phys. 23, 3623–3646 (2012) 27. G. Russo, J.A. Strain, Fast triangulated vortex methods for the 2D Euler equations. J. Comput. Phys. 111, 291–323 (1994) 28. L. Weynans, A. Magni, Consistency, accuracy and entropic behavior of remeshed particle methods. ESAIM Math. Model. Numer. Anal. 47, 57–81 (2013)
Convergence for PDEs with an Arbitrary Odd Order Spatial Derivative Term Clémentine Courtès
Abstract We compute the rate of convergence of forward, backward, and central finite difference θ -schemes for linear PDEs with an arbitrary odd order spatial derivative term. We prove convergence of the first or second order for smooth and less smooth initial data. Keywords Error estimates · Convergence rates · Finite difference schemes
1 Introduction We study in this paper linear partial differential equations with an arbitrary odd order spatial derivative term, which read ∂t u + ∂x2 p+1 u = 0,
(1)
with p ∈ N. The particular case p = 0 corresponds to the advection equation with a unit constant speed ∂t u + ∂x u = 0 and describes the passive advection of scalar field carried at constant speed. The case p = 1 leads to Airy equation ∂t u + ∂x3 u = 0 that models the propagation of long waves in shallow water [7] and derives from a linearization of the Korteweg–de Vries equation [4]. We especially focus on the initial value problem where (1) is considered with the initial condition u |t=0 = u 0 . We deal with the numerical approach of this Cauchy problem and study the convergence of several finite difference schemes. Our concern here is to find a rate of convergence without assuming the smoothness of the initial data.
C. Courtès (B) LMO, Université Paris-Sud, CNRS, 91405 Orsay, France e-mail: [email protected] C. Courtès University of Paris Saclay, Saint-Aubin, France © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_32
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For this purpose, we use the finite difference method to discretize (1) in R × [0, T ]. We choose to deal with a uniform time and space discretization. Let Δt > 0 and Δx > 0 be the time and space steps, we note t n = nΔt all n ∈ {0, ..., N } where for T n N = Δt and x j = jΔx for all j ∈ Z. We denote by v j the discrete unknowns ( j,n)
defined by
⎧ 2 p+1 n+1 2 p+1 n n+1 ⎪ v = vnj − (1 − θ )Δt D• v , ∀( j, n) ∈ Z × {0, .., N }, ⎪ ⎨ v j + θ Δt D• j j x j+1 (2) 1 ⎪ 0 ⎪ u 0 (y)dy, ∀ j ∈ Z, ⎩vj = Δx x j
with
2 p+1 2 p+1 (−1)k n 2 p+1 n 2 p+1 n k D• v = D+ v = v p−k+ j+1 (forward scheme), j j Δx 2 p+1 k=0
(3a) 2 p+1 2 p+1 (−1)k n 2 p+1 n 2 p+1 n k or D• v = D− v = v p−k+ j (backward scheme), j j Δx 2 p+1 k=0
or
1 2 p+1 2 p+1 2 p+1 2 p+1 n D+ v + D− v D• v = Dc v = (central scheme). j j j 2 n
n
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The parameter θ belongs to [0, 1] and we recover the explicit scheme for θ = 0 and the implicit scheme for θ = 1. Notations 1. We denote by Hs (R) (with s > 0) the Sobolev space defined with s
21 u (ξ )|2 dξ the norm ||u||Hs (R) = R 1 + |ξ |2 | , where u is the Fourier trans
form of u. Moreover, we use the standard ∞ 0, N ; 2Δ (Z) space whose norm is n 2 ||v||∞ (0,N ;2Δ (Z)) = sup j∈Z Δx|v j | . Lastly, we note A B when A ≤ C B n∈{0,..,N }
where C is a constant independent of Δx and Δt.
2 Order of Accuracy for an Initial Datum in H4 p+2 (R) We hereafter find some condition on θ , Δt and Δx for the schemes to be consistent and stable, and to conclude the convergence study according to the Lax–Richtmyer theorem [6].
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2.1 Consistency Estimate In Sect. 2, we suppose the initial datum regular enough to compute all the needed derivatives and the Taylor expansions up to the desired order. Indeed, supposing u 0 regular is sufficient to ensure the same regularity for u(t, .) for all t ∈ [0, T ] because of the following result. Remark 1. Let u be a solution of (1), then by linearity of the equation all the derivatives of u verify (1) too and by Fourier transform, the L2 –norm of all its derivatives are conserved : ||∂xk u(t, .)||L2 (R) = ||∂xk u 0 ||L2 (R) , for all k ∈ N. Thus, u 0 ∈ H4 p+2 (R) implies u(t, .) ∈ H4 p+2 (R), ∀t ∈ [0, T ]. x j+1 1 u(t n , y)dy Definition 1. For all ( j, n) ∈ Z × {0, .., N }, we note (u Δ )nj = Δx xj with u the exact solution of the Cauchy problem (1) from u 0 . For all ( j, n) ∈ Z × {0, .., N }, the consistency error is defined as εnj
=
2 p+1
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− (u Δ )nj (u Δ )n+1 j Δt
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+ θ D•2 p+1 u Δ j + (1 − θ ) D•2 p+1 u Δ j ,
defined by (3a)–(3c).
Proposition 1. Assume u 0 ∈ H4 p+2 (R) (and u 0 ∈ H6 p+3 (R) if θ = 21 ) then, for the forward or backward finite difference schemes (3a) and (3b), the following consistency inequality holds 1 4 p+2 6 p+3 2 p+2 ||ε||∞ 0,N ;2 (Z) Δt − θ ||∂x u 0 ||L2 (R) + Δx||∂x u 0 ||L2 (R) + Δt 2 ∂x u 0 2 . Δ L (R) 2
For the central finite difference scheme (3c), the consistency inequality is as follows 1 4 p+2 6 p+3 2 p+3 ||ε||∞ 0,N ;2 (Z) Δt − θ ||∂x u 0 ||L2 (R) + Δx 2 ||∂x u 0 ||L2 (R) + Δt 2 ∂x u 0 2 . Δ L (R) 2
Before proving the previous result, we state a useful lemma. Lemma 1. For all and p in N, there exists ξ ∈] − p, p + 1[ such that ⎧ 0 ⎪ ⎪ ⎪ ⎨ 2p + 1 ! (−1)k ( p − k + 1) = ⎪ k ! ⎪ ⎪ ⎩ ξ −2 p−1 ( − 2 p)!
2 p+1 k=0
if < 2 p + 1, if = 2 p + 1, if > 2 p + 1.
Proof (Lemma 1). Let (x j− p , ..., x j+ p+1 ) be 2 p + 2 points regularly spaced of h, we recall the divided difference of order 2 p + 2 of a smooth function f : 2 p+1 2 p+1 (−1)k f (x p−k+ j+1 ) k (2 p + 1)! f [x j− p , ..., x j+ p+1 ] = k=0 . h 2 p+1
(4)
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Moreover, we recall the existence of ξ ∈]min(x j− p , ..., x j+ p+1 ), max(x j− p , ..., x j+ p+1 )[ such as (2 p + 1)! f [x j− p , ..., x j+ p+1 ] = f (2 p+1) (ξ ). For more details, please refer to [1]. Lemma 1 is a consequence of the two previous equations with f : y → y , h = 1, j = 0 and xi = i for i ∈ Z. Proof (Proposition 1). For u 0 ∈ H4 p+2 (R) and for the forward finite difference scheme, one has
D•2 p+1 u Δ
n
=
j
n 2 p+1 D+ u Δ j
2 p+1 2 p+1 (−1)k k 2 Δx p+1 k=0
=
1 Δx
x j+1
u(t , y + ( p − k + 1)Δx)dy . n
xj
Using a Taylor expansion (in space) up to order 2 p + 2 and exchanging the two sums inside leads to
2 p+1
D+
n uΔ
j
1 = Δx
p+1 x j+1 2 xj
=0
∂x u(t n , y) !
2 p+1 2 p+1 (−1)k ( p k=0 k Δx 2 p+1
− k + 1) Δx
dy + (R+ )nj ,
(5)
where (R+ )nj =
1 Δx
x j+1 xj
y+( p−k+1)Δx y
2 p+2
∂x
u(t n , z) (2 p + 1)!
2 p+1 2 p+1 k=0
k
(−1)k (y + ( p − k + 1)Δx − z)2 p+1 Δx 2 p+1
dzdy.
2 p+2
n For simplicity, we will only use ||R+ ||2Δ Δx||∂x
u(t n , .)||L2 (R) . Equation (5) is n x j+1 2 p+1 2 p+1 1 simplified thanks to Lemma 1. Eventually, we obtain D+ u Δ = Δx ∂x xj
j
u(t n , y)dy + (R+ )nj . Similarly, by adapting the previous computation, one has
2 p+1
D+
n+1 uΔ
=
j
1 Δx
x j+1
2 p+1
∂x
x j+1
u(t n , y)dy+
xj
2 p+1
Δt∂t ∂x
u(t n , y)dy
xj
+
x j+1
t n+1 tn
xj
2 p+1
∂t2 ∂x
u(s, y)(t n+1 − s)dsdy + (R+ )n+1 . j
(u Δ )n+1 −(u Δ )n
j j In order to compute the difference that appears in Definition 1, we Δt perform a Taylor expansion (in time) up to order 3. Gathering all those results together yields (for θ = 21 for example)
εnj = +
Δt 2Δx θΔt Δx
x j+1 xj x j+1
∂t2 u(t n , x)d x + 2 p+1
∂t ∂ x
1 ΔtΔx
u(t n , y)dy +
xj
+ θ (R+ )n+1 + (1 − θ) (R+ )nj . j
θ Δx
x j+1
tn
xj
t n+1
x j+1 xj
∂t3 u(s, y)
t n+1 tn
(t n+1 − s)2 dsdy 2
2 p+1
∂t2 ∂x
u(s, y)(t n+1 − s)dsdy
Convergence for PDEs with an Arbitrary …
417 2 p+1
The conclusion comes from the relation ∂t u(t, x) = −∂x u(t, x), the Cauchy– Schwarz inequality and the conservation of the L2 -norm (cf. Remark 1). Remark 2. The regularity H4 p+2 (R) (or H6 p+3 (R) if θ = 21 ) comes from the Taylor expansion in time and is essential in this proof. Remark 3. We follow exactly the same guidelines for the backward finite difference scheme. For the central finite difference scheme, a Taylor expan we need n to perform x j+1 2 p+1 n 1 sion in space up to order 2 p + 3 to obtain Dc2 p+1 u Δ = Δx ∂ u(t , y)dy + x xj j
2 p+3
(Rc )nj , with ||Rcn ||2Δ Δx 2 ||∂x
u 0 ||L2 (R) .
2.2 Stability We note, for all vnj
n (ξ ) = k∈Z vkn e2iπkξ in L2 ([0, 1]) with and ξ ∈ [0, 1], V j∈Z 2 1 the equivalence of the norms : j∈Z Δx|vnj |2 = Δx 0 V n (ξ ) dξ. Eventually, we V n = e−2iπξ V n . define the shift operator S by S vn = (vn ) j∈Z thus, S j+
2 Definition 2. A scheme is said to be stable in Δ (Z), if there exists a constant C independent of Δt and Δx such that, for vnj verifying Eq. (2), ( j,n)
n+1 v
2Δ (Z)
≤ (1 + CΔt) vn
2Δ (Z)
, ∀n ∈ {0, ..., N }.
Proposition 2. For small Δt and Δx, the stability under the Courant–Friedrichs– Lewy condition (in short CFL cond.) is explained in Table 1. The following computation will simplify the proof of Proposition 2.
Table 1 Stability results for finite difference θ-schemes
p even
p odd
p=0 (Advection)
p = 0
Forward scheme
stable under the CFL cond. t(1 − 2 ) ≤ − x
unconditionally unstable
stable under the CFL cond. 2p+1 t(1 − 2 ) ≤ x22p
Backward scheme
stable under the CFL cond. t(1 − 2 ) ≤ x
stable under the CFL cond. 2p+1 t(1 − 2 ) ≤ x22p
unconditionally unstable
Central scheme
stable under the CFL cond. t(1 − 2 ) ≤ 2C x2
stable under the CFL cond. 4p+2 t(1 − 2 ) ≤ 2C x24p
stable under the CFL cond. 4p+2 t(1 − 2 ) ≤ 2C x24p
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Lemma 2. One has, for all ξ ∈ [0, 1], 2 p+1 2 p + 1 (−1)k e−2iπ( p−k+1)ξ = e−iπξ (−2isin(π ξ ))2 p+1 . k=0 k
Proof (Lemma 2). A proof may be found in Lemma 1.1 of [2]. Proof (Proposition 2). The forward finite difference scheme (3a) leads to ⎞ 2 p+1 2p + 1 θΔt k −2iπ( p−k+1)ξ ⎠ (−1) e k (Δx)2 p+1 k=0 ⎞ ⎛ 2 p+1 2 p + 1 (1 − θ)Δt n (ξ ) ⎝1 − (−1)k e−2iπ( p−k+1)ξ ⎠ , for any ξ in [0, 1]. =V k (Δx)2 p+1 ⎛
n+1 (ξ ) ⎝1 + V
k=0
n+1 (ξ ) = The two sums are simplified thanks to Lemma 2. We finally obtain V n A+ (ξ )V (ξ ), with A+ the amplification coefficient defined by, ∀ξ ∈ [0, 1] 1− A+ (ξ ) = 1+
(1−θ)Δt −iπξ e (−i)2 p+1 (Δx)2 p+1
(2sin(π ξ ))2 p+1
θΔt e−iπξ (−i)2 p+1 (Δx)2 p+1
(2sin(π ξ ))2 p+1
.
(6)
We are looking for a condition ensuring |A+ (ξ )|2 < (1 + CΔt)2 for any ξ in [0, 1]. Case 1 Assume that the parameter p of the spatial derivative is even. If p = 0, 22 p Δt 2p the stability condition leads to (Δx) (1 − 2θ ) ≤ −1, (cf. [2]) 2 p+1 (sin(π ξ )) which is impossible for all ξ ∈ [0, 1] : thus the forward finite difference scheme is unconditionally unstable for p even and non zero. On the contrary, assuming p = 0 means that the forward finite difference scheme is stable under CFL condition : Δt (1 − 2θ ) ≤ −Δx (which implies θ > 21 ). Case 2 In this case, the parameter p of the spatial derivative is odd, then the suffiΔt 2p (1 − 2θ ) ≤ 1 (cf. [2]). Then cient condition becomes (Δx) 2 p+1 (2sin(π ξ )) the forward finite difference scheme is stable under the CFL condition 2 p+1 Δt (1 − 2θ ) ≤ Δx22 p . Table 1 is a straightforward consequence. Remark 4. For the backward finite difference scheme, the only difference in the amplification coefficient is eiπξ instead of e−iπξ (in both the numerator and denominator). The parity needed for the stability changes because of that difference. For the central finite difference scheme, e−iπξ is replaced with cos(π ξ ) in the numerator and the denominator of the amplification coefficient.
Convergence for PDEs with an Arbitrary …
419
2.3 Error Estimates We define the convergence error as follows. Definition 3. For all j ∈ Z and n ∈ {0, ..., N }, for u the analytical solution of (1) from u 0 and (vnj )( j,n) the numerical solution of (2), the convergence error is denoted x j+1 1 by enj and defined by enj = Δx u(t n , y)dy − vnj . xj We are now able to state the main result of this section. Theorem 1. For an initial datum u 0 ∈ H4 p+2 (R) (and u 0 ∈ H6 p+3 (R) if θ = 21 ), the error estimate of the forward finite difference scheme (3a) (if p is odd) or of the backward finite difference scheme (3b) (if p is even) satisfies 4 p+2 1 2 p+2 6 p+3 ||e||∞ 0,N ;2 (Z) Δt − θ ∂x u 0 2 + Δx ∂x u 0 2 + Δt 2 ∂x u 0 2 . Δ L (R) L (R) L (R) 2
For the central finite difference scheme (3c), the convergence rate becomes 4 p+2 1 2 p+3 6 p+3 ||e||∞ 0,N ;2 (Z) Δt − θ ∂x u 0 2 + Δx 2 ∂x u 0 2 + Δt 2 ∂x u 0 2 . Δ L (R) L (R) L (R) 2
All those results are gathered in Table 4. Proof. We suppose p odd, so we work with the forward finite difference scheme. The case p even, with the backward scheme is similar. The definition of the convergence error implies en+1 (ξ ) = A+ (ξ )e n (ξ ) +
Δt 1+
θΔt e−iπξ (−i)2 p+1 (2sin(π ξ ))2 p+1 Δx 2 p+1
One has 1+ θ Δt e−iπξ (−i)12 p+1 (2sin(πξ ))2 p+1 Δx 2 p+1
L∞ ([0,1])
ε n (ξ ).
≤ 1 and the stability condition
gives ||A+ ||L∞ ([0,1]) ≤ 1 + CΔt. Thus, we obtain the following estimate, by discrete Grönwall lemma ||en+1 ||2 ≤ (1 + CΔt)||en ||2 + Δt||εn ||2 ≤ ... ≤ eC T ||e0 ||2 + ΔteC T Δ
Δ
Δ
Δ
n k=0
||εk ||2 . Δ
The initial condition v0j , Eq. (2), together with the consistency error conclude the proof. Remark 5. As expected, for the particular case θ = 21 (the so-called Crank–Nicolson case), the rate of convergence in time is better as illustrated in Table 4, provided u 0 ∈ H6 p+3 (R) (and not only in H4 p+2 (R)).
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3 Less Smooth Initial Data The previous order of accuracy is obtained for initial data u 0 at least in H4 p+2 (R) (or H6 p+3 (R) if θ = 21 ). In this section, our aim is to relax this hypothesis to obtain the rates of convergence for non-smooth initial data, for example, u 0 ∈ Hm (R) with m > 0. We detail only the case θ = 21 but state the Crank–Nicolson results in Table 4.
3.1 Initial Datum in Hm (R) with m ≥ 2 p + 2 As explained previously (Remark 2), the regularity of u 0 is determined by the Taylor expansion in time. A first step is then to deal with the time term in error estimates. The following proposition provides that the time error prevails until u 0 ∈ H2 p+2 (R), for which the spatial error becomes predominant. Proposition 3. Assume u 0 ∈ Hm (R) with m ≥ 2 p + 2, and let us fix M = min (m, 4 p + 2), then the error estimate for the forward (respectively backward) finite difference scheme, if p is odd (respectively even), yields M
||e||∞ (0,N ;2Δ (Z)) Δt 4 p+2 ||∂xM u 0 ||L2 (R) + Δx||∂x2 p+2 u 0 ||L2 (R) . For the central difference scheme, we suppose m ≥ 2 p + 3, and one has (for the same M) M
||e||∞ (0,N ;2Δ (Z)) Δt 4 p+2 ||∂xM u 0 ||L2 (R) + Δx 2 ||∂x2 p+3 u 0 ||L2 (R) . proving this result, we introduce a regularization of u 0 , thanks to mollifiers
Before ϕ δ δ>0 . Let χ be a C ∞ -function such that • 0 ≤ χ ≤ 1, • χ ≡ 1 in [− 21 , 21 ] and supp(χ ) ⊂ [−1, 1] (where supp is its support), • χ (−ξ ) = χ (ξ ), ∀ξ ∈ [−1, 1]. δ
δ Let ϕ be such that ϕ (ξ ) = χ (ξ ) and for all δ > 0, we define ϕ such that ϕ (ξ ) = 1 . δ χ (δξ ), which implies ϕ = δ ϕ δ . Eventually,
• let u δ be the solution of (1) with u δ0 = u 0 ϕ δ as initial data, where stands for the convolution product. • We denote then ((vδ )nj )(n, j)∈{0,...,N }×Z the numerical solution obtained by applying the numerical scheme (2) from u δ0 . • The unknowns u and (vnj )(n, j)∈{0,...,N }×Z are always the exact and numerical solutions starting from the initial data u 0 .
Convergence for PDEs with an Arbitrary …
421
Lemma 3. Assume u 0 ∈ Hr (R) with r > 0 then the following upper bound holds, for 0 ≤ ≤ r ≤ s, u 0 − u δ
0 H (R)
δr − ||u 0 ||Hr (R) and u δ0 Hs (R)
1 ||u 0 ||Hr (R) . δ s−r
Proof. Lemma 3 is proved in [3] (see also [5]).
Proof (Proposition 3). We are now able to prove Proposition 3. The triangular inequality applied to the convergence error gives ||en ||2Δ E 1 + E 2 + E 3 with ⎛ 2 ⎞ 21 x j+1 1 n δ n E1 = ⎝ Δx u(t , x) − u (t , x)d x ⎠ , Δx x j
(7)
j∈Z
⎛ 2 ⎞ 21 x j+1 1 E2 = ⎝ Δx u δ (t n , x)d x − (v δ )nj ⎠ , Δx x j
(8)
j∈Z
⎛ E3 = ⎝
Δx
(v δ )nj
− v nj
2
⎞1 2
⎠ .
(9)
j∈Z
Cauchy–Schwarz inequality together with the conservation of the L2 -norm (Remark 1) yield E 1 ≤ ||u 0 − u δ0 ||L2 (R) δ M ||∂xM u 0 ||L2 (R) . The latest inequality comes from Lemma 3 with = 0 and r = M. For the E 2 –term, we use the previous section (Sect. 2). Indeed, E 2 corresponds to the convergence error for a smooth initial data u δ0 . Hence, one has 4 p+2 δ 2 p+2 δ u 0 ||L2 (R) + Δx||∂x u 0 ||L2 (R)
E 2 Δt||∂x
Δt 2 p+2 ||∂xM u 0 ||L2 (R) + Δx||∂x u 0 ||L2 (R) , δ 4 p+2−M
where the latest inequality comes from Lemma 3 with (s, r ) = (4 p + 2, M) and (s, r ) = (2 p + 2, 2 p + 2). Finally, the stability of the scheme gives the following estimate for E 3 : E 3 = ||(v δ )n − v n ||2 ≤ ||u δ0 − u 0 ||L2 (R) . Δ
Thus, the convergence error is upper bounded by n e
2Δ
δ M ||∂xM u 0 ||L2 (R) +
Δt 2 p+2 ||∂ M u 0 ||L2 (R) + Δx||∂x u 0 ||L2 (R) . δ 4 p+2−M x 1
Proposition 3 comes from the optimal choice for δ : δ = Δt 4 p+2 .
Remark 6. The result for the central finite difference scheme is proved exactly in the same way, with the same s, r, and δ.
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3.2 Initial Datum in Hm (R) with m ≥ 0 The main result of this paper is summarized in the following theorem where the initial data u 0 has any Sobolev regularity. Theorem 2. Assume that u 0 ∈ Hm (R) with 0 ≤ m, then, the forward (respectively backward) finite difference scheme if p is odd (respectively even) has the following error estimate ||e||∞ 0,N ;2 (Z) Δt
min(m,4 p+2) 4 p+2
Δ
min(4 p+2,m)
||∂x
u 0 ||L2 (R) + Δx
min(m,2 p+2) 2 p+2
min(2 p+2,m)
||∂x
u 0 ||L2 (R) .
The previous inequality becomes for the central finite difference scheme ||e||∞ 0,N ;2 (Z) Δt Δ
min(m,4 p+2) 4 p+2
min(4 p+2,m)
||∂x
u 0 ||L2 (R) + Δx
p+3) 2 min(m,2 2 p+3
min(2 p+3,m)
||∂x
u 0 ||L2 (R) .
The previous results are summarized in Table 4. Proof. Here again, we suppose that p is odd, we thus detail the proof for the forward finite difference scheme. We have already proved the case m ≥ 4 p + 2 in Sect. 2 and the case 2 p + 2 ≤ m ≤ 4 p + 2 in Sect. 3.1. Let us now focus on the case 0 ≤ m ≤ 2 p + 2. The proof of Theorem 2 follows the same guidelines as the proof of Proposition 3. Let u 0 ∈ Hm (R), we regularize this initial data thanks to mollifiers ϕ δ δ>0 whose properties are listed in Sect. 3.1. This involves introducing the same new unknowns u δ , u δ0 and ((vδ )nj )( j,n) . is upper bounded by the same E 1 , E 2 and E 3 , The convergence error enj ( j,n)
defined in (7)–(9). Lemma 3 with =0 and r =m leads to E 1 + E 3 δ m ||∂xm u 0 ||L2 (R) . By definition u δ0 ∈ Hk (R), ∀k > 0, therefore Proposition 3 applies with k = 2 p + 2 for example and M = min(k, 2 p + 2) = 2 p + 2. It gives estimate for the following E2 :
p+1
2 p+2 δ u 0 ||L2 (R)
E 2 Δt 2 p+1 ||∂x
2 p+2 δ u 0 ||L2 (R)
+ Δx||∂x
p+1
2 p+2 δ u 0 ||L2 (R) .
Δt 2 p+1 +Δx ||∂x
We then apply Lemma 3 with s = 2 p + 2 and r = m. Finally, it yields n e
2Δ
δ m ||∂xm u 0 ||L2 (R) +
p+1 Δt 2 p+1 + Δx δ 2 p+2−m
||∂xm u||L2 (R) .
1 p+1 2 p+2 The conclusion comes from the optimal choice δ = Δt 2 p+1 + Δx .
Remark 7. The backward scheme, with p even, is very similar. The central finite difference scheme is proved with the same method except for the variable k, which is taken k = 2 p + 3 for that scheme.
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423
4 Numerical Results In order to illustrate numerically the previous results, we perform two sets of examples : on one hand, we compute the numerical rate of convergence of various equations for a fix initial data and on the other hand, the equation is fixed and we test different initial data. In all examples, the computational domain is set to [0, 50] subdivided into J cells with J ∈ {800, 1600, 3200, 6400, 12800, 25600, 51200, 102400} and the numerical simulation is performed up to time T = 0.1. In order to not have a too restricted Courant–Friedrich–Lewy condition, we implement the implicit scheme (θ = 1) and impose Δt = Δx. The convergence error is computed between the solution with J cells and a “reference” solution with 2J cells in space. Since the indicator function belongs to Hs (R) for all s < 21 , we build test functions in Hs (R) with s < 21 + k by integrating k-times the indicator function. Such functions 1 will be denoted in H 2 +k− (R). 3
Table 2 For a Sobolev regularity H 2 − Δx p = 0 (Advection) p = 1 (Airy) L2 -error 6.250.10−2 3.125.10−2 1.563.10−2 7.813.10−3 3.906.10−3 1.953.10−3 9.766.10−4 4.883.10−4 Theoretical
1.194.10−4 7.381.10−5 4.471.10−5 2.664.10−5 1.585.10−5
L2 -error
Order
1.348.10−3 1.125.10−3 9.670.10−4 7.968.10−4 6.572.10−4
0.694 0.723 0.747 0.749 0.750
p = 2 (Fifth-order derivative) Order
0.261 0.219 0.279 0.278 0.250
L2 -error 2.985.10−3 2.757.10−3 2.441.10−3 2.176.10−3 1.961.10−3
Order 0.115 0.175 0.166 0.149
0.150
5
Table 3 For a Sobolev regularity H 2 − Δx p = 0 (Advection) L2 -error
Order
p = 1 (Airy) L2 -error
Order
6.250.10−2 3.125.10−2 1.563.10−2 7.813.10−3 3.906.10−3 1.953.10−3 9.766.10−4 4.883.10−4 Theoretical
2.586.10−3 1.347.10−3 6.873.10−4 3.437.10−4 1.719.10−4
0.940 0.971 0.999 1.000 1.000
1.011.10−2 7.507.10−3 6.267.10−3 4.401.10−3 3.074.10−3
0.430 0.261 0.510 0.520 0.417
p = 2 (Fifth-order derivative) L2 -error 3.388.10−2 3.639.10−2 3.032.10−2 2.528.10−2 2.138.10−2
Order 0.093 0.263 0.262 0.242
0.250
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C. Courtès
Fig. 1 Numerical versus theoretical orders—left : Advection equation ( p = 0), right : Airy equation ( p = 1) Table 4 Error estimates for u 0 ∈ Hm (R) =
1 2
p=0
Forward scheme min(m,2) min(m,2) t 2 x u0 2 L min(m,2) min(m,2) + x 2 x u0
L2
Backward scheme min(m,2) min(m,2) t 2 x u 0 2 L min(m,2) min(m,2) + x 2 x u0
p even
t
(p = 0) p odd
t
min(m,4p+2) 4p+2
+ x =
1 2
p=0
min(m,2p+2) 2p+2
L2
min(m,4p+2) u0 2 x L min(m,2p+2) u0 x
t
(Crank-Nicolson case) min(m,3) min(m,3) min(m,3) min(m,3) t 2 3 x u0 2 t 2 3 x u0 2 L (R) L (R) min(m,2) min(m,2) min(m,2) min(m,2) + x 2 x u0 2 + x 2 x u0 2
L (R)
t2 +
min(m,4p+2) u0 2 x L min(m,2p+3) min(m,2p+3) 2 2p+3 x u0 2 x L
min(m,4p+2) 4p+2
+
L2
min(m,4p+2) u0 2 x L min(m,2p+3) min(m,2p+3) 2 2p+3 x u0 2 x L
min(m,4p+2) 4p+2
+ t
L (R)
min(m,3) u0 2 x L(R) min(m,3) min(m,3) 2 3 x u0 2 x L (R) min(m,3) 3
min(m,6p+3) min(m,6p+3) min(m,6p+3) u0 2 t 2 6p+3 x u0 2 x L (R) L(R) min(m,2p+2) min(m,2p+3) min(m,2p+2) min(m,2p+3) + x 2p+2 x u0 2 + x2 2p+3 x u0 2 t2
p even (p = 0) p odd
L2
min(m,4p+2) u0 2 x L min(m,2p+2) min(m,2p+2) 2p+2 u0 2 x x L
min(m,4p+2) 4p+2
+
Central scheme min(m,2) min(m,2) t 2 x u0 2 L min(m,3) min(m,3) + x 2 3 x u0
min(m,6p+3) 6p+3
L (R)
min(m,6p+3) t u0 2 x L (R) min(m,2p+2) min(m,2p+2) + x 2p+2 x u0 2 min(m,6p+3) 2 6p+3
L (R)
L (R)
t
min(m,6p+3) 2 6p+3
+ x2
min(m,2p+3) 2p+3
min(m,6p+3) u0 2 x L(R) min(m,2p+3) u0 2 x
L (R)
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425
The first test consists of fixing u 0 in H 2 − or H 2 − and computes the convergence rate for p = 0 (advection equation), p = 1 (Airy equation) and p = 2. The numerical results are gathered in Tables 2 and 3 and correctly match with the expected theoretical rates. For the second sample of examples, the equation is fixed ( p = 0 for Fig. 1-left and p = 1 for Fig. 1-right) whereas the Sobolev regularity of the initial data is fluctuating. As shown in Fig. 1, the theoretical rates are represented by the line and the numerical rates correspond to the dots. The exponent of the Sobolev regularity of u 0 is shown in the x-axis. Again, the different rates match very well, which tends to indicate that the convergence orders we have proven are optimal (Table 4). 3
5
5 Conclusion The theoretical estimates are principally determined by the Taylor expansion in time of the consistency error and the parity of p is needed in the stability study. Numerical tests confirm the theoretical results. Acknowledgements The author would thank F.Lagoutière and F.Rousset for their helpful remarks and the anonymous referees.
References 1. R.L. Burden, J.D. Faires, Numerical Analysis, 9th edn. (Brooks/Cole, 2011) 2. C. Courtès, Étude de l’équation d’Airy sous contrainte (2014), http://www.math.u-psud.fr/ ~santambr/Memoire-COURTES.pdf 3. C. Courtès, F. Lagoutière, F. Rousset, Numerical analysis with error estimates for the Kortewegde Vries equation. (submitted, 2018) 4. W. Craig, J. Goodman, Linear dispersive equations of Airy type. J. Differ. Equ. 87, 38–61 (1990) 5. B. Després, Finite volume transport schemes. Numer. Math. 108, 529–556 (2008) 6. P.D. Lax, R.D. Richtmyer, Survey of the stability of linear difference equations. Commun. Pure Appl. Math. 9, 267–293 (1956) 7. G.B. Witham, Linear and Nonlinear Waves (Wiley, New York, 1974)
A Cell-Centered Lagrangian Method for 2D Ideal MHD Equations Zihuan Dai
Abstract In this work, we present a Lagrangian cell-centered MHD scheme on unstructured quadrilateral grid which need neither corrector steps nor modifications to the original ideal MHD equations but preserve exactly the divergence constraint of the magnetic field. All primary variables in this scheme are cell centered. In order to compute the numerical fluxes through the cell interfaces, we introduce one velocity for each vertex, one subcell force and one subcell magnetic flux for each subcell of the mesh. We construct a nodal solver to compute the vertex velocity and the subcell force. The subcell magnetic fluxes in our scheme are assumed to remain unchanged all the time which guarantees the exact preservation of the divergence-free constraint. Several numerical tests are presented to demonstrate the robustness and the accuracy of this scheme. Keywords Lagrangian method · Magnetohydrodynamics · Divergence free
1 Introduction The system of magnetohydrodynamic (MHD) equations is an important model, which describes the macroscopic interaction between the motion of a conducting fluid and an electromagnetic field. It is indispensable for dealing with many important problems which arise in magneto-spheric, solar, astrophysical, and thermonuclear research, etc. These problems are so complicated that they often prohibit an analytical investigation. It is also very difficult to observe or measure the physical variables experimentally. So the researchers have to rely on numerical simulations. This motivates the development of the discrete schemes for these equations. In this work, we focus on the numerical methods for ideal MHD equations where the effects
Z. Dai (B) Institute of Applied Physics and Computational Mathematics, No. 6, Huayuan Road, Haidian District, Beijing 100088, China e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_33
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of resistivity, thermal conductivity, and viscosity are neglected. These equations consist of a set of nonlinear hyperbolic conservation laws. Many finite volume/difference schemes are established to solve the ideal MHD equations (see, e.g., [2, 4, 5, 9, 10, 12, 15] and their references). These schemes can well resolve the steep gradients, shock waves, contact discontinuities, and shear layers arising in the MHD flows. But they are all Eurerian schemes whose grids are fixed in the physical domains or ALE schemes with adaptive moving grids. Up to now, there has been only a little attention in the Lagrangian MHD schemes whose grids move along the flow of the fluid. Some existing Lagrangian MHD schemes decouple the system of hydrodynamic equations from the magnetic induction equation and treat the effect of magnetic field on the fluid as a source term on the momentum equation [16]. The main reason for doing so is that the structure of the MHD equations is more complex than that of the HD equation and thus HD schemes are relatively simple. However, this approach will obviously affect the accuracy of the calculation results. The main trouble in the pure finite volume/difference schemes for multidimensional MHD calculations is the violation of the divergence-free constraint of the magnetic field. Numerical evidence and some analysis indicate that the nonzero divergence of the computed magnetic field can be responsible for numerical instability or nonphysical features in approximated solutions. This has been driving the development of various divergence-cleaning or divergence-free numerical algorithms for MHD equations. Up to now, there are four popular approaches to treat this trouble. They are the projection method of Brackbill and Barnes [1], the 8-wave formulation of the MHD equations suggested by Powell [12], the hyperbolic divergence cleaning method introduced by Dedner et al. [6], and the constrained transport (CT) method introduced by Evans and Hawley [7]. Although these approaches have been used with great success, they have some common drawbacks: either some corrector steps are needed to reconstruct an exactly divergence-free magnetic field after the main finite volume or finite difference step, or the original ideal MHD equations are modified to damp or propagate the divergence error of the magnetic field but not preserve exactly the divergence constraint. Not only could this be computationally inefficient but also can not ensure the nondecreasing condition of entropy. In this work, we will present a Lagrangian cell-centered MHD scheme on unstructured quadrilateral grid which need neither corrector steps nor modifications to the original ideal MHD equations but preserve exactly the divergence constraint of the magnetic field. All primary variables in this scheme, including density, momentum, total energy, and magnetic field are cell centered. In order to compute the numerical fluxes through the cell interfaces, motivated by [11], we introduce one velocity for each vertex, one subcell force and one subcell magnetic flux for each subcell of the mesh. By using these variables, we construct our Lagrangian cell-centered MHD scheme. To compute the vertex velocity and the subcell force, we construct a nodal solver by using the jumping conditions for the discontinuous solution of the MHD equations, the divergence-free property of magnetic field, the conservation of momentum and total energy and the entropy inequality. In our scheme, the subcell magnetic fluxes are assumed to remain unchanged all the time and the magnetic field
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at the next time step are computed by using these subcell magnetic fluxes and the geometrical shape of each cell. The outline of this work is as follows: In Sect. 2, we introduce the ideal MHD equations and their properties. Section 3 proposes the numerical schemes. Full solution procedure will be outlined in Sect. 4. Many representative numerical tests are presented in Sect. 5 to demonstrate the robustness and the accuracy of this scheme.
2 Governing Equations and Their Properties The governing equations of ideal MHD can be written as the following set of nonlinear hyperbolic conservation laws of the mass, momentum, total energy of the plasma ∂ρ + ∇ · (ρv) = 0, ∂t 1 ∂ 1 2 |B| I3×3 − B B = 0, (ρv) + ∇ · ρvv + P + ∂t 2μ μ ∂ 1 1 1 2 2 ρ E + P + |B| v − v · ρE + |B| + ∇ · BB = 0, ∂t 2μ μ μ ∂B − ∇ · (Bv − vB) = 0, ∂t
(1) (2) (3) (4)
where ρ is the density, v is the velocity, P is the pressure, B is the magnetic field, E is the special total energy defined as e + 21 |v|2 , μ is the magnetic permeability, I3×3 is T ∂ the 3 × 3 identity matrix, and ∇ = ∂∂x , ∂∂y , ∂z . This system has seven eigenvalues in any direction n which read vn − c f , vn − c A , vn − cs , vn , vn + cs , vn + c A , vn + c f , where c f,s =
√
2 2
|B|2 μρ
+a
2
±
|B|2 μρ
+
a2
2
1/2 −
4Bn2 a 2 μρ
, cA =
Bn2 , μρ
vn = v ·
n, Bn = v · n and a is the acoustic speed. These characteristic speeds are ordered according to the sequence of inequalities 0 ≤ cs ≤ c A ≤ c f . Let us use the subscriptions 1 and 2 to signify the states on both sides of the discontinuity line with unit normal n in the solution of this system and denote [Q] = Q 1 − Q 2 , Q =
1 (Q 1 + Q 2 ) 2
for any physical quantity Q. We can easily write the jump conditions for the ideal magnetohydrodynamics as m [τ ] − [vn ] = 0,
(5)
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1 |B|2 = 0, m [vn ] + P + 2μ 1 m [vt ] − Bn [B t ] = 0, μ
1 1 τ m E+ |B|2 + [Pvn ] + vn |B|2 − Bn [v · B] = 0, 2μ 2μ μ mτ [B t ] − Bn [vt ] + B t [vn ] = 0,
(6) (7) (8) (9)
where m = ρ1 (v1 · n − δ) = ρ2 (v2 · n − δ) is the mass swept by the discontinuity, τ = 1/ρ is the special volume, vn and Bn are the components of v and B along the normal direction n of the discontinuity, respectively, vt and B t are the transverse parts of the velocity and the magnetic field defined by vt = v − vn n, B t = B − Bn n. In order to construct the Lagrangian cell-centered MHD scheme, we cast the ideal MHD equations into the following updated-type Lagrangian form dτ = ∇ · v, dt dv 1 1 ρ = −∇ · P+ |B|2 I − B B , dt 2μ μ 1 1 τ d P+ E+ |B|2 = −∇ · |B|2 v − v · BB , ρ dt 2μ 2μ μ d ρ (τ B) = ∇ · (Bv) , dt ρ
where dtd denotes the material derivative. This work focuses on the 2D case, where all dependent variables are assumed to be independent of z. For the convenience of T presentation, we rewrite the velocity and the magnetic field as v = u T , w , B = T T T B , Bz where u = (u, v)T and B = Bx , B y . The updated-type Lagrangian form of the 2D ideal MHD equations reads dτ = dt du ρ = dt dw = ρ dt τ d E+ |B|2 = ρ dt 2μ ρ
∇ · u, 1 1 −∇ · P+ |B|2 I − BB , 2μ μ 1 ∇· Bz B , μ 1 1 −∇ · P+ |B|2 u − (v · B) B , 2μ μ d ρ (τ B) = ∇ · (Bv) . dt
(10) (11) (12) (13) (14)
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Here and later I denotes the 2 × 2 unit matrix, ∇ and ∇· denote the gradient and diver T gence operators in the 2D space, i.e., ∇ = ∂∂x , ∂∂y , ∇· = ∂∂x + ∂∂y . The divergencefree constraint in 2D space is ∇ · B = 0. Integrating equations (10)–(13) on any domain ω which moves with the fluid and using Gauss’s divergence theorem, we get their integral forms as dτ ρ dω = u · nd S, (15) ω dt ∂ω ω
ρ
du dω = − dt ω
ω
ρ
d dt
E+
ρ
∂ω
P+
dw dω = dt
1 1 |B|2 I − BB · nd S, 2μ μ
∂ω
1 (B · n) Bz d S, μ
(16)
(17)
τ 1 1 |B|2 dω = − |B|2 n − (B · n) B d S u· P+ 2μ 2μ μ ∂ω 1 + w (B · n) Bz d S, (18) ∂ω μ
where ∂ω is the boundary of ω, n is the outward unit normal of ∂ω. By using the magnetic flux-freezing principle and Eq. (14), we can derive the integral form of the 2D magnetic induction equation as d B · ndl = 0, dt C d ρ (τ Bz ) dω = w (B · n) d S, dt ω
(19) (20)
∂ω
where C is a curve which moves with the fluid and n is its normal. The above integral forms will be used in the construction of our Lagrangian cell-centered MHD scheme on unstructured quadrilateral grid.
3 Numerical Schemes Suppose that the physical domain Ω which we consider is discretized into a set of quadrilateral cells at some initial time. Each cell is assigned a unique index c (Fig. 1). The domain which the cell c takes up at the initial time is denoted by Ωc . Suppose each cell moves with the fluid and denote the domain it takes up at time t by ωc (t). Let ω(t) = c ωc (t). Thus, we have ωc (0) = Ωc and ω(0) = Ω. A generic point (vertex) of these cells is labeled by the index p, its corresponding position vector is x p . For a cell c, we introduce the set P(c) which is the counterclockwise ordered
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p
Fig. 1 Notations of a cell and its subcells
pp
c cp
p
p
p p
list of vertexes of cell c. Conversely, for a given point p, we introduce the set C ( p) containing the cells that surround point p. Being given p ∈ P(c), p − and p + are the previous and next points with respect to p in this list. The length and the unit outward normal related to the edge [ p, p+] are l pp+ and n pp+ , Being given a polygonal cell c, for each vertex p ∈ P(c), we define the subcell (corner) pc (ω pc ) by connecting the centroid of c to the midpoints of edges [ p − , p] and [ p, p + ] impinging at node p. In two dimensions the subcell, as just defined, is always a quadrilateral. Using the subcell definition, cell ωc and its boundary ∂ωc can be decomposed as ω pc , ∂ωc = ∂ωc ∩ ∂ω pc . (21) ωc = p∈P (c)
p∈P (c)
We define the normal vector l pc n pc for the corner pc in terms of the two outward normals impinging at node p as l pc n pc =
1 l p− p n p− p + l pp+ n pp+ . 2
We call it the corner vector of the corner pc. It is one of the fundamental geometric objects in our scheme. By its definition, we have
l pc n pc = 0.
(22)
p∈P (c)
3.1 Semi-discrete Scheme of the 2D Ideal MHD Equations By using (15)–(18) and (21) on the cell c, we have ωc
dτ ρ dω = dt
p∈P (c)∂ω ∩∂ω c pc
u · nd S,
A Cell-Centered Lagrangian Method for 2D Ideal MHD Equations ρ
ρ
ρ ωc
d dt
ωc
E+
P+
p∈P (c)∂ω ∩∂ω c pc
ωc
du dω = − dt dw dω = dt
p∈P (c)∂ω ∩∂ω c pc
τ |B|2 dω = − 2μ
1 1 |B|2 n − (B · n) B d S, 2μ μ
1 (B · n) Bz d S, μ
u·
p∈P (c)∂ω ∩∂ω c pc
+
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w
p∈P (c)∂ω ∩∂ω c pc
P+
1 1 |B|2 n − (B · n) B d S 2μ μ
1 (B · n) Bz d S. μ
Approximating both hands of these formulations by the mean value theorem, we have dτc = u p · l pc n pc , (23) mc dt p∈P (c)
mc
duc =− F u, pc , dt
(24)
dwc =− Fw, pc , dt
(25)
p∈P (c)
mc
p∈P (c)
mc
d dt
Ec +
τc |B c |2 2μ
=−
u p · F u, pc + w p Fw, pc ,
(26)
p∈P (c)
where m c is the total mass in the cell c which remains unchanged all the time, τc , vc = (ucT , wc )T , E c , B c denotes the mean values of τ , v, E, B in the cell c, respectively, v p = (u Tp , w p )T denotes the velocity on the point p, F u, pc and Fw, pc are the u-component and w-component of the corner force related to subcell pc
F u, pc =
P+ ∂ωc ∩∂ω pc
Fw, pc = − ∂ωc ∩∂ω pc
If we denote F pc =
F u, pc
1 1 |B|2 n − (B · n) B d S, 2μ μ
1 (B · n) Bz d S, μ T
, Fw, pc mc
T , Eqs. (24)–(26) can be cast into
dvc =− F pc , dt p∈P (c)
(27)
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Fig. 2 Diagonals of a quadrilateral cell
d mc dt
τc |B c |2 Ec + 2μ
=−
v p · F pc .
(28)
p∈P (c)
Now, let us turn to the evolution of the cell-centered magnetic field B c . First, we deal with B. By using (19) on the two diagonals of cell c (dashed lines in Fig. 2), we can construct the following semi-discrete scheme for the evolution of the B field:
d dt d dt
(Bc · l13 n13 ) = 0, (Bc · l24 n24 ) = 0,
where l13 n13 and l24 n24 are the normal vectors of the two diagonals. We denote the two constants Bc · l13 n13 and Bc · l24 n24 by Φc,13 and Φc,24 respectively. Thus, at any time Bc can be obtained by solving the following algebraic system (y3 − y1 ) Bx,c + (x1 − x3 ) B y,c = Φc,13 , (y4 − y2 ) Bx,c + (x2 − x4 ) B y,c = Φc,24 , and reads
Bx,c = B y,c =
Φc,13 (x2 −x4 )−Φc,24 (x1 −x3 ) , 2|ωc | Φc,24 (y3 −y1 )−Φc,13 (y4 −y2 ) , 2|ωc |
(29)
where (x p , y p ), p = 1, 2, 3, 4, is the coordinate of the vertex p of the cell c and |ωc | is the area of the quadrilateral cell c. It is easy to prove that the numerical magnetic field thus obtained is exactly divergence free at any time. From (29), we can compute m c dtd (τc Bc ) which will be used in the discussion about the entropy inequality mc
d d d (τc Bc ) = (m c τc Bc ) = (|ωc |Bc ) = dt dt dt
p=1,2,3,4
u p Φ pc ,
(30)
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where Φ pc is the corner magnetic flux for the corner pc of cell c as Φ pc = Bc · l pc n pc , p = 1, 2, 3, 4. Since Φ1c = − 21 Φc,24 , Φ2c = 21 Φc,13 , Φ3c = 21 Φc,24 , Φ4c = − 21 Φc,13 , the corner magnetic fluxes thus defined remain unchanged and satisfy Φ pc = 0. (31) p=1,2,3,4
To construct the semi-discrete equation which evolves Bz,c , we use (20) and (21) on the cell c, d ρ (τ Bz ) dω = w (B · n) d S. dt p∈P (c)∂ω ∩∂ω c pc
ωc
Approximating both hands of the above equation by the mean value theorem, we have d τc Bz,c = mc wp (32) (B · n) d S. dt p∈P (c)
∂ωc ∩∂ω pc
Integrating the divergence-free constraint on the subcell pc gives 0=
∇ · Bdω = ω pc
B · nd S =
∂ω pc \(∂ωc ∩∂ω pc )
∂ω pc
(B · n) d S
B · nd S + ∂ωc ∩∂ω pc
which can be approximated to obtain
(B · n) d S = − ∂ωc ∩∂ω pc
B · nd S = −Bc ·
∂ω pc \(∂ωc ∩∂ω pc )
nd S = Φ pc .
∂ω pc \(∂ωc ∩∂ω pc )
Substituting the above equality into (32), we can obtain the semi-discrete equation which evolves the third component of the magnetic field Bz,c as follows mc
d τc Bz,c = w p Φ pc . dt
(33)
p∈P (c)
Combining (30) and (33), we can get the following relationship between the cellcentered magnetic field and the corner magnetic flux mc
d v p Φ pc . (τc B c ) = dt p∈P (c)
(34)
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3.2 Conservation Principles Considering the fluid in the domain Ω as a whole, we have d τc |B c |2 =− m c Ec + v p · F ∗p , dt c 2μ p∈∂Ω where F ∗p is the boundary force acting on the system through the part of boundary in the vicinity of p. The semi-discrete form of the energy equation yields d d τc |B c |2 τc |B c |2 Ec + m c Ec + mc = dt 2μ dt 2μ c c =− v p · F pc = − vp · F pc . p∈P (c)
c
p
c∈C ( p)
Hence, we have p v p · c∈C ( p) F pc = p∈∂Ω v p · F ∗p . Since v p is arbitrary, we have the following local balance condition of corner forces c∈C ( p)
F pc =
0, ∀ p ∈ D o , F ∗p , ∀ p ∈ ∂D.
(35)
3.3 Thermodynamic Consistency We derive a general form of the subcell force by requiring that the semi-discrete scheme satisfies a semi-discrete entropy inequality within cell c. This semi-discrete entropy inequality by mimicking its continuous counterpart will ensure that kinetic energy will be dissipated into internal energy through shock waves. Thanks to Gibbs formula, the time rate of change of entropy within cell c writes d d d ηc = m c εc + Pc τc dt dt dt
d d τc d 1 d 1 = mc Ec + |B c |2 − vc · vc − B c · |B c |2 τc . (τc B c ) + Pc + dt 2μ dt μ dt 2μ dt
m c Tc
By using Eqs. (22),(23),(27),(28),(31), and (34) we have m c Tc
d 1 1 ηc = |B c |2 l pc n pc , vc − v p · F pc + B c Φ pc − Pc + dt μ 2μ p∈P (c)
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T with n pc = nTpc , 0 . Thus, in order to satisfy the nondecreasing condition of entropy, a sufficient condition is 1 1 2 n pc ≥ 0, ∀ p ∈ P(c). vc − v p · F pc + B c Φ pc − Pc + |B c | l pc μ 2μ A sufficient condition for the above inequality to be satisfied is F pc =
Pc +
1 1 pc vc − v p , |B c |2 l pc n pc − B c Φ pc + M 2μ μ
(36)
pc is a semi-positive matrix which has the dimension of a length times a where M density times a velocity. We call this matrix the corner matrix of pc.
pc 3.4 Construction of the Semi-positive Matrix M Assuming that the wave which connects the physical states on the cell center and on the side p − p of the corner pc is a shock wave and signifying these states by the subscripts c and p − p, then according to (6), we have m − cp
1 1 2 2 v p − vc · n p− p + Π p− p + |B p− p | − Pc − |B c | = 0, 2μ 2μ
T where n p− p = nTp− p , 0 , Π p− p is the pressure on the side p − p, m − cp is the mass swept by the shock wave. Combining this equation with the relationship (7) and the divergence-free constraint in the vicinity of a shock, we have Π
p− p
1 1 2 − |B p p | l p− p n p− p − B p− p l p− p Bn, p− p + 2μ μ 1 1 2 |B c | l p− p n p− p − B c l p− p Bn, p− p = Pc + 2μ μ n p− p ⊗ + m− n p− p + t p− p ⊗ t p− p + ez ⊗ ez vc − v p . (37) cp l p− p
Applying the same derivation to the shock wave which connects the physical states on the cell center and on the side pp + of the corner pc, we have 1 1 |B pp+ |2 l pp+ Π pp+ + n pp+ − B pp+ l pp+ Bn, pp+ 2μ μ 1 1 |B c |2 l pp+ n pp+ − B c l pp+ Bn, pp+ = Pc + 2μ μ n pp+ ⊗ + m+ n pp+ + t pp+ ⊗ t pp+ + ez ⊗ ez vc − v p . (38) cp l pp+
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Adding Eqs. (37) and (38) yields (36) with F pc = + pc = M +
1 1 2 − − |B p p | l p− p n p− p − B p− p l p− p Bn, p− p Πp p + 2μ μ 1 1 2 + + n pp+ − B pp+ l pp+ Bn, pp+ , Π pp + |B pp | l pp+ 2μ μ n p− p ⊗ m − n p− p + t p− p ⊗ t p− p + ez ⊗ ez cp l p− p n pp+ ⊗ n pp+ + t pp+ ⊗ t pp+ + ez ⊗ ez . m + cp l pp+
(39)
Here, we have used the identities n p− p + l pp+ n pp+ = Φ pc , l p− p Bn, p− p + l pp+ Bn, pp+ = B c · l p− p n p− p + l pp+ n pp+ = l pc n pc . l p− p Now the unknowns Π pp+ , Π p− p , B pp+ , B p− p , and v p are all included in the formulation F pc except the nodal velocity v p which is in the right-hand side of this equation. pc in (39) is obviously a symmetric positive-defined matrix. Both m M − + cp and m cp can be chosen as the sonic impedance z c = ρc ac of the corresponding cell c where ac is the isentropic sonic velocity.
3.5 The 2D Nodal Solver at p Adding (36) for all corners which belong to the same vertex p and using the balance condition of corner force (35), we can easily obtain the following linear algebraic system satisfied by the nodal velocity v p : pc vc + Pc + 1 |B c |2 l pc M n pc − 2μ c∈C ( p) pc vc + Pc + 1 |B c |2 l pc pvp = M M n pc − 2μ pvp = M
c∈C ( p)
1 B c Φ pc μ 1 B c Φ pc μ
, ∀ p ∈ D o,
(40)
− F ∗p , ∀ p ∈ ∂D,
(41)
p = where M c∈C ( p) M pc is called the nodal matrix of the vertex p. Substituting the nodal velocities v p obtained from solving the above system into (36), we can get the corner forces F pc directly.
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4 Solution Procedure We discretize the system (23),(27),(28), and (33) in temporal direction by a standard forward Euler scheme to get the cell-centered Lagrangian algorithm for the 2D ideal MHD equations. Now we present the algorithm in detail as following: Step 0 Initialization at time t = t 0 . • Set up the list P(c) of all vertexes for each cell c and the list C ( p) of all neighbor cells for each node p. Both lists are counterclockwise ordered. • Set up the position r 0p of each node p, the fluid variables ρc0 , v0c , E c0 , and the magnetic field B 0c in each cell c. • Compute the geometric quantities l 0pp+ n0pp+ for all sides [ p, p + ] of each cell c, then evaluate l 0pc n0pc for each corner pc. • Compute τc0 , m c , ac0 , Pc0 , Φc,13 , Φc,24 for each cell c and Φ pc for each corner pc. In the whole run, the cell mass m c and the magnetic fluxes Φc,13 , Φc,24 , Φ pc will remain unchanged. Step 1 Nodal solver. • Evaluate the corner matrix Mnpc for each pc, then calculate the nodal matrix Mnp = n c∈C ( p) M pc . • Compute the nodal velocity vnp for each nodal p by solving the linear algebraic system (40),(41). • Evaluate the force F npc for each corner pc from (36). Step 2 Time step. We predict Δtn = tn − tn−1 from Δtn = min{Δt E , ΔtV , C M Δtn−1 } with Δt E = C E min c
d λnc n , Δt = C min {|ω (t )| | |ωc (t n )||}, V V c c acn dt
where C M = 1.01 is the coefficient which allows the time step to increase, C E = 0.5, λnc is the minimal value of the distance between two points of the cell c, C V = 0.1, |ωc (t)| is the area of the cell c. τ n+1 n+1 2 + c2 |B n+1 Step 3 Update of the fluid variables. We compute τcn+1 , vn+1 c | c , Ec from Δtn vn · l npc nnpc , p∈P (c) p mc Δtn vn+1 = vnc − F npc , c p∈P (c) mc 1 1 n n 2 Δtn 2 n E cn+1 + τcn+1 |B n+1 vn · F npc , c | = E c + τc |B c | − p∈P (c) p 2 2 mc τcn+1 = τcn +
which is the full discrete form of the fluid part of the 2D ideal MHD.
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Step 4 Update the geometric variables and the magnetic field. = • Compute the new position for each node p at the n + 1 time step by x n+1 p x np + Δtn+1 vnp . n+1 + • Compute the geometric quantities l n+1 pp+ n pp+ for all sides [ p, p ] of each cell c, n+1 n+1 then evaluate l pc n pc for each corner pc. • Evaluate B n+1 from c n+1 n+1 l13 n13 · Bcn+1 = Φc,13 , n+1 n+1 n24 · Bcn+1 = Φc,24 , l24 Step 5 Equation of state. Deduce the internal energy at the n + 1 time step for each cell c by cn+1 = E cn+1 − 21 vc 2 , then compute the pressure Pcn+1 and the isentropic sound speed acn+1 from the equation of state. Return to step 1.
5 Numerical Tests In this section, we will apply our 2D cell-centered Lagrangian MHD algorithm presented above to two ideal MHD problems, which are the 1D Noh’s problem with magnetic field and the rotor problems. The first problem is given to validate the accuracy of the proposed method. The second problem is given to check numerical resolution and efficiency in keeping the magnetic field divergence free, resolving strong shock wave, etc. In our computations, we take the CFL number as 0.5, the magnetic permeability μ = 1.
5.1 The 1D Noh’s Problem With Magnetic Field This problem describes collision of two plasmas in the magnetic field. We assume that the media is ideal gases with the adiabatic index γ = 5/3. The initial conditions are taken as (1, 1, 0, 0, 0, 1, 0, 0.01) , x < 0 ρ, u, v, w, Bx , B y , Bz , P |t=0 = (1, −1, 0, 0, 0, 1, 0, 0.01) , x > 0 The computational domain is [−1.6, 1.6] × [−0.016, 0.016] which is divided into 400 × 4 cells. The comparison between numerical solution and the exact solution is showed in Fig. 3. From this figure, we see that there is an excellent agreement between these two solutions, and discontinuities of the solution are sharply resolved.
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Fig. 3 Solutions of the 10 Noh’s problem with magnetic field at t = 0.6 on the 400 × 4 mesh Table 1 Numerical errors and convergence rate for the rotor problem Number of cells Numerical error Convergence rate 16 × 16 32 × 32 64 × 64 128 × 128
1.694E-01 1.178E-01 7.143E-02 3.370E-02
– 0.524 0.722 1.084
5.2 The Rotor Problem This is a truly two-dimensional problem taken from [4] and has been used in [17] to compare several numerical schemes. We use exactly the same setup of the problem as was described in [17]. The computational domain Ω = [−0.5, 0.5] × [−0.5, 0.5] is divided into N × N cells with zeroth-order extrapolation boundary conditions for all four sides. The fluid is ideal with √γ = 1.4. The initial pressure and magnetic field are uniform with P = 1, Bx = 5/ 4π , B y = Bz = 0. In the middle of the domain, there is a dense and rotating disk of fluid with radius r0 = 0.1. Initially, the rotor has a uniform initial density of ρ = 10 and rotates with such a uniform angular velocity that the toroidal velocity at r = r0 is u 0 = 2. The ambient fluid is at rest with ρ = 1 and v = 0 for r > r1 = 0.115. The fluid in the cirque, r0 ≤ r ≤ r1 , is specified by using a linear interpolation with respect to r which helps to reduce initial transients. The magnetic field, as it winds up, will confine the rotating dense fluid into an oblate shape.
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We estimate the relative numerical errors and the convergence rates as in [9] and show them in Table 1. Figure 4 shows the solutions for the rotor problem at t = 0.15 on the 300 × 300 Lagrangian mesh, where 35 equally spaced contour lines are used for contours of the density, fluid pressure, and magnetic pressure, respectively. These results demonstrate that our 2D cell-centered Lagrangian MHD algorithm is very robust, and the solutions are well resolved and also comparable with those fixed mesh solutions. Acknowledgements This work is supported by National Natural Science Foundation of China (11271053, 11671049, 91330107) and Defense Industrial Technology Development Program (B1520133015).
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References 1. J.U. Brackbill, D.C. Barnes, The effect of nonzero divB on the numerical solution of the magnetohydrodynamic equations. J. Comput. Phys. 35, 426–430 (1980) 2. D.S. Balsara, Second-order-accurate schemes for magnetohydrodynamics with divergence-free reconstruction. Astrophys. J. Suppl. 151, 149–184 (2004) 3. D.S. Balsara, J. Kim, A comparison between divergence-cleaning and staggered-mesh formulations for numerical magnetohydrodynamics. Astrophys. J. 602, 1079–1090 (2004) 4. D.S. Balsara, D. Spicer, A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations. J. Comput. Phys. 149(2), 270–292 (1999) 5. W. Dai, P.R. Woodward, A high-order Godunov-type scheme for shock interactions in ideal magnetohydrodynamics. SIAM J. Sci. Comput. 18(4), 957–981 (1997) 6. A. Dedner, F. Kemm, D. Kroer, ¨ C.-D. Munz, T. Schnitzer, M. Wesenberg, Hyperbolic divergence cleaning for the MHD equations. J. Comput. Phys. 175, 645–673 (2002) 7. C.R. Evans, J.F. Hawley, Simulation of magnetohydrodynamic flows: a constrained transport method. Astrophys. J. 332, 659–677 (1989) 8. M. Fey, M. Torrilhon, A constrained transport upwind scheme for divergence-free advection, ed. by T.Y. Hou, E. Tadmor. Hyperbolic Problems: Theory, Numerics, and Applications, (Springer, Heidelberg, 2003), pp. 529–538 9. Jianqiang Han, Huazhong Tang, An adaptive moving mesh method for two-dimensional ideal magnetohydrodynamics. J. Comput. Phys. 220, 791–812 (2007) 10. S.-T. Li, An HLLC Riemann solver for magnetohydrodynamics. J. Comput. Phys. 203, 344–357 (2005) 11. P.-H. Maire, R. Abgrall, J. Breil, J. Ovadia, A cell-centered Lagrangian scheme for two dimensional compressible flow problems. SIAM J. Sci. Comput. 29(4), 1781–1824 (2007) 12. K.G. Powell, An approximate Riemann solver for magnetohydrodynamics (that works in more than one dimensions), ICASE Report No. 94-24, Langley, VA (1994) 13. J.A. Rossmanith, An unstaggered, high-resolution constrained transport method for magnetohydrodynamic flows. SIAM J. Sci. Comput. 28, 1766–1797 (2006) 14. D.S. Ryu, F. Miniati, T.W. Jones, A. Frank, A divergence-free upwind code for multidimensional magnetohydrodynamic flows. Astrophys. J. 509(1), 244–255 (1998) 15. J.M. Stone, M.L. Norman, ZEUS-2D: A radiation magnetohydrodynamics code for astrophysical flows in two space dimensions, II: the magnetohydrodynamic algorithms and tests. Astrophys. J. Suppl. 80, 791–818 (1992) 16. R. E. Tipton, A 2D lagrange MHD code, Preprint UCRL-94277 (Lawrence Livermore National Laboratory, 1986) 17. G. Tóth, The ∇ · B = 0 constraint in shock-capturing magnetohydrodynamics codes. J. Comput. Phys. 161, 605–652 (2000)
The Riemann Problem for a General Phase Transition Model on Networks Edda Dal Santo, Massimiliano D. Rosini and Nikodem Dymski
Abstract We deal with phase transition models for vehicular traffic on road networks. The models consider two different traffic regimes and are given by a scalar conservation law in the free phase and by a system of two conservation laws in the congested phase. We focus on the Riemann problem at a junction as a preliminary step for the study of the Cauchy problem on a road network. Keywords Conservation laws · Phase transitions · Vehicular traffic Road network · Junction
1 Introduction These notes deal with phase transition models (PT for short) of hyperbolic conservation laws for vehicular traffic on road networks. The PT models belong to the class of macroscopic models that describe traffic flow in terms of macroscopic variables, such as the density and the average velocity. Their introduction is motivated by experimental observations that show two different behaviours in vehicular traffic, also called phases: at low densities and high speeds (free phase) the flow appears to be reasonably described by a function of the density, while at high densities and low speeds (congested phase) the flow is no more a single-valued function of the density and covers a two-dimensional domain in the fundamental diagram, see [7, Fig. 1.1]. Then, the dynamics are modelled differently depending on the traffic regime: in the free phase it is convenient to use a first-order E. Dal Santo (B) · M. D. Rosini · N. Dymski Instytut Matematyki, Fizyki i Informatyki, Uniwersytet Marii Curie-Skłodowskiej, pl. Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland e-mail: [email protected] M. D. Rosini e-mail: [email protected] N. Dymski e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_34
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model (one scalar conservation law), while the congested phase is better described by a second-order model (a 2 × 2 system of conservation laws). The two-phase approach was suggested for the first time in [7], where the free phase is governed by the classical LWR model due to Lighthill, Whitham and Richards [18, 19], which expresses the conservation of the number of vehicles, whereas the system in the congested phase includes one more equation for the conservation of the linearized momentum. This idea was then exploited by other authors in subsequent papers, see [2–4, 15]. In particular, it inspired the use in [15] of a PT model that couples the LWR equation for the free phase with the ARZ model due to Aw, Rascle and Zhang [1, 20] for the congested phase. Such a model improves the LWR and ARZ models taken separately and has been recently generalized in [2, 3, 11]. Another variant of the PT model of [7] is obtained in [4, 5], where the authors consider an extension of LWR that accounts for heterogeneous driving behaviours in the congested phase. In this paper, we consider the PTa and PT p models described in [11], which further generalize the models treated in [4, 5] and [2, 3, 15], respectively. We drop the superscripts a, p whenever they are not necessary. This generalization is obtained on the basis of two considerations: (1) no assumptions on the intersection of the two phases are imposed; (2) in order to avoid the loss of well posedness of Riemann problems in the case of intersecting phases (as noted in [7, Remark 2]), the free phase is characterized by a unique value of the velocity. However, while the main focus of [11] is on the Riemann problem in presence of a local point constraint on the flow (see [10] and references therein), here we are interested in the study of the Riemann problem on road networks. To describe the evolution of traffic on a network of roads, we first need to understand the behaviour at intersections. In general, the dynamics are underdetermined at junctions and the only conservation of the number of vehicles is not sufficient to obtain a unique solution to the Cauchy problem. Then, other relations must be found that take into account, for example, the drivers’ preferences (i.e. the distribution of traffic from the incoming to the outgoing roads) or the priorities among the incoming roads. A Riemann problem at a junction is a Cauchy problem with constant initial data on each arc entering and exiting the node and it is based on the notion of Riemann solver, namely a map providing solutions to Riemann problems as functions of the initial data. There exist already many Riemann solvers that deal with junctions on road networks, see [6, 12, 16, 17]. For instance, the Riemann solver proposed in [6] satisfies the two following requests: (i) there are some fixed coefficients α ji ∈ (0, 1), i = 1, . . . , n, j = n + 1, . . . , n + m, that represent the drivers’ preferences (i.e. the percentages of drivers who come from the ith incoming road and take the jth outgoing road) and verify n+m j=n+1 α ji = 1; (ii) under the previous condition, drivers behave so as to maximize the flow through the junction. Remark that the second rule (ii) can be thought of as an entropy-like condition used to single out a unique solution to the Riemann problem. However, rules (i) and (ii) do not suffice for nodes having more entering than exiting arcs and some priority parameters must be introduced. Another possible way out of this problem is proposed in [12, 13] for telecommunication
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networks: the authors design a Riemann solver that inverts the order of rules (i) and (ii), which means that, first, the through flow at the junction is maximized and, then, priority and traffic distribution rules are imposed. A Riemann solver similar to [12, 13] is used to handle the Riemann problem at a junction in this paper, where a PTa or a PT p model describes the evolution of traffic along the roads. For completeness, we recall that other Riemann solvers at junctions have been proposed for phase transition models: see [5, 9] for a solver resembling that of [6] in the case of PTa and of the model of [7]; see also [8] for a variant of the solver designed in [12] applied to the model of [7]. The paper is organized as follows. In Sect. 2, we introduce the PT models with the related notations and the main assumptions. In Sect. 3, we recall the Riemann solvers used to construct the solutions to the Riemann problems along the roads. Finally, in Sect. 4 we describe our Riemann solver at a junction.
2 The PT Models In this section, we introduce the models with the related notations and assumptions. We first observe that both PTa and PT p can be rewritten as Free ⎧ .flow ⎪ ⎨u = (ρ, q) ∈ Ωf , ρt + f (u)x = 0, ⎪ ⎩ v(u) = Vf ,
Congested flow ⎧ . ⎪ u = (ρ, q) ∈ Ωc , ⎨ ρt + f (u)x = 0, ⎪ ⎩ qt + [q v(u)]x = 0,
(1)
where ρ ∈ [0, R] is the density and q is the (linearized) momentum of the vehicles, while Ωf and Ωc are the domains of the free phase and congested phase, respectively. . We write Ω = Ωf ∪ Ωc and we refer to Fig. 1 for a detailed picture of Ω a and Ω p in the two cases of intersecting and non-intersecting phases. Moreover, Vf > 0 is the unique velocity in the free phase and we denote Vc ∈ (0, Vf ] the maximal velocity in the congested phase. The (average) speed v ≥ 0 and the flow f ≥ 0 of the vehicles are defined as . a (ρ) (1 + q) for PTa , . va (u) = veq . v(u) = f (u) = ρ v(u), . q for PT p , v p (u) = ρ − p(ρ) a ∈ C2 ((0, R]; R) is the equilibrium velocity in PTa given by where veq
ρ →
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the parameters a ∈ R, σ ∈ (0, R) are fixed and q is a perturbation which provides a thick fundamental diagram in the congested phase; on the other hand, p ∈ C2 ((0, R]; R) is a pressure function in PT p , that satisfies
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Fig. 1 Above we represent in the (ρ, f )-plane, the domains Ωf and Ωc for the PT models. The first column refers to PTa and the second one refers to PT p ; the first row refers to Ωf ∩ Ωc = ∅, namely Vc = Vf , while the second row refers to Ωf ∩ Ωc = ∅, namely Vc < Vf . We also denote . . . . ± f . f u ∗ = u ∗ (u − , u + ), u f± = (σ±f , w± σ±f ), u c± = (σ±c , w± σ±c ), u ± 2 = ψ2 (u + ), u 1 = ψ1 (u − ) defined . in this section, and u c1 = ψ1c (u − ) defined in (4)
p (ρ) > 0,
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for every ρ ∈ (0, R] (e.g. a typical choice is p(ρ) = ρ γ , γ > 0; see [1]). Let q− < q+ and σ±f ≤ σ±c be such that 0 < σ−f < σ+f < R, 0 < σ−c < σ+c < R,
v(σ±f , σ±f q± /R) = Vf , v(σ±c , σ±c q± /R) = Vc .
(2)
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We can explicitly characterize the free and congested domains as
. Ωf = u ∈ [0, σ+f ] × R : q = Q(ρ) ,
. Ωc = u ∈ [σ−c , R] × R : 0 ≤ v(u) ≤ Vc , q− ≤ q ≤ q+ , ⎧ ⎨ (ρ − σ )[Vf R + a(R − ρ)(R − σ )] for PTa , . Q(ρ) = (R − ρ)[Vf σ + a (σ − ρ)(R − σ )] ⎩ ρ [Vf + p(ρ)] for PT p .
where
Notice that v(u) = Vf for all u ∈ Ωf . Furthermore, we denote
. . Ωf+ = u ∈ Ωf : ρ ∈ [σ−f , σ+f ] , Ωf− = u ∈ Ωf : ρ ∈ [0, σ−f ) ,
. . Ωc− = Ωc \ Ωf+ , Ωcex = u ∈ (0, R] × R : v(u) ∈ [0, Vf ], q ∈ [q− , q+ ] , and we point out that Vc = Vf
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A PT model as (1) consists of a scalar conservation law in the free phase and of a 2 × 2 system of conservation laws in the congested phase. The eigenvalues, eigenvectors and Riemann invariants for the system in Ωc are . λ1 (u) = v(u) + u · ∇v(u), . ρ , r1 (u) = q . w1 (u) = q/ρ
. λ2 (u) = v(u), ∂q v(u) . , r2 (u) = −∂ p vρ(u) . w2 (u) = v(u).
For later convenience, we consider the natural extension of the above functions to Ωcex and we denote . w(u) =
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ex if u ∈ Ωc , ρ − 1 if u ∈ Ωf− , σ−f
. w± = q± /R.
We observe that λ2 (u) ≥ 0 and ∇λ2 (u) · r2 (u) = 0 for all u ∈ Ωcex , namely the waves of the second family have non-negative speed and the second characteristic field is linearly degenerate. Moreover, for simplicity we assume (H1)
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the first characteristic field is genuinely nonlinear in Ωcex , except for PTa when a = 0.
We point out that for PT0 (i.e. PTa when a = 0) the first characteristic field is linearly degenerate along the Lax curve of the first family corresponding to w = 0. The Lax curve of the first characteristic family through a point u o ∈ Ωcex is described . in the (ρ, f )-plane by the graph of the map ρ → L w(u o ) (ρ) = f (ρ, w(u o ) ρ) and assumptions (H1) and (H2) can be restated in terms of such curves. Indeed, (H1) is equivalent to require that the first Lax curves are nonincreasing, since L w (ρ) = λ1 (ρ, w ρ); on the other hand, (H2) is equivalent to require that the first Lax curves are strictly concave or convex, since L w (ρ) = ρ1 ∇λ1 (ρ, w ρ) · r1 (ρ, w ρ) (except for PT0 ). In particular, by (2) we have that the first Lax curves for PT p are strictly concave. Notice also that the first Lax curves corresponding to w± give the upper and lower boundaries of the set Ωc in the (ρ, f )-plane. We conclude this section by recalling some other useful notations (see Fig. 1): • the point ψ1f,c (u o ) is the intersection of the Lax curve of the first family through u o with {u ∈ Ω : v(u) = Vf,c }; • the point ψ2± (u o ) is the intersection of the Lax curve of the second family through u o (represented by the graph of the map ρ → v(u o )ρ in the (ρ, f )-plane) with {u ∈ Ω : w(u) = w± }; • the point u ∗ (u , u r ) is the intersection between the Lax curve of the first family through u and the Lax curve of the second family through u r ; • Λ(u , u r ) is the Rankine–Hugoniot speed connecting two states u , u r ∈ Ω.
3 The Riemann Problem on a Single Unidirectional Road In this section, we consider the Riemann problem for PTa and PT p along a single unidirectional road with no entrances and no exits, i.e. the Cauchy problem for (1) with initial datum u if x < 0, (3) u(0, x) = u , u r ∈ Ω. u r if x > 0, Admissible solutions to (1), (3) are defined as in [7, 11]. Definition 1. Let u , u r ∈ Ω. An admissible solution to the Riemann problem (1), . (3) is a self-similar function u = (ρ, q) : R+ × R → Ω that fulfills the following requirements. • If u , u r ∈ Ωf or u , u r ∈ Ωc , then u is the usual Lax solution to (1), (3). • If u ∈ Ωf \ Ωc and u r ∈ Ωc− , then there exists Λ ∈ R such that
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– u(t, (−∞, Λ t)) ⊆ Ωf and u(t, (Λ t, +∞)) ⊆ Ωc for all t > 0; – for all t > 0 the following Rankine–Hugoniot jump condition is satisfied Λ [ρ(t, Λ t + ) − ρ(t, Λ t − )] = f (u(t, Λ t + )) − f (u(t, Λ t − )); – the functions
u(t, x) if x < Λ t, (t, x) → − u(t, Λ t ) if x > Λ t,
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u(t, Λ t + ) if x < Λ t, u(t, x) if x > Λ t,
are the Lax solutions to the Riemann problems, respectively, given by ⎧ ρt + f (u)x = 0, ⎪ ⎪ ⎪ ⎨v(u) = V , f ⎪ u
if x < 0, ⎪ ⎪ ⎩u(0, x) = − u(t, Λ t ) if x > 0,
⎧ ρt + f (u)x = 0, ⎪ ⎪ ⎪ ⎨q + [q v(u)] = 0, t x ⎪ u(t, Λ t + ) if x < 0, ⎪ ⎪ ⎩u(0, x) = if x > 0. ur
• If u ∈ Ωc− and u r ∈ Ωf \ Ωc , conditions entirely analogous to the previous case are required. We denote by R and S the Riemann solvers associated to the Riemann problem (1), (3) in the case Ωf ∩ Ωc = ∅ and Ωf ∩ Ωc = ∅, respectively. The Riemann solvers R and S are valid for both PTa and PT p and are defined according to Definition 1, in the sense that (t, x) → R[u , u r ](x/t) and (t, x) → S [u , u r ](x/t) are admissible solutions to (1), (3). . Definition 2 (The Riemann solver R). Assume Ωf ∩ Ωc = ∅ and call V = Vf = . f Vc , ψ1 = ψ1 = ψ1c . The Riemann solver R : Ω 2 → L∞ (R; Ω) for (1), (3) is defined as follows. 1. If u , u r ∈ Ωf , then the solution is a contact discontinuity from u to u r with speed V . 2. If u , u r ∈ Ωc , then the solution is a 1-wave from u to u ∗ (u , u r ), followed by a 2-contact discontinuity from u ∗ (u , u r ) to u r . 3. If u ∈ Ωc− and u r ∈ Ωf− , then the solution is a 1-wave from u to ψ1 (u ), followed by a contact discontinuity from ψ1 (u ) to u r . 4. If u ∈ Ωf− , u r ∈ Ωc− and Λ(u , ψ2− (u r )) ≥ λ1 (ψ2− (u r )), then the solution is a phase transition from u to ψ2− (u r ), followed by a 2-contact discontinuity from ψ2− (u r ) to u r . 5. If u ∈ Ωf− , u r ∈ Ωc− and Λ(u , ψ2− (u r )) < λ1 (ψ2− (u r )), then let u p = u p (u ) be the state satisfying w(u p ) = w− and Λ(u , u p ) = λ1 (u p ). In this case, the solution is a phase transition from u to u p , followed by a 1-rarefaction from u p to ψ2− (u r ) and, then, by a 2-contact discontinuity from ψ2− (u r ) to u r .
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Notice that, if L w− (σ− ) ≤ 0, then Λ(u , ψ2− (u r )) ≥ λ1 (ψ2− (u r )) for all u ∈ Ωf− , u r ∈ Ωc− and, hence, the last case in Definition 2 never occurs. In particular, this holds true for PT p by (2). Definition 3 (The Riemann solver S ). Assume Ωf ∩ Ωc = ∅, namely Vf >Vc >0. Call u c− ∈ Ωc− the point defined by . u c− = (σ−c , w− σ−c ).
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The Riemann solver S : Ω 2 → L∞ (R; Ω) associated to (1), (3) is defined as follows. . 1. We let S [u , u r ] = R[u , u r ] whenever (u , u r ) ∈ Ωf2 ∪ Ωc2 ∪{(u , u r ) ∈ Ωc × Ωf : L w(u ) (ρ ) ≥ 0} ∪{(u , u r ) ∈ Ωf− × Ωc : Λ(u , u c− ) ≥ λ1 (u c− )} ∪{(u , u r ) ∈ Ωf+ × Ωc : L w(u ) (ρ ) ≤ 0}. 2. If u ∈ Ωc , u r ∈ Ωf and L w(u ) (ρ ) < 0, then we let . R[u , ψ1c (u )](x) for x < Λ(ψ1c (u ), ψ1f (u )), S [u , u r ](x) = R[ψ1f (u ), u r ](x) for x > Λ(ψ1c (u ), ψ1f (u )). 3. If u ∈ Ωf− , u r ∈ Ωc and Λ(u , u c− ) < λ1 (u c− ), then we let for x < Λ(u , u c− ), . u
S [u , u r ](x) = c R[u − , u r ](x) for x > Λ(u , u c− ). 4. If u ∈ Ωf+ , u r ∈ Ωc and L w(u ) (ρ ) > 0, then we let for x < Λ(u , ψ1c (u )), . u
S [u , u r ](x) = c R[ψ1 (u ), u r ](x) for x > Λ(u , ψ1c (u )). Summarizing, we have that S [u , u r ] differs from R[u , u r ] if and only if (u , u r ) satisfies one of the following conditions: u ∈ Ωc ,
u r ∈ Ωf ,
u ∈ Ωf+ ,
L w(u ) (ρ ) < 0,
u r ∈ Ωc ,
u ∈ Ωf− ,
L w(u ) (ρ ) > 0,
u r ∈ Ωc ,
Λ(u , u c− ) < λ1 (u c− ).
In particular, for PT p we have that S [u , u r ] differs from R[u , u r ] if and only if (u , u r ) ∈ Ωc × Ωf ; this is also the case for PTa when L w− (σ−f ) < 0.
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4 The Riemann Problem at a Junction In this section, we deal with the Riemann problem at a junction of roads, on which either PTa or PT p is used to model the evolution of traffic. As in [14, Definition 4.1.1], by road network we mean a couple of finite sets (I , J ), where I is a collection of unidirectional roads and J is a set of junctions. Fix a node J ∈J with n incoming arcs Ii = ] − ∞, 0[ ∈ I , i = 1, . . . , n, and m outgoing arcs I j = ]0, +∞[ ∈ I , j = n + 1, . . . , n + m. The traffic evolution along each road Ih meeting at J , h = 1, . . . , n + m, is governed by (1) for a specific choice of PTa or PT p . A Riemann problem at J is a Cauchy problem for (1) with initial datum constant on each road, namely it is given by n + m Riemann problems each of the form (1) for (t, x) ∈ R+ × Ih , u h,0 ∈ Ω, h = 1, . . . , n + m. (5) u h (0, x) = u h,0 for x ∈ Ih , For simplicity, we use T to indicate both the Riemann solvers R and S , when we do not need to distinguish between the two cases. As in the previous section, we first give the definition of admissible solutions for the Riemann problem at J . Definition 4. Let u h,0 ∈ Ω be given for all h = 1, . . . , n + m. An admissible solution to (5) is a (n + m)-tuple of self-similar functions (u 1 , . . . , u n+m ) : R+ × R → Ω n+m such that for some uˆ 1 , . . . , uˆ n+m ∈ Ω the following conditions hold. • For every i = 1, . . . , n and for t > 0, . T [u i,0 , uˆ i ](x/t) for x < 0, u i (t, x) = for x ≥ 0, uˆ i namely the first n components of the admissible solution consist of waves with non-positive speed. • For every j = n + 1, . . . , n + m and for t > 0, for x ≤ 0, . uˆ j u j (t, x) = T [uˆ j , u j,0 ](x/t) for x > 0, namely the last m components of the admissible solution consist of waves with non-negative speed. n f (uˆ i ) = n+m ˆ j ), namely the number of vehicles is conserved at J . • i=1 j=n+1 f (u The above definition states that the solution is uniquely determined once the values of the traces at the junction are known. Thus, a Riemann solver at the junction J is a map (u 1,0 , . . . , u n+m,0 ) → (uˆ 1 , . . . , uˆ n+m ), where uˆ 1 , . . . , uˆ n+m are the traces of an admissible solution at J .
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In the next section, we introduce the Riemann solvers R J and S J for PTa and PT p , which are inspired by [12, 13] and refer to the case of intersecting and nonintersecting phases, respectively. For simplicity, we denote both the solvers by T J , when we do not need to distinguish between the two cases. We start by determining the possible traces at x = 0 and the corresponding flows for an admissible solution at J . In particular, by Definition 4 we determine the maximum flow γhmax (h = 1, . . . , n + m) that can be reached from an initial datum u h,0 = (ρh,0 , qh,0 ) ∈ Ω by means of waves with either non-positive speed (if h = 1, . . . , n) or non-negative speed (if h = n + 1, . . . , n + m). Proposition 1 (Incoming roads). Let i = 1, . . . , n and fix u i,0 ∈ Ω. If Ωf ∩ Ωc = ∅, then the set of reachable flows is if u i,0 ∈ Ωf , . [0, f (u i,0 )] Oi (u i,0 ) = [0, f (ψ1f (u i,0 ))] if u i,0 ∈ Ωc . Otherwise, if Ωf ∩ Ωc = ∅, then the set of reachable flows is ⎧ [0, ⎪ ⎪ ⎪ ⎨[0, . Oi (u i,0 ) = ⎪[0, ⎪ ⎪ ⎩ [0,
f (u i,0 )] f (u c− )] f (ψ1c (u i,0 ))] ∪ { f (u i,0 )} f (ψ1c (u i,0 ))] ∪ { f (ψ1f (u i,0 ))}
if u i,0 if u i,0 if u i,0 if u i,0
∈ Ωf− and f (u i,0 ) < f (u c− ), ∈ Ωf− and f (u i,0 ) ≥ f (u c− ), ∈ Ωf+ , ∈ Ωc .
Proof. It is sufficient to observe that waves with negative speed can be of three types: waves of the first characteristic family, phase transitions (u , u r ) ∈ Ωf− × Ωc− with f (u ) > f (u r ) and w(u ) < w(u r ) = w− , or (in the case Ωf ∩ Ωc = ∅) phase transitions (u , u r ) ∈ (Ωf+ × Ωc− ) ∪ (Ωc− × Ωf+ ) with w(u ) = w(u r ). Since Oi (u i,0 ) is non-convex, we do not take into account the metastable states in the definition of the corresponding maximum flows. Hence, we let ⎧ f (u i,0 ) ⎪ ⎪ ⎪ ⎪ f (ψ f (u )) ⎪ ⎪ 1 i,0 ⎪ ⎪ ⎨ . γimax (u i,0 ) = ⎧ ⎪ ⎪ ⎪ ⎨ f (u i,0 ) ⎪ ⎪ ⎪ ⎪ f (u c− ) ⎪ ⎪ ⎩⎪ ⎩ f (ψ1c (u i,0 ))
if u i,0 ∈ Ωf , if u i,0 ∈ Ωc ,
if Ωf ∩ Ωc = ∅,
if u i,0 ∈ Ωf− and f (u i,0 ) < f (u c− ), if u i,0 ∈ Ωf− and f (u i,0 ) ≥ f (u c− ), if Ωf ∩ Ωc = ∅. if u i,0 ∈ Ωf+ ∪ Ωc .
Corollary 1. Given u i,0 ∈ Ω on an incoming road Ii = ] − ∞, 0] and γˆi ∈ Oi (u i,0 ), there exists a unique uˆ i ∈ Ω such that the Riemann problem with initial data (u i,0 , uˆ i ) has an admissible solution u i = u i (t, x) consisting of waves with non-positive speed and that satisfies f (u i (t, 0)) = f (uˆ i ) = γˆi .
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Proposition 2 (Outgoing roads). Let j = n + 1, . . . , n + m and fix u j,0 ∈ Ω. Then, the set of reachable flows is if u j,0 ∈ Ωf , . [0, σ+f Vf ] O j (u j,0 ) = [0, f (ψ2+ (u j,0 ))] if u j,0 ∈ Ωc . Proof. It is sufficient to observe that waves with positive speed can be of three types: waves with speed equal to Vf , contact discontinuities of the second characteristic family or phase transitions (u , u r ) ∈ Ωf− × Ωc− with f (u ) < f (u r ) and w(u ) < w(u r ) = w− . Notice that now we do not have to distinguish between the cases of intersecting and non-intersecting phases. Since the set O j (u j,0 ) is convex, the corresponding maximum flow is defined as if u j,0 ∈ Ωf , . σ+f Vf max γ j (u j,0 ) = + f (ψ2 (u j,0 )) if u j,0 ∈ Ωc . Corollary 2. Given u j,0 ∈ Ω on an outgoing road I j = [0, +∞[ and γˆ j ∈ O j (u j,0 ), there exists a unique uˆ j ∈ Ω such that the Riemann problem with initial data (uˆ j , u j,0 ) has an admissible solution u j = u j (t, x) consisting of waves with nonnegative speed and that satisfies f (u j (t, 0)) = f (uˆ j ) = γˆ j . The construction of an admissible solution at the junction J by means of T J proceeds as follows. n+m such that 1. Fix an (n + m)-tuple of priority parameters (θ 1 , . . . , θn+m ) ∈ R n n+m θh > 0 for all h = 1, . . . , n + m and i=1 θi = j=n+1 θ j = 1. 2. Define n n+m . max . max Γinc = γi , Γout = γj . i=1
j=n+1
. Thus, the maximal possible through flow at the junction is Γ = min{Γinc , Γout }. 3. Consider the closed, convex and non-empty sets: Cinc Cout
n n . max = (γ1 , . . . , γn ) ∈ [0, γi ] : γi = Γ , . =
⎧ ⎨ ⎩
i=1
(γn+1 , . . . , γn+m ) ∈
i=1 n+m j=n+1
[0, γ jmax ] :
n+m j=n+1
γj = Γ
⎫ ⎬ ⎭
.
4. Denote by (γˆ1 , . . . , γˆn ) the projection of the point (Γ θ1 , . . . , Γ θn ) on the convex set Cinc and by (γˆn+1 , . . . , γˆn+m ) the projection of (Γ θn+1 , . . . , Γ θn+m ) on the convex set Cout .
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5. For every h = 1, . . . , n + m, set . u h,0 uˆ h = uh
if f (u h,0 ) = γˆh , otherwise,
where u h is the unique solution to the equation f (u) = γˆh provided in either Corollary 1 or 2. . 6. Finally, let T J (u 1,0 , . . . , u n+m,0 ) = (uˆ 1 , . . . , uˆ n+m ). Remark that T J satisfies the consistency condition T J T J (u 1,0 , . . . , u n+m,0 ) = T J (u 1,0 , . . . , u n+m,0 ) for every (u 1,0 , . . . , u n+m,0 ) ∈ Ω n+m . This ensures the well definiteness of the Riemann solver at the junction J and it can be proved as in [13, Lemma 3]. We conclude the section with an example. Example 1. Consider a simple network made of two incoming roads i = 1, 2 and one outgoing road j = 3 that meet at x = 0. Assume that the traffic on each road is described by either PTa or PT p . Once the initial data u 1,0 , u 2,0 , u 3,0 ∈ Ω are chosen, we can determine the maximum flows γ1max , γ2max , γ3max . Then, we define . Γ = min{γ1max + γ2max , γ3max } and fix (θ1 , θ2 , θ3 ) = (θ, 1 − θ, 1), for some θ > 0. We have
Cinc = (γ1 , γ2 ) ∈ [0, γ1max ] × [0, γ2max ] : γ1 + γ2 = Γ ,
Cout = {Γ }.
Then, we take γˆ3 = Γ . We have to distinguish two possible cases: either γ1max + . γ2max = Γ ≤ γ3max or γ1max + γ2max > Γ = γ3max . In the first case, we take (γˆ1 , γˆ2 ) = max max (γ1 , γ2 ). In the second case, let (γ¯1 , γ¯2 ) be the point of intersection of the sets {(γ1 , γ2 ) ∈ R+ × R+ : γ1 + γ2 = Γ } and {(γ1 , γ2 ) ∈ R+ × R+ : γ2 = ((1 − θ )/ θ )γ1 }. If (γ¯1 , γ¯2 ) ∈ [0, γ1max ] × [0, γ2max ], then we take (γˆ1 , γˆ2 ) = (γ¯1 , γ¯2 ). Otherwise, (γˆ1 , γˆ2 ) is the projection of (γ¯1 , γ¯2 ) on Cinc . Finally, we can find the traces at the junction (uˆ 1 , uˆ 2 , uˆ 3 ) by Corollary 1 and 2. Acknowledgements The authors were partially supported by the organizers of HYP2016; the first two authors were also supported by the INdAM – GNAMPA Project 2017 —“Equazioni iperboliche con termini nonlocali: teoria e modelli”.
References 1. A. Aw, M. Rascle, Resurrection of “second order” models of traffic flow. SIAM J. Appl. Math. 60(3), 916–938 (2000) 2. M. Benyahia, M.D. Rosini, Entropy solutions for a traffic model with phase transitions. Nonlinear Anal. Theory Methods Appl. 141, 167–190 (2016)
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3. M. Benyahia, M.D. Rosini, A macroscopic traffic model with phase transitions and local point constraints on the flow. Netw. Heterog. Media 12(2), 297–317 (2017) 4. S. Blandin, D. Work, P. Goatin, B. Piccoli, A. Bayen, A general phase transition model for vehicular traffic. SIAM J. Appl. Math. 71(1), 107–127 (2011) 5. S. Blandin, P. Goatin, B. Piccoli, A. Bayen, D. Work, A general phase transition model for traffic flow on networks. Proc. Soc. Behav. Sci. 00, 1–10 (2012) 6. G.M. Coclite, M. Garavello, B. Piccoli, Traffic flow on a road network. SIAM J. Math. Anal. 36(6), 1862–1886 (2005) 7. R.M. Colombo, Hyperbolic phase transitions in traffic flow. SIAM J. Appl. Math. 63(2), 708– 721 (2002) 8. R.M. Colombo, M. Garavello, Phase transition model for traffic at a junction. J. Math. Sci. (N. Y.) 196(1), 30–36 (2014) 9. R.M. Colombo, P. Goatin, B. Piccoli, Road networks with phase transitions. J. Hyperbolic Differ. Equ. 7(1), 85–106 (2010) 10. R.M. Colombo, P. Goatin, M.D. Rosini, On the modelling and management of traffic. ESAIM Math. Model. Numer. Anal. 45, 853–872 (2011) 11. E. Dal Santo, M.D. Rosini, N. Dymski, M. Benyahia, General phase transition models for vehicular traffic with point constraints on the flow. Math. Method. Appl. Sci. 40(18), 6623– 6641 (2017) 12. C. D’Apice, R. Manzo, B. Piccoli, Packet flow on telecommunication networks. SIAM J. Math. Anal. 38(3), 717–740 (2006) 13. M. Garavello, B. Piccoli, Conservation laws on complex networks. Ann. I. H. Poincarè 26, 1925–1951 (2009) 14. M. Garavello, B. Piccoli, Traffic flow on networks, in Volume 1 of AIMS Series on Applied Mathematics (American Institute of Mathematical Sciences AIMS, Springfield, 2006) 15. P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions. Math. Comput. Model. 44(3), 287–303 (2006) 16. M. Herty, A. Klar, Modeling, simulation, and optimization of traffic flow networks. SIAM J. Sci. Comput. 25(3), 1066–1087 (2003) 17. H. Holden, N.H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads. SIAM J. Math. Anal. 26(4), 999–1017 (1995) 18. M.J. Lighthill, G.B. Whitham, On kinematic waves II. a theory of traffic flow on long crowded roads. Proc. Roy. Soc. Lond. Ser. A. 229, 317–345 (1955) 19. P.I. Richards, Shock waves on the highway. Oper. Res. 4, 42–51 (1956) 20. H.M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior. Transp. Res. part B Methodol. 36(3), 275–290 (2002)
Residual Error Indicators for Discontinuous Galerkin Schemes for Discontinuous Solutions to Systems of Conservation Laws Andreas Dedner and Jan Giesselmann
Abstract This contribution is devoted to fully discrete discontinuous Galerkin approximations of systems of hyperbolic conservation laws in one space dimension. Its focus is on a posteriori error estimators which are obtained by a combination of a reconstruction approach with the relative entropy stability framework. It was shown in earlier works that for certain numerical fluxes, the error estimators are of the same order as the true error before shock formation. For discontinuous solutions, the use of the relative entropy methodology prevents convergence of the error estimator. We investigate whether a part of the error estimator (related to residuals) is convergent post-shock and whether it is useful as an error indicator or a smoothness indicator. Keywords Discontinuous galerkin · A posteriori estimates Hyperbolic conservation laws · Relative entropy
1 Introduction This contribution is concerned with a posteriori analysis of numerical schemes for hyperbolic systems of conservation laws endowed with a strictly convex entropy pair in one space dimension. We study schemes based on a method of lines paradigm where the problem is first discretized in space by a discontinuous Galerkin (dG) approach and the resultant ordinary differential equation is discretized using some single- or multistep method. The main a priori analysis for such methods can be found in [18].
A. Dedner Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK e-mail: [email protected] J. Giesselmann (B) University of Stuttgart, Institute of Applied Analysis and Numerical Simulation, Pfaffenwaldring 57, 70569 Stuttgart, Germany e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_35
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A posteriori results for systems of hyperbolic conservation laws were derived in [15, 16] for front tracking and Glimm’s schemes, see also [14]. An a posteriori error estimator for semi-discrete dG schemes was introduced in [9]. In [11], the authors derive a posteriori estimates for space–time dG schemes in a goal-oriented framework, provided certain dual problems are well posed. All these results deal mainly with the pre-shock case. The contribution at hand is based on a reliable a posteriori error estimator which was introduced in [6]. It was derived using explicitly computable reconstructions of the numerical solution. The reconstructions are Lipschitz continuous in space and time and approximately solve the underlying hyperbolic conservation law with a computable residual. For a general exposition of the ideas of reconstruction-based error estimates, we refer to [17]. Since, the reconstruction is Lipschitz continuous the relative entropy stability framework, originally introduced in [3, 8], gives rise to an upper bound for the difference between any entropy solution and the reconstruction of the numerical solution in terms of the residual and the Lipschitz constant of the reconstruction. Other a priori and a posteriori results using the relative entropy method include [1, 12, 13]. The exponential dependence of the error estimator on the Lipschitz constant of the reconstruction leads to the latter blowing up under mesh refinement, in case the exact solution is discontinuous. This is arguably related to the fact that for generic systems of hyperbolic conservation laws and arbitrary initial data, little is known about uniqueness of entropy solutions, although the known ill-posedness results require more than one dimension [2, 5]. Still, the simulation of hyperbolic conservation laws with solutions containing discontinuities is an active and important research field in mathematics and engineering alike, and in many important examples, there is good agreement between numerical simulations and experimental data. In the contribution at hand, we will investigate to which extent the residual can be used as an error indicator, driving mesh adaptivity, once a discontinuity has formed and whether it is able to distinguish between shocks, contact discontinuities and kinks. The outline of this paper is as follows: In Sect. 2, we describe the numerical schemes under consideration, including the limiting strategy from [7]. We also recall the reconstruction approach and a posteriori error analysis presented in [6]. Section 3 is devoted to the numerical study of the behaviour of the residual, in problems whose solutions contain discontinuities or kinks.
2 Numerical Method and Reconstruction We consider a spatially one-dimensional, hyperbolic system of m ∈ N conservation laws on the flat one-dimensional torus T endowed with initial data u0 : T → U ⊂ Rm where the state space U is an open set: ∂t u + ∂x g(u) = 0 on (0, T ) × T, u(0, ·) = u0 on T.
(1)
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We assume the flux function g to be in C 2 (U , Rm ). We restrict ourselves, to the case that (1) admits a strictly convex entropy, entropy flux pair, i.e. there exists a strictly convex η ∈ C 1 (U , R) and Q ∈ C 1 (U , R) satisfying (D η) D g = D Q. It is well known that weak solutions to (1) are not unique and selection criteria are needed to weed out non-physical solutions. The following condition is inspired by the second law of thermodynamics, see [4, e.g.]. Definition 1 (Entropy solution). A weak solution u ∈ L∞ ((0, T ) × T, U ) to (1) is called an entropy solution with respect to (η, Q), if it weakly satisfies ∂t η(u) + ∂x Q(u) ≤ 0.
(2)
Entropy solutions to scalar problems (satisfying a more discriminating entropy condition) are unique. In contrast, in multiple space dimensions entropy solutions to systems, e.g. the Euler equations are not unique [2, 5]. It is not entirely clear whether entropy solutions to (general) systems in one space dimension with generic initial data are unique. Nevertheless, the entropy inequality (2) gives rise to some stability results, which in particular imply weak–strong uniqueness, see [3, 8].
2.1 Numerical Scheme In the sequel, we study fully discrete schemes approximating (1) based on a method of lines approach. We assume that the spatial discretization is done using a dG method with q-th order polynomials and that the temporal discretization is based on some single- or multistep method of order r . We decompose the space–time domain by choosing −1 = x0 < x1 < · · · < x M−1 < x M = 1 and 0 = t0 < t1 < · · · < t N −1 < t N = T . We account for the periodic boundary conditions by identifying x0 and x M . We define time steps τn := tn+1 − tn , a maximal time step τ := maxn τn , spatial mesh sizes h k+ 21 := xk+1 − xk , and a maximal and minimal spatial step h := maxk h k+ 21 , h min := mink h k+ 21 where h we assume that h min is uniformly bounded for h → 0. For brevity, we write T instead M xi of i=1 xi−1 . Let us define the piecewise polynomial dG ansatz and test spaces: Vqs := {w : [x0 , x M ] → Rm : w|(xi−1 ,xi ) ∈ Pq ((xi−1 , xi ), Rm ) for 1 ≤ i ≤ M} (3) where for any q ∈ N we denote by Pq the space of polynomials of degree q. Then, the fully discrete scheme results as a single- or multistep discretization of the semidiscrete scheme (4) ∂t uh = − f (uh ) where the (non-linear) map f : Vqs → Vqs is defined by requiring that for all vh , ψ ∈ Vqs it holds
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T
f (vh )ψdx = −
T
g(vh )∂x ψdx +
M−1
G(vh (xi− ), vh (xi+ ))[[ψ]]i ,
(5)
i=0
where G : U × U → Rm is a numerical flux function and jumps are denoted [[ψ]]i := ψ(xi− ) − ψ(xi+ ) := lim ψ(xi − s) − lim ψ(xi + s). s0
s0
(6)
Any single- or multistep method applied to (4) computes a sequence of approxiN in time: u0h , u1h , u2h , . . . , uhN ∈ Vqs (or it fails in a way mate solutions at points {tn }n=0 which makes a posteriori error estimates superfluous).
2.2 Limiting For the fluxes under consideration, applying explicit time discretizations to (4) leads to unstable methods. Thus, some stabilization mechanism is needed. We consider the limiting strategy described in [7] which is based on a two-step procedure. In the first step, a smoothness indicator is used to flag cells where the solution needs to be limited. In the second step, a moments limiter is applied to the solution within these ‘troubled cells’. The smoothness indicator is described in detail in [7] and, for the Euler system, is based solely on the pressure. In the simplest approach used here a linear approximation is used on a troubled cell Tk of the form u¯ k + Dk (x − ωk ), where u¯ k is the average of the DG solution in the cell, ωk is the cell’s barycenter. The gradient Dk of the limited linear approximation is computed by Dk =
1 h k+ 21
minmod h k+ 21 ∇u k (ωk ), u¯ k+1 − u¯ k , u¯ k − u¯ k−1 .
2.3 Reconstruction Our reconstruction approach proceeds in two steps. From the discrete values u0h , u1h , ut which is Lipschitz in u2h , . . . , uhN ∈ Vqs we compute a temporal reconstruction time and piecewise polynomial (but discontinuous) in space. In a second step, we ust which is Lipschitz in space and compute from ut a space–time reconstruction time. In order to define the temporal reconstruction, we consider for any vector space V a space of piecewise polynomials in time Vrt (0, T ; V ) := {w : [0, T ] → V : w|(tn ,tn+1 ) ∈ Pr ((tn , tn+1 ), V )}.
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We define the reconstruction ut as a C 0 or even C 1 -function which is piecewise polynomial and whose polynomial degree matches the convergence order of the time n+1 n discretization method. To define ut |[tn ,tn+1 ] , the information unh , un+1 h , f (uh ), f (uh ) is readily available, but for polynomial degrees above 3, we need additional condiut on [tn , tn+1 ]. Note that tions. Let un denote the polynomial which coincides with n t u |[tn ,tn+1 ] by being defined on all of R. Then, ut |[tn ,tn+1 ] can be defined u differs from n n u ) at additional points, which preferably lie in the by prescribing values of u , ( past. In [6], alternative approaches are discussed, as well. Definition 2 (Reconstruction in time). For p ∈ N the reconstruction ut ∈ Vt2 p+3 s (0, T ; Vq ) is determined by ut |[tn ,tn+1 ] = un |[tn ,tn+1 ] for n = 0, . . . , N − 1, j
un (t j ) = uh , for j = n − p, . . . , n + 1, dt u (tj ) = n
j f (uh ),
(8)
for j = n − p, . . . , n + 1.
For a time discretization of order r , a reconstruction with p ≥
r −1 2
should be used.
Remark 1 (Start-up). Note that (strictly speaking) the reconstruction ut is not defined on [t0 , t p ]. However, computing the numerical solution for the first p + 1 time steps ut |[t0 ,t p ] in an analogous way. we may use conditions at {t0 , . . . , t p } to define The following Lemma is proven in [6]. Lemma 1 (Properties of reconstruction in time). Any reconstruction ut as given 1,∞ in time. in Definition 2 is well-defined, computable and W For the remainder of this section, we suppose that there is some compact and convex K ⊂ U such that ut (t, x) ∈ K ∀(t, x) ∈ [0, T ] × T.
(A1)
Remark 2 (Bounded reconstruction). Note that Assumption (A1) is verifiable in an a posteriori fashion, since ut is explicitly computable. It is, however, not sufficient n to verify uh (x) ∈ K for all n = 0, . . . , N and x ∈ T. Remark 3 (Bounds on flux and entropy). Due to the regularity of g and η and the compactness of K, there exist constants 0 < C g¯ < ∞ and 0 < Cη < Cη < ∞, which can be explicitly computed from K, g and η, such that |v T H g(u)v| ≤ C g¯ |v|2 , Cη |v|2 ≤ v T H η(u)v ≤ Cη |v|2 ∀ v ∈ Rm , u ∈ K,
(9)
where |·| is the Euclidean norm for vectors and H denotes Hessian matrices. Our spatial reconstruction of ut is based on [9]. To this end, we restrict ourselves to two types of numerical fluxes G.
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Remark 4 (Condition on the numerical flux). We impose that there exists a locally Lipschitz continuous function w : U × U → U such that for any compact K ⊂ U there exists a constant Cw (K ) > 0 with |w(a, b) − a| + |w(a, b) − b| ≤ Cw (K )|a − b| ∀ a, b ∈ K .
(10)
With this function, the numerical flux G is of one of the two following types: (i) G(a, b) = g(w(a, b)) ∀ a, b ∈ U ; (ii) G(a, b) = g(w(a, b)) − μ(a, b; h)h ν (b − a) ∀ a, b ∈ U for some ν ∈ N0 and some matrix-valued function μ which has the property that for any compact K ⊂ U , there exists a constant μ K > 0 so that |μ(a, b; h)| ≤ for h small enough. μ K 1 + |b−a| h Remark 5 (Restrictions on the numerical flux). 1. The conditions imposed in Remark 4 are stronger than the classical Lipschitz and consistency conditions. 2. The conditions do not guarantee stability of the numerical scheme. In practical computations, interest is restricted to numerical fluxes satisfying one of the assumptions and leading to a stable numerical scheme. 3. The Lax–Wendroff (LW) and Richtmyer numerical fluxes, e.g. G(a, b) = g(w(a, b)), w(a, b) =
a+b λ − (g(b) − g(a)), 2 2
(11)
fit into the frame of Remark 4 (i). 4. The Lax–Friedrichs (LF) flux G(a, b) =
1 g(a) + g(b) − λ(b − a) 2
(12)
corresponds to Remark 4 (ii) with w(a, b) = 21 (a + b), ν = 0, and μ(a, b; h) = λI −
g(a) − 2g(w(a, b)) + g(b) ⊗ (b − a) , 2 b − a 2
where I denotes the m × m identity matrix. The space–time reconstruction ust is constructed by applying a spatial reconstruct tion operator to u (t, ·) for all t. It takes into account the details of the numerical scheme via the map w from Remark 4. Definition 3 (Space–time reconstruction). Let ut be the temporal reconstruction, as in Definition 2. Then, the space–time reconstruction ust (t, ·) is defined as the s unique element of Vq+1 satisfying
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T
s ( ust (t, ·) − ut (t, ·)) · ψ = 0 ∀ψ ∈ Vq−1
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For a proof of the following lemma, we refer to [6, 9]. Lemma 2 (Properties of space–time reconstruction). Let ust be the space–time reconstruction defined by (13). Then, for each t ∈ [0, T ] the function ust (t, ·) is well-defined and locally computable. Moreover, s ∩ C 0 (T)). ust ∈ W1,∞ (0, T ; Vq+1
The reconstruction ust is Lipschitz continuous in space since it is a piecewise polynomial and continuous in space. We define the space–time residual by ust + ∂x g( ust ) ∈ L2 ((0, T ) × T, Rm ). Rst := ∂t
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It is well defined and computable since ust is computable and Lipschitz continuous in space and time. We can now formulate the main a posteriori estimate from [6]: Theorem 1 (A posteriori error bound). Let u be an entropy solution of (1). Let N of a fully discrete dG ust be the space–time reconstruction of the solution {unh }n=0 scheme, defined according to Definition 3. Provided u takes only values in K, then for n = 0, . . . , N the error between the numerical solution unh and u(tn , ·) satisfies u(tn , ·) − unh 2L2 (T) ≤ 2 ust (tn , ·) − unh 2L2 (T) + 2Cη−1 Rst 2L2 ((0,tn )×T) + Cη u0 − ust (0, ·) 2L2 (T) (15)
st 2 tn C C ∂ η g x u (s, ·) L∞ (T) + C η × exp ds . Cη 0 Remark 6 (Computation of the estimator). If w is not any smoother than Lipschitz continuous, then ust |(tn ,tn+1 ) is also only Lipschitz continuous in time, although t u |(tn ,tn+1 ) is a polynomial. In this case, the evaluation of Rst 2L2 ((0,tn )×T) with high precision is numerically extremely expensive. Thus, for our numerical experiments, we use Lax–Wendroff and Lax–Friedrichs fluxes for which w is smooth. Note alut |(tn ,tn+1 ) ) for the so that ust |(tn ,tn+1 ) is a polynomial in time (of the same degree as Lax–Friedrichs flux, since in this case w is linear. Remark 7 (Discontinuous entropy solutions). The estimate in Theorem 1 does not require the entropy solution u to be continuous. However, in case u is discontinuous, ust (s, ·)) L∞ (T) is expected to scale like h −1 and the estimator in (15) is expected ∂x ( to diverge for h → 0. This is a direct consequence of using the relative entropy method and might be connected to a lack of uniqueness of entropy solutions.
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It is shown in [6] that if the time step satisfies a CFL-like restriction and if the numerical flux satisfies Remark 4 (i) or Remark 4 (ii) with ν ≥ 1, then the error estimator in (15) is of optimal order for smooth solutions. This means that, in case the numerical solution converges to a smooth solution with some order, then the error estimator converges to zero with the same order.
3 Numerical Experiments We conduct a sequence of numerical experiments in order to highlight different features of the behaviour of the residual for non-smooth solutions. We study Riemann problems for the full Euler system with ideal gas law and adiabatic constant γ = 1.4 such that exact solutions are available. We use dG schemes with third-order polynomials and a three-stage third-order explicit SSP Runge–Kutta time discretization together with limiters, see [7, 10] for details on the limiter and the time stepping method. We have used the Lax–Friedrichs flux (cf. Remark 5, 4) and the Lax–Wendroff flux (cf. Remark 5, 3). It is quite visible from the figures below that in case the exact solution is not smooth the scheme with Lax–Friedrichs flux yields better results, since less (spurious) oscillations are created, than with Lax–Wendroff flux. For all our experiments the final time is T = 0.6 and the computational domain is Ω = (−1, 2). Our initial grid contains 89 elements and from one refinement level to the next we double the number of elements.
3.1 Two Rarefaction Wave Problems We started the dG scheme with the exact solution at time t1 = 0.2 to the Riemann problem having the initial1data (1, − 2 , 1) x < 21 (ρ0 , v0 , p0 )(x) = . (1, 21 , 1) otherwise at t0 = 0. Its solution contains one left and one right moving rarefaction wave and no contact discontinuity. Note that this solution is not smooth but only continuous, due to the kinks at the boundaries of the rarefaction waves. Results in Fig. 1 show that the error converges to zero with some order between 1 and 1.5 while the residual converges with some order between 0.5 and 1. It should be noted that in this case, in contrast to the analysis and numerical experiments for the smooth case presented in [6], the residual does not have the same convergence rate as the true error for the Lax–Wendroff flux. For the Lax–Friedrichs flux, such a behaviour was expected. The results in Fig. 2 show that for a continuous, but not C1 , solution, the maximal residual converges with rate 0.5. Figure 3 shows that the residual picks up the structure of the solution quite well. The positions of the kinks are visible in the residual. It is notable (and slightly surprising) that the residual is larger in the constant state between the rarefaction waves than inside the rarefaction waves.
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3.2 Sod Problem We consider the Sod shock tube problem1 with initial data (1, 0, 1) x 0, siL ,R = O(1): R(¯x P ) := x ∈ Ω ε , xi ∈ [x¯ P,i − εsiL /2, x¯ P,i + εsiR /2], i = 1, . . . , d, i = 2 . (1) We assume for simplicity siL = siR = si , i = 1, . . . , d, i = 2, but emphasize that symmetry is not essential. Using the microscale variables y := (x − x¯ P )/ε, for x ∈ R(¯x P ), any periodic roughness element is related to the cell domain Y := {y(x) , x ∈ R(¯x P )}, see Fig. 1 (right), with Lipschitz boundary ∂Y .
Fig. 1 Macroscale domain Ω ε (left). Roughness element: in the macroscale domain Ω ε (center) and microscale reference domain Y (right)
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The flow field in the rough domain Ω ε is assumed to be compressible and turbulent that is modeled by the RANS equations. To determine the eddy viscosity [31], we use the Spalart–Allmaras model [28], but emphasize that any algebraic or one-equation model could be used. Given the density ρ, the velocity u = (u 1 , . . . , u d )T , and the pressure p, we define ⎛
⎞ (u · ∇)ρ + ρ∇ · u ⎠ L (u) := ⎝ (u · ∇)u + ρ1 ∇ p − ∇ · σ t (u · ∇) p + γ p(∇ · u) − (γ − 1)((σ t · ∇)u − ∇ · qt )
(2)
with u := (ρ, u, p). The turbulent viscous stress tensor σ t and the turbulent heat flux q t for an isentropic Newtonian fluid are defined by η + ηt σt = Re
2 κ κt γ T ∇T . (3) − (∇ · u)I + ∇u + (∇u) , q t = + 3 Re Pr Prt
Here, η and ηt denote the molecular, respectively, turbulent dynamic viscosity, where the value of ηt is given by the Spalart–Allmaras model. Moreover, κ and κt = c P ηt stand for the laminar and turbulent heat conductivity, respectively, c P is the specific gas constant, while the temperature is denoted by T . Furthermore, γ denotes the ratio of specific heats at constant pressure and volume. The system (2) is closed by the thermal equation of state for a perfect gas p = ρ RT with R the specific gas constant. Note that throughout this work we will use dimensionless quantities. The flow is characterized by the Reynolds number Re and the laminar and turbulent Prandtl number Pr and Prt . The values will be given in Sect. 3. In this paper, we assume that the flow field satisfies the following assumptions. Hypothesis 2.1
√ (i) The flow is subsonic, i.e., the Mach number M := |u|/c, with c := γ p/ρ the sound speed, is less than one in the flow field, ensuring smoothness of the flow field. (ii) The Reynolds number is of the order 106 and large enough to ensure that the flow is turbulent. In view of Hypothesis 2.1, we impose at the various boundary portions of Ω ε subsonic free-stream conditions ρ∞ , p∞ , u∞ = u ∞ e1 with M∞ < 1 at the inflow boundary Γin , and subsonic outflow conditions with pressure pout at the outflow boundary Γout . The rough wall Γε is assumed to be isothermal and we impose no-slip conditions for the velocity: ε (4) ρ ε = ρwall , ∂∂np = 0, uε = 0 on Γε . Only the wall boundary conditions at Γε will be affected by the upscaling strategy. To distinguish them, we use the notation Biso (u ε ) = 0. All other boundary conditions are denoted by B(u ε ) = 0. Then, the exact problem reads
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Biso (u ) = 0 on Γε , B(u ε ) = 0 on Γ ⊂ ∂Ω ε \Γε . Since solving the exact problem (5) numerically requires resolving the roughness by a discretization, the computational cost might be prohibitively high. Therefore, it is of interest to find an approximate solution that reduces the computational load to an acceptable level but captures the effect of the roughness on the solution without resolving it. This so-called effective problem is defined on an effective domain Ω σ ⊂ Ω ε , Fig. 2 (right), with smooth boundary Γσ := {x ∈ Ω ε , x2 = σ } located on top of the roughness Γε : L (u e f f ) = 0 in Ω σ , = 0 on Γσ
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In the effective problem, we have to choose an appropriate smooth effective boundary Γσ on top of the roughness, i.e., σ > 0, and so-called effective boundary conditions B e f f defined on Γσ . The solution strategy is detailed in the following steps. For details, we refer to [10]. Step 1: Asymptotic expansion of true solution. The starting point is an asymptotic expansion of the solution u ε of problem (5) in terms of powers of ε: x − x¯ P + O(ε2 ) , u ε (x) = u 0 (x) + εu 1 x, ε
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Fig. 2 Macroscale domains: rough domain Ω ε (left), smooth domain Ω 0 (center), and effective domain Ω σ (right)
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with u = (ρ, u, p) ∈ Rd+2 . This expansion is assumed to exist for any x ∈ R(¯x P ) ⊂ Ω ε close to the rough surface, i.e., dist(x, Γε ) 1. Here, u 0 = (ρ 0 , u0 , p 0 ) ∈ Rd+2 is the so-called zeroth-order solution, which is the solution of the following system of equations defined on Ω 0 , see Fig. 2 (center): L (u 0 ) = 0 in Ω 0 , Biso (u 0 ) = 0 on Γ0 ,
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B(u 0 ) = 0 on Γ ⊂ ∂Ω 0 \Γ0 . Step 2: Ansatz for upscaling function u 1 . In contrast to classical perturbation theory [21], we assume that the upscaling function u 1 depends on both the macroscale variable x and the microscale variable y = (x − x¯ P )/ε. More precisely, x − x¯ P x − x¯ P =β , x ∈ R(¯x P ) , u 1 x, ε ε
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where the so-called cell function β = (φ, χ , π ) ∈ Rd+2 is assumed to be sufficiently smooth in the cell domain Y . Step 3: Derivation of the cell problem. The term u 1 in (7) is not explicitly known and therefore has to be determined via suitable closing conditions. For each roughness element, we proceed as follows: we plug (7) and the upscaling function (9) into the exact problem (5) and assume that ηt and κt are locally constant. Then, from (2), neglecting higher order terms, we obtain the so-called cell problem defined in Y : (u0 · ∇ y )φ + ρ 0 ∇ y · χ = 0,
t Δ χ + 13 ∇ y (∇ y · χ ) , ρ 0 (u0 · ∇ y )χ + ∇ y π = η+η ε Re y
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Analogously, we derive the boundary conditions of the cell problem. Since in (4) Dirichlet boundary conditions are applied for the velocity uε and the density ρ ε and Neumann conditions for the pressure p ε , respectively, the following boundary conditions are applied on W , see Fig. 1 (right): χ (y) = −y2
∂u0 ∂ρ 0 (¯x P ), φ(y) = −y2 (¯x P ), ∂ x2 ∂ x2
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where n εy is the normal derivative to W in y. Since the roughness is assumed to be periodic, we impose periodic boundary conditions for χ, φ and π along y1 and y3 .
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To solve the cell problem (10), (11) numerically, we cut the unbounded domain Y in the wall-normal direction, introducing a fixed cross section Γup , see Fig. 1 (right). On Γup we prescribe the following boundary conditions: η
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This choice is motivated by the weak formulation of the cell problem, as described in [9]. We define the effective constants as follows: 1 χ1 := |Γup |
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Note that the cell problem and the zeroth-order problem are linked. In contrast to previous work [1, 15, 16, 18, 19], the macroscale variables u 0 , ηt , and κt enter the cell problem. In [9], where the laminar problem is analyzed, it has been numerically verified that for our flow range, the zeroth-order solution cannot be discarded because it significantly affects the effective coefficients and, thus, the effective problem. Step 4: Effective problem. The effective problem is determined by (6) with effective boundary conditions B¯ iso (u e f f ) = 0 on Γσ ∂ pe f f ∂ x2
(x) = 0, ue f f (x) = u0 (x) + ε χ , ρ e f f (x) = ρ 0 (x) + ε φ , x ∈ Γσ . (14) To perform computations, we finally need to specify σ because it enters the effective boundary conditions (14), via the position x ∈ Γσ . For this purpose, we consider the error introduced between the exact solution u ε and the effective boundary conditions on Γσ . As detailed in [9, 10], a reasonable choice is σ = ε.
3 Numerical Results For an application, we consider a three-dimensional turbulent, subsonic flow over a rough surface, where the roughness is modeled by three periodic riblets in spanwise direction. The roughness height ε and the spacing εs3 are chosen as ε = 3.682602 × 10−5 and εs3 = 1.018821 × 10−4 , respectively, for L ∗ = 0.14452. The shape of the riblets can be seen in Fig. 4 (center and right). The dimensionless representation of the state variables is described in [9]. The flow field is characterized by the Reynolds number Re∞ = 106 . In the viscous stress tensor and the heat flux, see Eq. (3), the dynamic viscosity and the heat conductivity are chosen as η = 1 and κ = 1. The gas is assumed to be air; thus, we use Pr = 0.72 and Prt = 1 for the Prandtl numbers and
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Fig. 3 Sketch of the computational domains: exact domain Ωε (left) and effective domain Ωσ (right)
γ = 1.4 for the ratio of specific heats and R = 0.4 for the specific gas constant. The free-stream conditions are determined by the free-stream Mach number M∞ = 0.3, 2 γ ). Moreover, we assume that Twall = T∞ = 1. ρ∞ = 1, u∞ = e1 , and p∞ = 1/(M∞ As starting point, we perform an RANS simulation over the flat plate Γ0 , Γ0 = x ∈ R3 : x1 ∈ [0.899529, 1.383891], x2 = 0, x3 ∈ [0, 1.176307 × 10−4 ] , which will be used as zeroth-order approximation. We assume that in the exact problem the roughness is located in the streamwise direction only in the subinterval [1, 1.383891]. The exact and the smooth effective domains, Ω ε and Ω σ , respectively, are sketched in Fig. 3.
3.1 Cell Problem As described in [10], since in the riblets case the roughness is constant in the streamwise direction, the three-dimensional cell problem (10) decouples into two independent two-dimensional cell problems. Moreover, it can be shown that χ2 = χ3 = π = 0. Thus, given the y¯1 -section of Y sketched in Fig. 4 (left), one has to solve the following two independent problems for χ1 and φ: ∂ 2 χ1 ∂ 2 χ1 (y) + (y) = 0, y ∈ Y y¯1 , 2 ∂ y2 ∂ y32 ∂u 01 (x¯P )y2 , y ∈ W y¯1 , ∂ x2 ∇ y χ1 (y) · n(y) = 0, y ∈ Γup, y¯1 , y3 − periodicit y χ1 (y) = −
and
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∂ 2φ ∂ 2φ (y) + 2 (y) = 0, y ∈ Y y¯1 , 2 ∂ y2 ∂ y3 ∂ρ 0 (x¯P )y2 , y ∈ W y¯1 , ∂ x2 ∇ y φ(y) · n(y) = 0, y ∈ Γup, y¯1 , y3 − periodicit y . φ(y) = −
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These cell problems are solved numerically using the finite element software package deal.II, cf. [3]. 0 ∂u 0 (x¯P ), i.e., on the posiObserve that each cell problem depends on ∂ x21 (x¯P ) and ∂ρ ∂ x2 tion x¯P . In this paper, we will simply approximate χ1,¯x P and φx¯ P by piecewise linear functions χ1,¯x P ∗ and φx¯ P ∗ obtained by solving the cell problems at uni¯ (i) formly distributed locations x¯ (i) P , i = 1, . . . , N , x P ∈ Γ0 , as exemplified in Fig. 5 for N = 5. The cell functions computed at x1 = 1.383891 are given in Fig. 4. Observe that they converge to a constant in the wall-normal direction.
Fig. 4 Longitudinal riblets: sketch of the domain Y y¯1 (left). Effective functions: χ1 (center) and φ (right) computed at x1 = 1.383891
Fig. 5 Effective coefficients: χ1 ∗ (left) and φ ∗ (right)
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3.2 Effective Problem Having computed the effective constants, we can solve the effective problem (6). The RANS equations are discretized by a mixed centered-upwind advective upstream splitting method (AUSM) at second-order accuracy [24] for the convective inviscid terms. The second-order terms are discretized by a second-order central scheme. The integration in time is performed by an explicit five-stage Runge–Kutta method with coefficients optimized for maximum stability at second-order accuracy. For the RANS equations, the Spalart–Allmaras [28] one-equation model is used to compute the eddy viscosity. For a detailed description of the flow solver, the reader is referred to Meinke et al. [25]. Additional results of the solver, showing the quality achieved for solutions based on RANS one-equation models, can be found in Fares et al. [14]. The effect of microstructured surfaces on laminar–turbulent transition was analyzed by Klumpp et al. [22]. To validate the effective solution, we compare the streamwise velocity u 1 and density ρ for the exact, effective, and zeroth-order problem, at Γσ , both in the spanwise and the streamwise directions, cf. Figs. 6 and 7, respectively. Both figures show that the effective solution is a much better approximation of the flat plate, since the effect of the roughness is taken into account. In particular, in Fig. 6, while the zeroth-order approximations are far off all the riblet solutions and the graphs do not even intersect, the effective solutions are significantly closer to the average of the RANS riblet solution. In Fig. 7, since the RANS riblet solution varies spatially with the roughness, we depict it at three different spanwise locations in a roughness element, at the peak, at the trough, and at the midpoint of a riblet, respectively. The effective solutions approximate the average of the riblet solutions while the zeroth-order approximation stays either strictly above (u 1 ) or below (ρ) the exact solutions. Figure 8 illustrates the difference between the derivative of the effective solution and the averaged derivative of the riblets solution on Γσ . It clearly indicates a much better agreement in comparison with the derivative of the zeroth-order solution. The computational cost of solving the effective problem is less costly than that of computing a fully resolved riblet solution. In fact, the number of grid points used
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to compute the structured surface setup is ≈2.4 × 106 in contrast to the flat plate solution which required ≈200, 000 grid points. The effective solution was computed on a domain with ≈185, 000 mesh points. Consequently, the number of grid points between the structured surface and the computation of the effective solution was reduced by a factor of 13. Since the flat plate and the effective solutions are constant in spanwise direction, they can be computed on a two-dimensional domain. As a consequence, the gain will be larger in real scenarios, where more than three riblets structures are considered in the fully three-dimensional exact problem.
4 Conclusions We have extended the concept from [9, 10] for laminar flows, modeled by the compressible Navier–Stokes equations, to the turbulent RANS equations closed by the one-equation Spalart–Allmaras model. This corresponds to a regime of much larger Reynolds numbers and thus stronger convection. As demonstrated by the numerical
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results, the methodology extends in a robust manner to the turbulent regime and correspondingly more complex flow scenarios. In contrast to the laminar case, one has to compute a flat plate RANS solution to obtain the relevant parameters entering the new cell problem. Fortunately, the principal structure of the cell problem remains the same as in the laminar case. In particular, even in the 3D case, one has to deal only with a system of two twodimensional cell problems. However, due to the spatial dependence of the parameters in the cell problem, they need to be solved at different locations x¯ P ∈ Γ0 . To retain the complexity reduction offered by effective boundary conditions, one needs an efficient—essentially online—way of solving the cell problems for frequent parameter queries. This will be addressed in forthcoming work using the reduced basis method. Acknowledgements This work has been supported in part by the German Research Council (DFG) within the DFG Research Unit FOR 1779, by grant DA 117/22-1 and the DFG Collaborative Research Center SFB-TR-40, TP A1, and by the Excellence Initiative of the German Federal and State Governments (RWTH Aachen Distinguished Professorship, Graduate School AICES). Furthermore, the computing resources made available by the High-Performance Computing Center in Stuttgart (HLRS) along with the continued support are gratefully acknowledged.
References 1. Y. Achdou, O. Pironneau, F. Valentin, Effective boundary conditions for laminar flows over periodic rough boundaries. J. Comp. Phys. 147, 187–218 (1998) 2. J.D. Anderson, Hypersonic and High Temperature Gas Dynamics. McGraw-Hill Series in Aeronautical and Aerospace Engineering (1989) 3. W. Bangerth, R. Hartmann, G. Kanschat, deal.II—a general-purpose object-oriented finite element library. ACM Trans. Math. Softw. 33(4), 24/1–24/27 (2007) 4. D.W. Bechert, M. Bruse, W. Hage, R. Meyer, Biological surfaces and their technological application—laboratory and flight experiments on drag reduction and separation control. AIAA Paper 97-1960 (1997) 5. A. Bensoussan, J.L. Lions, G. Papanicolaou, Asymptotic Analysis for Periodic Structures (North-Holland, Amsterdam, 1978) 6. J. Boris, F. Grinstein, E. Oran, R. Kolbe, New insights into large Eddy simulation. Fluid Dyn. Res. 10(4–6), 199 (1992) 7. F. Bramkamp, Ph Lamby, S. Müller, An adaptive multiscale finite volume solver for unsteady and steady state flow computations. J. Comp. Phys. 197(2), 460–490 (2004) 8. M. Bruse, D.W. Bechert, J.G.T.V. der Hoeven, W. Hage, G. Hoppe, in Near-Wall Turbulent Flows, Experiments with Conventional and with Novel Adjustable Drag-reducing Surfaces, ed. by R.M. So, C.G. Speziale, B.E. Launder (Elsevier, Amsterdam, The Netherlands, 1993), pp. 719–738 9. G. Deolmi, W. Dahmen, S. Müller, Effective boundary conditions for compressible flows over rough surface. Math. Models Methods Appl. Sci. 25, 1257–1297 (2015) 10. G. Deolmi, W. Dahmen, S. Müller, Effective boundary conditions: a general strategy and application to compressible flows over rough boundaries. Commun. Comput. Phys. 21(2), 358–400 (2017) 11. Y. Efendiev, T.Y. Hou, Multiscale Finite Element Methods: Theory and Applications (Springer, 2009)
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12. W.E.B. Engquist, The heterogeneous multiscale methods. Commun. Math. Sci. 1(1), 87–132 (2003) 13. W.E.B. Engquist, The heterogeneous multi-scale method for homogenization problems, in Multiscale Methods in Science and Engineering. Lecture Notes Computer Sciences Engineering, vol. 44. (Springer, Berlin, 2005), pp. 89–110 14. E. Fares, W. Schröder, A general one-equation turbulence model for free shear andwall-bounded flows. Flow Turbul. Combust. 73(3–4), 187–215 (2005) 15. E. Friedmann, The optimal shape of riblets in the viscous sublayer. J. Math. Fluid Mech. 12, 243–265 (2010) 16. E. Friedmann, T. Richter, Optimal microstructures drag reducing mechanism of riblets. J. Math. Fluid Mech. 13, 429–447 (2011) 17. M. Itoh, S. Tamano, R. Igushi, K. Yokota, N. Akino, R. Hino, S. Kubo, Turbulent drag reduction by the seal fur surface. Phys. Fluids 18, 065102 (2006) 18. W. Jaeger, A. Mikelic, On the roughness-induced effective boundary conditions for an incompressible viscous flow. J. Differ. Equ. 170, 96–122 (2001) 19. W. Jaeger, A. Mikelic, Couette flows over a rough boundary and drag reduction. Commun. Math. Phys. 232, 429–455 (2003) 20. V. Jikov, S. Kozlov, O. Oleinik, Homogenization of Differential Operators and Integral Functionals (Springer, Berlin, 1995) 21. J. Kevorkian, J.D. Cole, Perturbation Methods in Applied Mathematics. Applied Mathematical Sciences, vol. 34 (Springer-Verlag, New York-Heidelberg-Berlin, 1981) 22. S. Klumpp, M. Meinke, W. Schröder, Numerical simulation of riblet controlled spatial transition in a zero-pressure-gradient boundary layer. Flow Turbul. Combust. 85(1), 57–71 (2010) 23. S.-J. Lee, Y.-G. Jang, Control of flow around a NACA 0012 airfoil with a micro-riblet film. J. Fluids Struct. 20, 659–672 (2005) 24. M.-S. Liou, C. Steffen, A new flux splitting scheme. J. Comput. Phys. 107, 23–39 (1993) 25. M. Meinke, W. Schröder, E. Krause, T. Rister, A comparison of second-and sixth-order methods for large-eddy simulations. Comp. Fluids 31(4), 695–718 (2002) 26. W.-E. Reif, A. Dinkelacker, Hydrodynamics of the squamation in fast swimming sharks. Neues Jahrbuch für Geologie und Paläontologie Abhandlungen 164, 184–187 (1982) 27. B. Roidl, M. Meinke, W. Schröder, A zonal RANS/LES method for compressible flows. Comp. Fluids 67, 1–15 (2012) 28. P.R. Spalart, S.R. Allmaras, A one-equation turbulence model for aerodynamic flows. AIAA paper 92-0439 (1992) 29. L. Tartar, The General Theory of Homogenization. A Personalized Introduction. Lecture Notes of the Unione Matematica Italiana, vol. 7. (Springer, Berlin; UMI, Bologna, 2009) 30. P.R. Viswanath, Aircraft viscous drag reduction using riblets. Prog. Aerosp. Sci. 38, 571–600 (2002) 31. D.C. Wilcox, Turbulence Modeling for CFD, 1st edn. (DCW Industries Inc, La Canada CA, 1993)
A Deterministic Particle Approximation for Non-linear Conservation Laws Marco Di Francesco, Simone Fagioli, Massimiliano D. Rosini and Giovanni Russo
Abstract We review our analytical and numerical results obtained on the microscopic Follow-The-Leader (FTL) many particle approximation of one-dimensional conservation laws. More precisely, we introduce deterministic particle schemes for the Hughes model for pedestrian movements and for two vehicular traffic models that are the scalar Lighthill–Whitham–Richards model (LWR) and the 2 × 2 system Aw–Rascle–Zhang model (ARZ). Their approximation is performed by a set of ODEs, determining the motion of platoons of possible fractional vehicles or pedestrians seen as particles. Convergence results of the schemes in the many particle limit are stated. The numerical simulations suggest the consistency of the schemes. Keywords Conservation laws · Follow-the-leader system · Particle approximation
M. Di Francesco · S. Fagioli DISIM, Università degli Studi dell’Aquila, via Vetoio 1 (Coppito), 67100 L’Aquila, Italy e-mail: [email protected] S. Fagioli e-mail: [email protected] M. D. Rosini (B) Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, pl. Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland e-mail: [email protected] G. Russo Dipartimento di Matematica ed Informatica, Università di Catania, viale Andrea Doria 6, 95125 Catania, Italy e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_37
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1 Introduction The purpose of this paper is to give a brief overview of the convergence results for the FTL schemes of one-dimensional conservation laws recently considered in [11, 12, 14–16]. We point out that such schemes are more theoretical rather than numerical tools. Indeed, for instance, wavefront tracking algorithms are more efficient than FTL schemes which, on the other hand, greatly simplify the proofs of existence results and validate macroscopic models in cases in which the microscopic dynamics are easier to justify compared to the macroscopic ones. The deterministic particle approach started in [14] has seen significant extensions. A first one is in [15], which provides a particle approximation for ARZ [6, 22]. Despite the second-order nature of ARZ, the analytical tools developed in [14] for the first-order LWR [20, 21] applies also in this case. Let us point out that our approach deeply differs from the one proposed in [5], which is implemented via a time discretization and suitable space–time scaling and essentially works away from the vacuum state. Our result in [15] works also in presence of the vacuum and no scaling is performed. Unlike previous numerical attempts (e.g. [8]) our method is conservative and is able to cope with the vacuum. Another extension of our particle approach has been performed in [16] on a onedimensional version of the Hughes model [18]. In this model, the crowd is modelled as a continuum medium, with Eulerian velocity computed via a non-local constitutive law of the overall distribution of pedestrians. Such a non-local dependence is encoded in the weighted distance function φ, computed via a non-linear running cost function c(ρ) and interpreted as an estimated exit time for a given distribution of pedestrians. We refer to [1, 2, 7, 13, 17] for analytical and numerical results available in the literature on the Hughes model. This note is structured as follows. In Sect. 2, we shortly review the results in [11, 14] on the convergence of the FTL scheme towards entropy solutions to LWR. Section 3 deals with the convergence of the FTL scheme proposed in [12] for the IBVP for LWR. In Sect. 4, we review the results in [16] on the particle approximation of the Hughes model. In Sect. 5, we review the results in [15] on the FTL scheme for ARZ. In Sect. 6, we collect some of the numerical simulations performed in [12] for the particle methods introduced for the aforementioned models.
2 LWR In this section, we review the results obtained in [11, 14] on the Cauchy problem for LWR [20, 21] ρt + f (ρ)x = 0, t > 0, x ∈ R, (1) ρ(0, x) = ρ(x), ¯ x ∈ R,
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. where f (ρ) = ρ v(ρ). If ρmax > 0 is the maximal density corresponding to the bumper-to-bumper situation, and vmax is the maximal speed corresponding to the free road, then the initial datum ρ¯ and the velocity map v are assumed to satisfy ρ¯ ∈ L∞ ∩ L1 (R; [0, ρmax ]),
(I1)
v ∈ C1 ([0, ρmax ]; [0, vmax ]), v < 0, v(0) = vmax , v(ρmax ) = 0.
(V1)
In some cases, we require also one of the following conditions: ρ¯ ∈ BV(R; [0, ρmax ]),
(I2)
[0, ρmax ] ρ → ρ v (ρ) ∈ R− is non increasing.
(V2)
Definition 1. Assume (I1)–(V1). We say that ρ ∈ L∞ (R+ × R) is an entropy solution to the Cauchy problem (1) if ρ(t) → ρ¯ in the weak∗ L∞ sense as t ↓ 0 and R+ ×R
|ρ(t, x) − k| ϕt (t, x) + sign(ρ(t, x) − k) f (ρ(t, x)) − f (k) ϕx (t, x) dx dt ≥ 0
for all ϕ ∈ C∞ c (R+ × R) with ϕ ≥ 0 and for all k ∈ [0, ρmax ]. We point out that the above definition is slightly weaker than the definition in [19]. The next theorem collects the uniqueness result in [19] and its variant in [9]. Theorem 1. ([9, 19]). Assume (I1)–(V1). Then there exists a unique entropy solution to the Cauchy problem (1) in the sense of Definition 1. . . We now introduce our FTL scheme for (1). Let L = ρ ¯ L1 (R) , R = ρ ¯ L∞ (R) and . n n fix n ∈ N sufficiently large. Let n = L/n and x¯1 , . . . , x¯n−1 be defined recursively by
⎧ x . ⎨x¯1n = sup x ∈ R : −∞ ρ(x) ¯ dx < n ,
x . ⎩x¯ n = sup x ∈ R : , i ∈ {2, . . . , n − 1}. ρ(x) ¯ dx < n n i x¯ i−1
n It follows that x¯1n < x¯2n < . . . < x¯n−1 and
x¯1n −∞
ρ(x) ¯ dx =
x¯in n x¯i−1
ρ(x) ¯ dx =
∞ n x¯n−1
n ρ(x) ¯ dx = n ≤ (x¯in − x¯i−1 )R,
We let (n − 1) particles evolve according to the FTL system
i ∈ {2, . . . , n − 1}.
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⎧ n n ⎪ ⎨x˙i (t) = v(Ri (t)), t > 0, i ∈ {1, . . . , n − 2}, n (t) = vmax , t > 0, x˙n−1 ⎪ ⎩ n i ∈ {1, . . . , n − 1}, xi (0) = x¯in ,
. Rin (t) =
n . (2) n xi+1 (t) − xin (t)
The discrete maximum principle stated in [14, Lemma 1] ensures that the solution to (2) is well defined and satisfies for any t ≥ 0 n (t) − xin (t) ≥ n /R, xi+1
i ∈ {1, . . . , n − 2}.
We then introduce the discrete density . n n ρ (t, x) = Ri (t) χ[xin (t),xi+1 (t)) (x) = n−1
n−1
i=0
i=0
n
n n χ[xin (t),xi+1 (t)) (x), (3) n xi+1 (t) − xin (t)
. . n . . n n n = Rn−2 where R0n = R1n , Rn−1 , x0n = 2x1n − x2n , and xnn = 2xn−1 − xn−2 . It is easy n n n to prove that ρ (t) L1 (R) = L , ρ (t) L∞ (R) ≤ R and ρ (t) has compact support for all t ≥ 0. The main result of [11, 14] reads as follows. Theorem 2. ([11, Theorem 2.3], [14, Theorem 3]). Assume (I1)–(V1). Moreover, assume at least one of the two conditions (I2) and (V2). Then, (ρ n )n converges (up 1 on R+ × R to the unique entropy solution to the to a subsequence) a.e. and in Lloc Cauchy problem (1) in the sense of Definition 1.
3 The IBVP for LWR with Dirichlet Boundary Conditions In this section, we review the results obtained in [12] on the IBVP for LWR ⎧ ⎪ t > 0, x ∈ (0, 1), ⎨ρt + [ρ v(ρ)]x = 0, ρ(0, x) = ρ(x), ¯ x ∈ (0, 1), ⎪ ⎩ ρ(t, 0) = ρ¯0 (t), ρ(t, 1) = ρ¯1 (t), t > 0.
(4)
We assume that the velocity map satisfies (V1); further, we assume that there exists δ > 0 such that the initial and the boundary data satisfy ρ¯ ∈ L∞ ∩ BV((0, 1); [δ, ρmax ]),
(I3)
ρ¯0 , ρ¯1 ∈ L∞ ∩ Lip ∩ BV(R+ ; [δ, ρmax ]).
(B)
We recall the definition of entropy solution given in [10, Definition 2.1].
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∞ Definition 2. Assume (I3)–(B)–(V1). We say that ρ ∈ C0 (R+ ; Lloc ([0, 1]; [0, ρmax ])) is an entropy solution to the IBVP (4) if
• for any test function φ ∈ C∞ c (R+ × (0, 1)) with φ ≥ 0 and for any k ∈ [0, ρmax ] 0≤
1 |ρ − k| φt + sign(ρ − k) [ f (ρ) − f (k)] φx dx dt + |ρ¯ − k| φ(0) dx; 0
R+ ×(0,1)
• for a.e. τ ≥ 0 we have ρ(τ, 0+ ) = u(1, x) for all x > 0, where u is the self-similar Lax solution to the Riemann problem ⎧ t > 0, x ∈ R, ⎪ ⎨u t + f (u)x= 0, ρ¯0 (τ ) if x < 0, ⎪ x ∈ R; ⎩u(0, x) = + ρ(τ, 0 ) if x > 0, • conditions entirely analogous to the previous case are required at x = 1. We now introduce our FTL scheme for (4). Fix T > 0 and m ∈ N sufficiently . large. Let τm = T /m and approximate the boundary data ρ¯0 , ρ¯1 with . k ρ¯i,m = ρ¯i χ[k τm ,(k+1) τm ) , m−1
. ρ¯ik = ρ¯i (k τm ),
i ∈ {0, 1}.
k=0
. . . Let again L = ρ ¯ L1 (0,1) , R = ρ ¯ L∞ (0,1) and fix n ∈ N sufficiently large. Let n = 0 0 L/n and x¯0 , . . . , x¯n be defined recursively by x¯00
. =0
and
x¯i0
. = sup x ∈ (0, 1) :
x 0 x¯i−1
ρ¯δ (x) dx < n , i ∈ {1, . . . , n}.
By construction x¯n0 ≤ 1. Let the artificial queuing mass Q and the number of queuing particles N be defined by . Q = 2 T vmax ρmax ,
. N = Q/n .
0 0 Let the initial positions of the queuing particles x¯−N , . . . , x¯−1 be defined by
. n x¯i0 = i 0 , ρ¯0
i ∈ {−N + 1, . . . , −1},
qn . 0 0 = x¯−N x¯−N +1 − 0 , ρ¯0
. . where qn = Q − n (N − 1) ∈ [0, n ] and ρ¯00 = ρ¯0 (0). The queuing particles are set in R− , with equal distances from each other in order to match the density ρ¯00 , with the 0 , which carries a mass qn (possibly less only exception of the leftmost particle x¯−N than n ) in order to have a fixed total mass Q for the whole set of queuing particles.
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We let the (n + N + 1) particles evolve according to the FTL system ⎧ 0 0 ⎪ ⎨x˙i (t) = v(Ri (t)), t ∈ [0, τm ], i ∈ {−N , . . . , n − 1}, 0 0 t ∈ [0, τm ], x˙n (t) = v(ρ¯1 ), ⎪ ⎩ 0 0 i ∈ {−N , . . . , n}, xi (0) = x¯i , where for any i ∈ {−N + 1, . . . , n − 1} and t ∈ [0, τm ] we let . Ri0 (t) =
n , xi+1 (t) − xi (t)
. 0 R−N (t) =
qn . x−N +1 (t) − x−N (t)
We then extend the above definitions to [0, T ] recursively as follows. For any integer k ≥ 1, let h k0 be the number of particles that strictly crossed 0 during the time interval (0, k τm ], and by h k1 the number of particles that crossed 1 during the same time interval. We rearrange the particles positions at time t = k τm by setting ⎧ xi (k τm ), ⎪ ⎪ ⎪ ⎪ ⎨xn−h k +1 (k τm ) + (i − n + h k1 − 1) nk , . ρ¯1 1 x¯ik = n k k ⎪ (k τ ) + (i + h + 1) , x m k 0 ⎪ −h 0 −1 ρ¯0 ⎪ ⎪ ⎩x¯ k − qn , −N +1
ρ¯0k
i ∈ {−h k0 − 1, . . . , n − h k1 + 1}, i ∈ {n − h k1 + 2, . . . , n}, i ∈ {−N + 1, . . . , −h k0 − 2}, i = −N .
Then, the (n + N + 1) particles evolve according to the FTL system ⎧ k k ⎪ ⎨x˙i (t) = v(Ri (t)), t ∈ [k τm , (k + 1) τm ], i ∈ {−N , . . . , n − 1}, k k t ∈ [k τm , (k + 1) τm ], x˙n (t) = v(ρ¯1 ), ⎪ ⎩ k k i ∈ {−N , . . . , n}, xi (k τm ) = x¯i ,
(5)
where for any i ∈ {−N + 1, . . . , n − 1} and t ∈ [k τm , (k + 1) τm ] we let . Rik (t) =
n , xi+1 (t) − xi (t)
. k R−N (t) =
qn . x−N +1 (t) − x−N (t)
The discrete maximum–minimum principle stated in [12, Lemma 1.2] ensures that the solution to (5) is well defined and satisfies for any t ≥ 0 n n ≤ xi+1 (t) − xi (t) ≤ , R δ
i ∈ {−N , . . . , n − 1}.
We then introduce the discrete density for t ∈ (0, T ] as m−1 n−1 . k ρ n,m (t, x) = Ri (t) χ[xi (t),xi+1 (t)) (x) χ(k τm ,(k+1) τm ] (t). k=0 i=−N
The main result of [12] reads as follows.
(6)
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Theorem 3. ([12, Theorem 1.4]). Assume (I3)–(V1)–(B). Then, (ρ n,m |(0,1) )n,m converges (up to a subsequence) a.e. and in L1 on R+ × (0, 1) to a weak solution to the IBVP (4) in the sense of Definition 2.
4 The Hughes Model In this section, we review the results obtained in [16] on the one-dimensional version of the Hughes model [18] ⎧ φx ⎪ ρ − f (ρ) = 0, t > 0, x ∈ (−1, 1), ⎪ t |φx | x ⎪ ⎪ ⎨ t > 0, x ∈ (−1, 1), |φx | = c(ρ), ⎪ ⎪(ρ, φ)(t, −1) = (ρ, φ)(t, 1) = (0, 0), t > 0, ⎪ ⎪ ⎩ ρ(0, x) = ρ(x), ¯ x ∈ (−1, 1),
(7)
. where f (ρ) = ρ v(ρ) and c is the cost function. Beside (I1)–(V1) we assume ˆ (V3) ∃ρˆ ∈ (0, ρmax ) s.t. [v(ρ) + ρ v (ρ)](ρˆ − ρ) > 0 for all ρ ∈ (0, ρmax ) \ {ρ}, c : [0, ρmax ] → [1, ∞] is C2 , c ≥ 0, c > 0, c(0) = 1, and c(R) < ∞,
(C)
where now ρmax is the maximal crowd density and vmax is the maximal pedestrian . . ¯ L∞ (−1,1) . The maximum principle velocity. Let again L = ρ ¯ L1 (−1,1) and R = ρ in [17, Proposition 2.5] shows that ρ never exceeds the range [0, R]. As observed in [1, 2, 17], the differential equations in (7) can be rewritten as ρt + F(t, x, ρ)x = 0,
ξ(t) −1
c(ρ(t, y)) dy =
1 ξ(t)
c(ρ(t, y)) dy, t > 0, x ∈ (−1, 1),
(8) . with F(t, x, ρ) = sign(x − ξ(t)) f (ρ). The form (8) highlights that Hughes’ model can be seen as a two-sided LWR, with the turning point ξ(t) splitting the whole interval (−1, 1) into two subintervals and implicitly defined in (8). Definition 3. ([16, Definition 1.1]). Assume (I1)–(V1)–(V3)–(C). We say that ρ ∈ L∞ (R+ × R; [0, R]) is a (well-separated) entropy solution to (7) if . • ρ ≡ 0 on an open cone C = {(t, x) ∈ R+ × R : |x − ξ¯ | < ε t} with ε > 0. • ρ χ(−∞,ξ¯ ) and ρ χ(ξ¯ ,∞) are the entropy solutions in the sense of Definition 1 to ⎧ ⎪ ⎨ρt − f (ρ)x= 0,
ρ(x) ¯ if x ∈ (−1, ξ¯ ), ⎪ ⎩ρ(0, x) = 0 otherwise,
⎧ ⎪ ⎨ρt + f (ρ)x= 0,
t > 0, x ∈ R,
ρ(x) ¯ if x ∈ (ξ¯ , 1), ⎪ ⎩ρ(0, x) = 0 otherwise,
respectively, where x → χ A (x) is the indicator function of a set A ⊆ R.
x ∈ R,
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. • The turning curve T = {(t, x) ∈ R+ × (−1, 1) : x = ξ(t)} is continuous and con 1 ξ(t) tained in C . Moreover, (0, ξ¯ ) ∈ T and −1 c(ρ(t, y)) dy = ξ(t) c(ρ(t, y)) dy for a.e. t ≥ 0. We now introduce our FTL scheme for (7). Fix n ∈ N sufficiently large. Let . n = L/n and x¯0n , . . . , x¯nn be defined recursively by x¯0n
. n . n = min {supp(ρ)} ¯ , x¯i = inf x > x¯i−1 :
x n x¯i−1
ρ(y) ¯ dy ≥ m , i ∈ {1, . . . , n} .
x¯ n n n By definition −1 ≤ x¯in < x¯i+1 ≤ 1 and x¯ ni+1 ρ(y) ¯ dy = n ≤ (x¯i+1 − x¯in )R for all i n i ∈ {0, . . . , n − 1}. We introduce the discretized initial density ρ¯ : R → [0, ρmax ] . ¯n n n ρ¯ n (x) = Ri χ[x¯i ,x¯i+1 ) (x), n−1
. R¯ in =
i=0
n ∈ (0, R]. − x¯in
n x¯i+1
The initial approximate turning point ξ¯ n ∈ (−1, 1) is implicitly defined by
ξ¯ n −1
c ρ¯ n (y) dy =
1 ξ¯ n
c ρ¯ n (y) dy.
It is not restrictive to assume that there exists I0 ∈ {0, . . . , n} such that ξ¯ n ∈ (x¯ In0 , x¯ In0 +1 ). We let (n + 1) particles evolve according to the FTL system ⎧ n ⎪ , ⎪ ⎪x˙0 (t) = −vmax n ⎪ n ⎪ ⎪ (t) = −v R x ˙ ⎨ i i−1 (t) , x˙in (t) = v Rin (t) , ⎪ ⎪ ⎪x˙nn (t) = vmax , ⎪ ⎪ ⎪ ⎩x n (0) = x¯ n , i i
t t t t
> 0, > 0, i ∈ {1, . . . , I0 }, > 0, i ∈ {I0 + 1, . . . , n − 1}, > 0, i ∈ {0, . . . , n},
(9)
. n where Rin (t) = n /[xi+1 (t) − xin (t)], i ∈ {0, . . . , n − 1} \ {I0 }. The approximated n turning point ξ (t) is implicitly defined by
ξ n (t) −1
c(ρ (t, y)) dy =
1
n
ξ n (t)
c(ρ n (t, y)) dy,
where ρ n : R+ × R → [0, ρmax ] is the discretized density . n n ρ n (t, x) = Ri (t) χ[xin (t),xi+1 (t)) (x), n−1
i=0
. R nI0 (t) = 0.
(10)
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Notice that the density has been set to equal zero outside [x0n (t), xnn (t)) and in [x In0 (t), x In0 +1 (t)). The latter is meant to simplify the numerical computation of ξ n , but it introduces an error n in the total mass. Finally, the solution to (9) is well defined and R nI0 (t) = 0 until the turning point does not collide with a particle. In the next theorem, we state the main result in [16], but first, we need to introduce . L = max c (ρ) ρ : ρ ∈ [0, R] ,
. C = c (R) R.
Theorem 4. ([16, Theorem 1.3]). Assume (I1)–(I2)–(V1)–(V3)–(C). If . R = ρ ¯ L∞ (−1,1) < ρmax
and
vmax [L TV(ρ) ¯ + 3 C] < 2v(R),
(11)
then (ρ n )n converges (up to a subsequence) a.e. and in L1 on R+ × (−1, 1) to the unique entropy solution to (7) in the sense of Definition 3. Let us remark that the assumption R < ρmax in (11) is essential in order to have the right-hand side in the inequality in (11) strictly positive. Although the analytical results are restricted to cases in which each particle keeps the same direction for all times, the numerical simulation in Example 3 covers a case with direction switching.
5 ARZ Consider the Cauchy problem for ARZ [6, 22] ⎧ ⎪ t > 0, x ∈ R, ⎨ρt + (ρ v)x = 0, t > 0, x ∈ R, (ρ w)t + (ρ v w)x = 0, ⎪ ⎩ (v, w)(0, x) = (¯v , w)(x), ¯ x ∈ R,
(12)
where v = w − p(ρ) is the velocity, w is a Lagrangian marker identifying the maximum speed of the driver ¯ is the initial datum. For any (v, w) in and (¯v , w) . −1 . ¯ 2+ : v ≤ w , the corresponding density is ρ = p (w − v) ≥ 0, W = (v, w) ∈ R ¯ +; R ¯ + ) ∩ C2 (R+ ; R ¯ + ) satisfies where the “pressure” function p ∈ C0 (R p(0+ ) = 0,
p (ρ) > 0 and 2 p (ρ) + ρ p (ρ) > 0 for every ρ > 0.
(P)
. The typical choice for the pressure function is p(ρ) = ρ γ , γ > 0. Definition 4. ([4, Definition 2.3] and [3, Definition 2.2]). Let (¯v , w) ¯ ∈ L∞ (R; W ). 1 ∞ ¯ 0 ¯ We say that a function (v, w) ∈ L (R+ × R; W ) ∩ C (R+ ; Lloc (R; W )) is a weak solution to (12) if it satisfies the initial condition (v(0, x), w(0, x)) = (¯v (x), w(x)) ¯ (R × R) for a.e. x ∈ R and for any test function φ ∈ C∞ + c
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R+ ×R
p −1 (w − v) (φt + v φx )
1 0 dx dt = . w 0
We now introduce our FTL scheme for (12). Let (¯v, w) ¯ ∈ BV(R; W ) be such that . ρ¯ = p −1 (w¯ − v¯ ) belongs to L1 (R) and ρ¯ has compact support. Let x¯min < x¯max be the extremal points of the convex hull of supp(ρ). ¯ Fix n ∈ N sufficiently large. Let . . L = ρ ¯ L1 (R) > 0, n = L/n and define recursively x¯0n
. = x¯min ,
x¯in
. = sup x ∈ R :
x n x¯i−1
ρ(x) ¯ dx < n ,
i ∈ {1, . . . , n}.
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. . n where vin (ρ) = w¯ in − p(ρ) and Rin (t) = n /[xi+1 (t) − xin (t)]. The existence of a global solution to (13) follows from [15, Lemma 2.3]. Finally, since vin is decreasn , we have x0n (t) ≥ x¯min + ing, and its argument Rin (t) is bounded above by ρi,max n v0 (R0 ) t = x¯min . Define the discretized velocity–Lagrangian marker couple
⎧ n n T ⎪ if x ∈ (−∞, x0n (t)), ⎨(w¯ 0 , w¯ 0 ) V n (t, x) . n = (vin (Rin (t)), w¯ in )T if x ∈ [xin (t), xi+1 (t)), i ∈ {0, . . . , n − 1}, W n (t, x) ⎪ ⎩ n n T n if x ∈ [xn (t), ∞). (w¯ n−1 , w¯ n−1 )
We now state the main result proved in [15]. Theorem 5. ([15, Theorem 3.2]). Assume (P). Let (¯v, w) ¯ ∈ BV(R ; W ) be such that . ρ¯ = p −1 (w¯ − v¯ ) is compactly supported and belongs to L1 (R). Then, (V n , W n )n 1 ¯ + × R; W ) to a weak solution to the converges (up to a subsequence) in Lloc (R Cauchy problem (12) in the sense of Definition 4.
6 Numerical Simulations In this section, we present numerical simulations for our many particle schemes. The particle system is solved by using the Runge–Kutta MATLAB solver ODE23. The initial mesh size is determined by the total number of particles N and the data. For . . simplicity, we take v(ρ) = 1 − ρ and f (ρ) = ρ (1 − ρ).
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Example 1. In Fig. 1 we plot the evolution of ρ n given by (3) and corresponding to the Cauchy problem (1) for LWR with initial datum ρ(x) ¯ = 0.4 χ[−1,0] (x) + 0.8 χ(0,1] (x). Example 2. In Fig. 2 we plot the evolution of ρ n,m given in (6) and corresponding to the IBVP (4) for LWR with data ρ(x) ¯ = 0.8 χ[0,0.5] (x) + 0.1 χ(0.5,1] (x), ρ¯0 = 0.3, ρ¯1 = 0.1. This simulation shows that the actual entropy solution does not match the approximate solution obtained without reupdating the boundary condition at every time step. Example 3. In Fig. 3 we show the evolution of ρ n given in (10) and corresponding . to the Cauchy problem (7) for the Hughes model with c(ρ) = v(ρ)−1 and initial datum ρ(x) ¯ = 0.8 χ(−0.8,−0.5] (x) + 0.6 χ(−0.3,0.3] (x) + 0.9 χ(0.4,0.75] (x). As shown in Fig. 3, this example exhibits the typical mass transfer phenomenon occurring when the turning point ξ(t) is not surrounded by a vacuum region. In this case, particles are crossing ξ(t) and an undercompressive shock starts from ξ(t). Example 4. For ARZ (12), we consider two Riemann problems: the one shown in [8, Sect. 4] and used to check the ability of the scheme to deal with contact discontinuities, and the one given in [5, Sect. 5] and used to check the ability of the scheme to deal with vacuum. The results are presented in Fig. 4 and Fig. 5, respectively.
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References 1. D. Amadori, M. Di Francesco. The one-dimensional Hughes model for pedestrian flow: Riemann-type solutions. Acta Math. Sci. Ser. B Engl. Ed. 32(1), 259–280 (2012) 2. D. Amadori, P. Goatin, M.D. Rosini, Existence results for Hughes’ model for pedestrian flows. J. Math. Anal. Appl. 420(1), 387–406 (2014) 3. B. Andreianov, C. Donadello, M.D. Rosini, A second-order model for vehicular traffics with local point constraints on the flow. Math. Models Methods Appl. Sci. 26(4), 751–802 (2016) 4. B.P. Andreianov, C. Donadello, U. Razafison, J.Y. Rolland, Massimiliano D. Rosini, Solutions of the Aw-Rascle-Zhang system with point constraints. Netw. Heterog. Media 11(1), 29–47 (2016) 5. A. Aw, A. Klar, T. Materne, M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models. SIAM J. Appl. Math. 63(1), 259–278 (2002) 6. A. Aw, M. Rascle, Resurrection of “second order” models of traffic flow. SIAM J. Appl. Math. 60(3), 916–938 (2000) 7. M. Burger, M. Di Francesco, P.A. Markowich, M.-T. Wolfram, Mean field games with nonlinear mobilities in pedestrian dynamics. Discrete Contin. Dyn. Syst. Ser. B 19(5), 1311–1333 (2014) 8. C. Chalons, P. Goatin, Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling. Commun. Math. Sci. 5(3), 533–551 (2007) 9. G.-Q. Chen, M. Rascle, Initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws. Arch. Ration. Mech. Anal. 153(3), 205–220 (2000) 10. R.M. Colombo, M.D. Rosini, Well posedness of balance laws with boundary. J. Math. Anal. Appl. 311(2), 683–702 (2005) 11. M. Di Francesco, S. Fagioli, M.D. Rosini, Deterministic particle approximation of scalar conservation laws. Bollettino dell’Unione Matematica Italiana 10, 487–501 (2017) 12. M. Di Francesco, S. Fagioli, M. D. Rosini, G. Russo, Follow-the-Leader Approximations of Macroscopic Models for Vehicular and Pedestrian Flows (Springer International Publishing, Cham, 2017), pp. 333–378 13. M. Di Francesco, P.A. Markowich, J.-F. Pietschmann, M.-T. Wolfram, On the Hughes’ model for pedestrian flow: the one-dimensional case. J. Differ. Equ. 250(3), 1334–1362 (2011) 14. M. Di Francesco, M.D. Rosini, Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit. Arch. Ration. Mech. Anal. 217(3), 831– 871 (2015) 15. M. Di Francesco, S. Fagioli, M.D. Rosini, Many particle approximation of the Aw-RascleZhang second order model for vehicular traffic. Math. Biosci. Eng. 14(1), 127–141 (2017) 16. M. Di Francesco, S. Fagioli, M.D. Rosini, G. Russo, Deterministic particle approximation of the Hughes model in one space dimension. Kinetic Relat. Models 10(1), 215–237 (2017) 17. N. El-Khatib, P. Goatin, M.D. Rosini, On entropy weak solutions of Hughes’ model for pedestrian motion. Z. Angew. Math. Phys. 64(2), 223–251 (2013) 18. R.L. Hughes, A continuum theory for the flow of pedestrians. Transp. Res. Part B: Method. 36(6), 507–535 (2002) 19. S.N. Kruzhkov, First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81 (123):228–255 (1970) 20. M.J. Lighthill, G.B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. R. Soc. Lond. Ser. A. 229, 317–345 (1955) 21. P.I. Richards, Shock waves on the highway. Operat. Res. 4, 42–51 (1956) 22. H.M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior. Transp. Res. Part B: Method. 36(3), 275–290 (2002)
Splash Singularity for a Free-Boundary Incompressible Viscoelastic Fluid Model Elena Di Iorio, Pierangelo Marcati and Stefano Spirito
Abstract Numerical computations in viscoelasticity show the failure of many numerical schemes when the Weissenberg number is beyond a critical value Keunings (J Non-Newtonian Fluid Mech 20:209–226, 1986, [6]). The existence of singularities in the continuum model could be the way to explain instability appearing in numerical simulations. We consider here a 2D Oldroyd-B type model at high Weissenberg number, and we show the existence of the so-called splash singularities (namely, points where the free boundary remains smooth but self-intersects). In our case, we assume physically realistic boundary conditions given by the static equilibrium of all the force fields acting at the interface. Our strategy is based on local existence and stability results applied to a family of smooth suitable initial configurations, we show they will evolve into a self-intersecting configuration, and then necessarily there exists a positive time t = t ∗ , where the configuration has a splash singularity. To prove local existence and stability, we first apply a conformal transformation to the 2D domain, in order to separate the contact point with splash, and then we pass into Lagrangian coordinates to fix our domain, inspired by a Thomas Beale’s paper on the initial value problem for the Navier–Stokes equations with a free surface. Keywords Splash singularity · Viscoelasticity · Oldroyd model
E. Di Iorio (B) · P. Marcati Gran Sasso Science Institute, L’Aquila, Italy e-mail: [email protected] P. Marcati e-mail: [email protected] S. Spirito University of L’Aquila, L’Aquila, Italy e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_38
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1 Introduction In this paper, we study a viscoelastic fluid model of Oldroyd-B type at high Weissenberg number. The goal of this paper is to study the existence of splash singularity for the following system: ⎧ ⎨ ∂t F + u · ∇ F = ∇u F ∂t u + u · ∇u − Δu + ∇ p = div(F F T ) ⎩ divu = 0,
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with appropriate boundary conditions, which will be explained later. Incompressible viscoelastic fluids are described classically by the following momentum balance equations: ρ(∂ ¯ t u + (u · ∇)u) + ∇ p = divτ, where τ = νs (∇u + ∇u T ) + τ p denotes the stress, νs is the solvent viscosity, and τ p is the stress related to the elastic part. From now on, we assume ρ¯ = 1. The stress tensor satisfies the well-known Oldroyd-B model τ + λ∂tuc τ = ν0 ((∇u + ∇u T ) + λs ∂tuc (∇u + ∇u T )),
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where • ∂tuc τ = ∂t τ + (u · ∇)τ − ∇u T τ − τ ∇u denotes the upper convective time derivative, • ν0 = νs + ν p denotes the material viscosity, νs the solvent viscosity, and ν p the polymeric viscosity, respectively, • λ the relaxation time, and • λs = νν0s λ. By separating the solvent and the polymeric contributions to the stress, we get that the stress τ p satisfies λ∂tuc τ p + τ p = ν p (∇u + ∇u T ). Moreover, combining the equations for the stress together with the equations for the mass and the momentum balance, it follows ⎧ ⎨ ∂t u + u · ∇u + ∇ p = νs Δu + divτ p ν (3) ∂ uc τ = − λ1 τ p + λp (∇u + ∇u T ) ⎩ t p divu = 0. In the study of this type of non-Newtonian fluids, an important role is played by the relaxation time λ, which in our case turns out to be proportional to the Weissenberg number We, a number which measures the ratio between the viscous and the elastic forces, see [11]. For very high Weissenberg number (We → ∞), the system (3)
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reduces to the following one: ⎧ ⎨ ∂t u + u · ∇u + ∇ p = νs Δu + divτ p ∂t τ p + (u · ∇)τ p − ∇u T τ p − τ p ∇u = 0 ⎩ divu = 0.
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Let us denote with α ∈ R2 the Lagrangian coordinate and let X (t, α) be the flux associated with the velocity u. Then, by applying the chain rule, the deformation gradient F(t, X ) = ∂∂αX satisfies the following transport equation: ∂t F + u · ∇ F = ∇u F. If the initial condition τ p (0, X ) = τ0 (X ) is positive definite, then τ p (t, X ) = Fτ0 F T is also positive definite and satisfies the equation ∂t τ p + (u · ∇)τ p − (∇u)τ p − τ p (∇u)T = 0. This allows us to solve the system (1), instead of (4). By imposing that the physical boundary conditions are given by the static equilibrium of the force fields at the interface, the free-boundary problem for the system (1) is ⎧ ∂t F + u · ∇ F = ∇u F ⎪ ⎪ ⎪ ⎪ ⎨ ∂t u + u · ∇u − Δu + ∇ p = div(F F T ) divu = 0, ⎪ ⎪ (− pId + (∇u + ∇u T ) + (F F T − Id))n = 0 ⎪ ⎪ ⎩ u(t)|t=0 = u 0 , F(t)|t=0 = F0
in Ω(t) (5) on ∂Ω(t) in Ω0 .
We also assume divF0 = 0, therefore divF = 0, for all t. The variable domain Ω(t) ⊂ R2 denotes the region occupied by the fluid. Our main result is stated in the following theorem. Theorem 1. There exists a time t ∗ ∈ [0, T ] such that the interface ∂Ω(t ∗ ) selfintersects in one point. Similar results for the Navier–Stokes equations are obtained by Castro, Córdoba, Fefferman, Gancedo and Gómez-Serrano in [2] and by Coutand and Shkoller in [3]. In our paper, the main problem is the presence of the elastic components, which could prevent the development of splash singularity. We prove that this is not the case. One of the important ingredients in the proof is the use of a conformal map that has been introduced for this specific problem in [2]. The map P(z) = z˜ , for z ∈ C \ Γ , √ is defined as a branch of z, where Γ is a line, passed through the splash point. We √ take z ∈ C \ Γ in order to make z an analytic function. The key idea, to prove our theorem is to make the analysis into the Lagrangian framework, in order to have a fixed boundary, as done in the paper of Beale [1] to analyze the free boundary of the
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˜ Fig. 1 Possibilities for P −1 (Ω(t))
Navier–Stokes equations. The geometric ideas behind the proof are inspired from the construction in [2]. The main steps are explained below with the help of Fig. 1. • Let the initial domain Ω0 be a non-regular domain as (b); for this reason, we use the conformal map P and by projection we get Ω˜ 0 , a non-splash domain. ˜ ˜ ·), p(0, ˜ ·), F(0, ·)} are smooth, we can prove the existence of a local • If {Ω˜ 0 , u(0, ˜ ·)}, t ∈ [0, T ]. ˜ solution {Ω(t), u(t, ˜ ·), p(t, ˜ ·), F(t, ˜ z˜ 2 ) · • By a suitable choice of the initial velocity, in particular u(0, ˜ z˜ 1 ) · n > 0, u(0, ˜ t¯)) is as (c). This solution lives in n > 0 such that there exists t¯ > 0 and P −1 (Ω( the tilde complex plane and cannot be transformed, by P −1 , into a solution in the non-tilde complex plane. • To solve the problem in the non-tilde domain, we take a one-parameter family {Ω˜ ε (0), u˜ ε (0), F˜ε (0)}, with Ω˜ ε (0) = Ω˜ 0 + εb and |b| = 1, such that P −1 (Ω˜ ε (0)) is regular, and there exists a local in time smooth solution {Ω˜ ε (t), u˜ ε (t, ·), p˜ ε (t, ·), F˜ε (t, ·)}, which can be inverted in the non-tilde complex plane. • By stability we get
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˜ t¯)) ≤ Cε dist(∂ Ω˜ ε (t¯), ∂ Ω( ˜ t¯)) and so P −1 (Ω˜ ε (t¯)) self-intersects. hence P −1 (Ω˜ ε (t¯)) ∼ P −1 (Ω( −1 ˜ • Since P (Ωε (0)) is regular of type (a) and P −1 (Ω˜ ε (t¯)) is self-intersecting domain of type (c), and then there exists a time t ∗ such that P −1 (Ω˜ ε (t ∗ )) has a splash singularity.
2 Conformal and Lagrangian Transformations The free-boundary incompressible viscoelastic fluid model in Eulerian coordinates that we intend to study is given in (5). Because of the geometrical singularity induced by the self-intersection point, to start with an initial domain Ω0 which is a splash √ domain, as in Fig. 1b, we use the conformal transformation P(z) = z to set our problem inside a regular domain. The new velocity field is defined as follows ˜ P(X )). u(t, ˜ X˜ ) = u(t, P −1 ( X˜ )) then u(t, X ) = u(t, The same for the deformation gradient F ˜ P(X )). ˜ X˜ ) = F(t, P −1 ( X˜ )) then F(t, X ) = F(t, F(t, Remark 1. Defining JkPj = ∂ X j Pk (P −1 ( X˜ )), we have the following derivation formulas: ∂ X j u i (t, X ) = ∂ X˜ u˜ i (t, P(X ))∂ X j Pk (X ), hence k
∂ X j u i (t, P −1 ( X˜ )) = JkPj ∂ X˜ u˜ i (t, X˜ ). k
The system in Ω˜ takes the following form: ⎧ ∂t F˜ + (J P u˜ · ∇ X˜ ) F˜ = J P ∇ X˜ u˜ F˜ ⎪ ⎪ ⎪ ⎪ ⎨ ∂t u˜ + (J P u˜ · ∇ X˜ )u˜ − Q 2 Δu˜ + J P ∇ X˜ p˜ = (J P F˜ · ∇ X˜ ) F˜ Tr(∇ u˜ J P ) = 0 ⎪ ⎪ ⎪ (− pId ˜ + (∇ u˜ J P + (∇ u˜ J P )T ) + ( F˜ F˜ T − Id))(J P )−1 n˜ = 0 ⎪ ⎩ ˜ |t=0 = F˜0 u(t) ˜ |t=0 = u˜ 0 , F(t)
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The next step is to move form Eulerian into Lagrangian coordinates so that we transform a free-boundary problem into a fixed boundary problem. Then, the equation for the flux becomes ⎧ ⎨ d X˜ (t, α) ˜ ˜ u(t, ˜ X˜ (t, α)) ˜ in Ω(t) ˜ = J P ( X˜ (t, α)) dt (6) ⎩ ˜ ˜ X (0, α) ˜ = α˜ in Ω(0).
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Therefore, the Lagrangian variables are given by ⎧ ˜ = u(t, ˜ X˜ (t, α)) ˜ ⎨ v˜ (t, α) ˜ q(t, ˜ α) ˜ = p(t, ˜ X (t, α)) ˜ ⎩ ˜ ˜ X˜ (t, α)). G(t, α) ˜ = F(t, ˜ The new system in [0, T ] × Ω˜ 0 that we are going to study becomes ⎧ ⎪ ∂t G˜ = J P ( X˜ )ζ˜ ∇α˜ v˜ G˜ ⎪ ⎪ ⎪ 2 ˜ ˜ ⎪ ∇α˜ (ζ˜ ∇α˜ v˜ ) + (J P ( X˜ ))T ζ˜ ∇α˜ q˜ = J P ( X˜ )G˜ ζ˜ ∇α˜ G˜ ⎪ ⎪ ∂t v˜ − Q ( X )ζ−1 ⎨ P ˜ ˜ Tr(∇α˜ v˜ (∇α˜ X ) J ( X ))−1= P0 ⎪ − qId ˜ + ((∇α˜ v˜ (∇α˜ X˜ ) J ( X˜ )) + (∇α˜ v˜ (∇α˜ X˜ )−1 J P ( X˜ ))T + ⎪ ⎪ ⎪ ⎪ +(G˜ G˜ T − Id) (J P )−1 ( X˜ )∇Λ X˜ n˜0 = 0 ⎪ ⎪ ⎩ ˜ v˜ (0, α) ˜ = v˜ 0 (α) ˜ = u˜ 0 (α), ˜ G(0, α) ˜ = G˜ 0 (α) ˜ = F˜0 (α), ˜
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0 −1 where ζ˜ = (∇ X˜ )−1 and ∇Λ X˜ = −Λ∇ X˜ Λ, with Λ = , since n˜ = 1 0 −ΛJ|∂PΩ(t) ˜ Λn.
3 Local Existence of Smooth Solutions The idea to prove the local existence is based on a fixed point argument. The iteration ˜ In particular, G˜ satisfies an will separate the equation in v˜ from the equation in G. ODE, and then it will not interfere in the boundary conditions.
3.1 Iterative Scheme The iterative scheme is given by the following two steps: STEP 1
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STEP 2 ⎧ (n+1) ∂t v˜ − Q 2 Δ˜v(n+1) + (J P )T ∇ q˜ (n+1) = f˜(n) ⎪ ⎪ ⎨ (n+1) P Tr(∇ v˜ J ) = g˜ (n) (n+1) ⎪ (−q˜ Id + ((∇ v˜ (n+1) J P ) + (∇ v˜ (n+1) J P )T ))(J P )−1 n˜ 0 ) = h˜ (n) ⎪ ⎩ v˜ (0, α) ˜ = v˜ 0 (α). ˜
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where f˜(n) = − Q 2 Δ˜v(n) + (J P )T ∇ q˜ (n) + Q 2 ( X˜ (n) )ζ˜ (n) ∇(ζ˜ (n) ∇ v˜ (n) ) − (J P ( X˜ (n) ))T ζ˜ (n) ∇ q˜ (n) + J P ( X˜ (n) )G˜ (n) ζ˜ (n) ∇ G˜ (n) , g˜ (n) = Tr(∇ v˜ (n) J P ) − Tr(∇ v˜ (n) ζ˜ (n) J P ( X˜ (n) )), h˜ (n) = −q˜ (n) (J P )−1 n˜ 0 + q˜ (n) (J P ( X˜ (n) ))−1 ∇Λ X˜ (n) n˜ 0 − [∇ v˜ (n) ζ˜ (n) J P ( X˜ (n) ) + (∇ v˜ (n) ζ˜ (n) J P ( X˜ (n) ))T ](J P ( X˜ (n) ))−1 ∇Λ X˜ (n) n˜ 0 + ((∇ v˜ (n) J P ) + (∇ v˜ (n) J P )T )(J P )−1 n˜0 − (G˜ (n) G˜ T (n) − Id)(J P ( X˜ (n) ))−1 ∇Λ X˜ (n) n˜ 0 .
The system regarding the flux X˜ is given by ⎧ d ⎨ X˜ (n+1) (t, α) ˜ = J P ( X˜ (n) (t, α))˜ ˜ v(n) (t, α) ˜ dt ⎩ X˜ (0, α) ˜ = α˜ in Ω˜ 0 ,
(10)
thus X˜ (n+1) satisfies X˜ (n+1) (t, α) ˜ = α˜ +
t
J P ( X˜ (n) )˜v(n) (τ, α) ˜ dτ
(11)
0
3.2 Analysis of the System (9) As we mentioned before, we will analyze the linearized system related to STEP 2 by means of the techniques introduced by T. Beale in [1] and their improvements in [2]. Namely, we study ⎧ ∂t v˜ − Q 2 Δ˜v + (J P )T ∇ q˜ = f˜ ⎪ ⎪ ⎨ Tr(∇ v˜ J P ) = g˜ ⎪ (−qId ˜ + (∇ v˜ J P ) + (∇ v˜ J P )T )(J P )−1 n˜ = h˜ ⎪ ⎩ v˜ (0, α) ˜ = v˜ 0 ,
(12)
where the compatibility conditions for the initial data are as follows:
˜ Tr(∇ v˜ 0 J P ) = g(0) P −1 ⊥ ˜ ((J P )−1 n) ˜ ⊥ (∇ v˜ 0 J P + (∇ v˜ 0 J P )T )(J P )−1 n˜ = h(0)((J ) n) ˜
in Ω˜ 0 on ∂ Ω˜ 0 . (13)
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We are going to use the following functional spaces: ˜ ∈K X 0 := (˜v , q)
s+1
× K prs : v˜ (0) = 0, ∂t v˜ (0) = 0, q(0) ˜ =0 ,
˜ 0) ∈ K s−1 × K¯ s × K s− 21 ([0, T ] × ∂ Ω) ˜ : ˜ h, Y0 :={( f˜, g, ˜ f˜(0) = 0, g(0) ˜ = 0, ∂t g(0) ˜ = 0, h(0) = 0 and (13) are satisfied}. Therefore on the linearized problem, we recall the following result obtained by Beale, used later on in order to prove the local existence. Theorem 2. Let 2 < s < 25 and let L : X 0 → Y0 be the operator associated with the system (12), then L is invertible and the norm of the inverse is uniformly bounded for any 0 < T < T¯ .
3.3 The Fixed Point Argument In order to apply Theorem 2 and hence to get bounds independent of T , we need v˜ |t=0 = 0 and ∂t v˜ |t=0 = 0. For this reason, we replace the initial condition v˜ 0 as follows: φ = v˜ 0 + t exp(−t 2 )(Q 2 Δ˜v0 − (J P )T ∇ q˜φ ), (n) where q˜φ is chosen in such a way that for all n, ∂t v˜ |t=0 = ∂t φ|t=0 = 0. The velocity is defined by (14) w˜ (n) = v˜ (n) − φ.
Therefore, we can rewrite the system (9) in the following way: ⎧ ∂t w˜ (n+1) − Q 2 Δw˜ (n+1) + (J P )T ∇ q˜w(n+1) = f˜(n) − ∂t φ ⎪ ⎪ ⎪ ⎪ ⎪ +Q 2 Δφ − (J P )T ∇ q˜φ ⎪ ⎪ ⎪ ⎨ Tr(∇ w˜ (n+1) J P ) = g˜ (n) − Tr(∇φ J P ) (n+1) (n+1) P ⎪ J ) + (∇ w˜ (n+1) J P )T )](J P )−1 n˜0 = ⎪ [−q˜w Id + ((∇ w˜ ⎪ ⎪ ⎪ ⎪ = h˜ (n) + q˜φ (J P )−1 n˜ 0 − ((∇φ J P ) + (∇φ J P )T )(J P )−1 n˜ 0 ⎪ ⎪ ⎩ (n+1) w˜ |t=0 = 0.
(15)
Our main local existence result is then equivalent to show the following theorem. Theorem 3. Let 2 < s < 25 and 1 < γ < s − 1. If (˜v(0), ∂t v˜ (0)) = (0, 0) and (q(0), ˜ s ˜ ˜ ˜ ˜ ˜ h(0)) = (0, 0, 0, 0, 0), moreover G(0) = G 0 ∈ H , then there f (0), g(0), ˜ ∂t g(0), ˜ exist T (sufficiently small) and a solution { X˜ (·), v˜ (·), q(·), ˜ G(·)} ∈ F s+1,γ × s+1 s s,γ −1 × K pr × F on [0, T ]. K In order to prove this theorem, we need the following technical results.
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Proposition 1. 1. Let G˜ (n) − G˜ 0 ∈ F s,γ −1 , X˜ (n) − α˜ ∈ F s+1,γ , and w˜ (n) ∈ K and such that (i) G˜ (n) − G˜ 0 ∈ G˜ − G˜ 0 ∈ F s,γ −1 : t P ∇φ G G˜ − G˜ 0 − ˜ 0 dτ J 0
(ii) w˜ (n) K
s+1
F s,γ −1
t P ∇φ G ˜ 0 dτ ≤ J 0
s+1
F s,γ −1
≡ B,
≤ N.
Then, for T > 0 small enough, depending only on N , v˜ 0 , G˜ 0 , G˜ (n+1) − G˜ 0 ∈ B. ˜ X˜ (n−1) − α˜ ∈ F s+1,γ and 2. Let G˜ (n) − G˜ 0 , G˜ (n−1) − G˜ 0 ∈ F s,γ −1 , with X˜ (n) − α, w˜ (n) , w˜ (n−1) ∈ K s+1 and such that (n) w˜ s+1 ≤ M, w˜ (n−1) s+1 ≤ M, (i) K K (ii) X˜ (n) − α˜ s+1,γ ≤ M, X˜ (n−1) − α˜ s+1,γ ≤ M, F F (iii) G˜ (n) − G˜ 0 s,γ −1 ≤ M, G˜ (n−1) − G˜ 0 s,γ −1 ≤ M, F
F
for some M > 0. Then ˜ (n+1) − G˜ (n) G
F s,γ −1
≤ C T δ G˜ (n) − G˜ (n−1) s,γ −1 + w˜ (n) − w˜ (n−1) K F
˜ (n) (n−1) ˜ + X − X s+1,γ , F
for a suitable δ > 0. Proof. The proof is given in [4]. Proposition 2. 1. Let X˜ (n) − α˜ ∈ F s+1,γ , w˜ (n) ∈ K s+1 and such that
t P ˜ − α˜ − X (i) X˜ (n) − α˜ ∈ X˜ − α˜ ∈ F s+1,γ : J φ dτ s+1,γ ≤ 0 F t P ≡ BJ P φ , J φ dτ (ii) w˜ (n) K
s+1
≤ N.
0
F s+1,γ
Then, for T > 0 small enough, depending only on N and v˜ 0 , X˜ (n+1) − α˜ ∈ B J P φ . ˜ X˜ (n−1) − α˜ ∈ F s+1,γ , with w˜ (n) , w˜ (n−1) ∈ K 2. Let X˜ (n) − α, (i) w˜ (n) K s+1 ≤ M, w˜ (n−1) K s+1 ≤ M,
s+1
and such that
s+1
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(ii) X˜ (n) − α˜
F s+1,γ
≤ M, X˜ (n−1) − α˜
F s+1,γ
≤M
for some M > 0. Then ˜ (n+1) − X˜ (n) X
F s+1,γ
≤ C T δ X˜ (n) − X˜ (n−1)
F s+1,γ
+ w˜ (n) − w˜ (n−1) K
s+1
,
for a suitable δ > 0. Proof. The proof of this theorem is given in [2]. Proposition 3. 1. Let X˜ (n) − α˜ ∈ F s+1,γ , q˜w(n) ∈ K prs and w˜ (n) ∈ K that (i) X˜ (n) − α ˜ F s+1,γ ≤ N , (n) (ii) G˜ − G˜ 0 F s,γ −1 ≤ N , ˜ q) ˜ ∈K (iii) (w˜ (n) , q˜w(n) ) ∈ (w,
s+1
s+1
, and such
× K prs : w˜ |t=0 = 0, ∂t w˜ |t=0 = 0,
(w, ˜ q) ˜ − L −1 ( f˜φL , g˜ φL , h˜ φL ) K
s+1 ×K s pr
≤ L −1 ( f˜φL , g˜ φL , h˜ φL ) K
s+1 ×K s pr
≡ B L −1 ( f˜L ,g˜ L ,h˜ L ) . φ
φ
Then
φ
(w˜ (n+1) , q˜w(n+1) ) ∈ B L −1 ( f˜φL ,g˜φL ,h˜ φL ) .
2. Let X˜ (n) − α, ˜ X˜ (n−1) − α˜ ∈ F s+1,γ , G˜ (n) − G˜ 0 , G˜ (n−1) − G˜ 0 ∈ F s,γ −1 , (n) (n−1) (n) (n−1) (n) (n−1) s+1 ∈K , with w˜ |t=0 = w˜ |t=0 = 0, ∂t w˜ |t=0 = ∂t w˜ |t=0 = 0, q˜w(n) , w˜ , w˜ (n−1) s ∈ K pr , and such that q˜w (n−1) (n) w˜ w˜ s+1 ≤ M, (i) K K s+1 ≤ M, ˜ (n) ˜ (n−1) (ii) X − α˜ s+1,γ ≤ M, − α˜ s+1,γ ≤ M, X F F ˜ (n−1) (iii) G˜ (n) − G˜ 0 s,γ −1 ≤ M, − G˜ 0 s,γ −1 ≤ M, G F
(iv) q˜w(n) K prs , ≤ M,
F
q˜w(n−1) K prs , ≤ M.
for some M > 0. Then (n+1) w˜ − w˜ (n) K s+1 + q˜w(n+1) − q˜w(n) K s ≤ C T δ X˜ (n) − X˜ (n−1) s+1,γ pr F
(n) (n) (n−1) (n−1) (n) (n−1) s+1 + q˜ − q˜ s + G˜ − G˜ , + w˜ − w˜ w w K K s,γ −1 pr
for a suitable δ > 0. Proof. For the proof of this theorem, we will use Theorem 2.
F
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By putting together the results of Proposition 1, Proposition 2 and Proposition 3 and by applying the contraction mapping principle we get the proof of Theorem 3.
4 Stability In this section, we want to prove what we described in the Introduction. So we pick a one-parameter family Ω˜ ε (0) = Ω˜ 0 + εb, where |b| = 1 and such that P −1 (Ω˜ ε (0)) is a regular domain as in Fig. 1a. We take the difference between the solution ˜ and the perturbed solution (w˜ ε , q˜ε , X˜ ε , G˜ ε ), which is as follows: (w, ˜ q, ˜ X˜ , G) ⎧ ∂t (w˜ − w˜ ε ) − Q 2 Δ(w˜ − w˜ ε ) + (J P )T ∇(q˜w − q˜w,ε ) = F˜ε ⎪ ⎪ ⎨ Tr(∇( w˜ − w˜ ε )J P ) = K˜ ε ⎪ −(q˜w − q˜w,ε )Id + ∇(w˜ − w˜ ε )J P + (∇(w˜ − w˜ ε )J P )T (J P )−1 n˜ 0 = H˜ ε ⎪ ⎩ w˜ 0 − w˜ ε,0 = 0.
(16)
The main estimates we prove, for a suitable δ > 0 and for 2 < s < 25 , are • G˜ − G˜ ε L ∞ H s + G˜ − G˜ ε H 2 H γ −1 ≤ Cε + C T δ w˜ − w˜ ε K s+1 + G˜ − G˜ ε L ∞ H s
˜ G˜ ε 2 γ −1 + X˜ − X˜ ε ∞ s+1 + X˜ − X˜ ε 2 γ , + G− H H L H H H • w˜ − w˜ ε K s+1 + q˜w − q˜w,ε K prs ≤ Cε + C T δ ( w˜ − w˜ ε K s+1 + q˜w − q˜w,ε K prs + G˜ − G˜ ε L ∞ H s + G˜ − G˜ ε H 2 H γ −1 + X˜ − X˜ ε L ∞ H s+1 + X˜ − X˜ ε H 2 H γ ), • X˜ − X˜ ε L ∞ H s+1 + X˜ − X˜ ε H 2 H γ ≤ Cε + C T δ w˜ − w˜ ε K
+ X˜ − X˜ ε H 2 H γ .
s+1
+ X˜ − X˜ ε L ∞ H s+1
5 Existence of Splash Singularity (Proof of Theorem 1) From the stability estimates above, we obtain
X˜ − X˜ ε L ∞ H s+1 ≤ 3Cε, by choosing 0 < T
0, and then we get a domain Ω( ˜ t¯)) is a self-intersecting domain, for t¯ > 0. By using (17), it folmapping, P −1 (Ω( lows that P −1 (Ω˜ ε (t¯)) is also a self-intersecting domain. In conclusion, we have that P −1 (Ω˜ ε (0)) is a regular domain of type (a) and P −1 (Ω˜ ε (t¯)) is a self-intersecting domain of type (c), for some t¯ ∈ (0, T ], then there exists a time t ∗ ∈ (0, t¯) such ˜ ∗ )) self-intersects in one point, so it forms a splash singularity. Thus, that P −1 (Ω(t Theorem 1 holds.
Appendix The spaces we used in our proof are of the type H s,r ([0, T ]; Ω) = L 2t Hxs ∩ Htr L 2x . For our purposes, we shall always take r = 2s , and we introduce the following notations: s
K s ([0, T ]; Ω) = L 2t Hxs ∩ Ht 2 L 2x , ˙1 K prs ([0, T ]; Ω)={q ∈ L ∞ t Hx : ∇q ∈ K K¯ s ([0, T ]; Ω) = L 2t Hxs ∩ Ht
s+1 2
s−1
([0, T];Ω), q ∈ K
s− 21
([0, T ]; ∂Ω)},
Hx−1 ,
s+1 F s+1,γ ([0, T ]; Ω) = L ∞ ∩ Ht2 Hxγ , 1 Hx ,t 4
for s − 1 − ε < γ < s − 1, with
f L ∞1
4 ,t
= sup t − 4 f (t) Hxs . 1
Hxs
t∈[0,T ]
References 1. T. Beale, The initial value problem for the Navier-Stokes equations with a free surface. Comm. Pure Appl. Math. 34, 359–392 (1981) 2. A. Castro, D. Córdoba, C. Fefferman, F. Gancedo, J. Gómez-Serrano, Splash Singularities for the Free Boundary Navier-Stokes Equations, arXiv: 1504.02775 3. D. Coutand, S. Shkoller, On the Splash Singularity for the Free-surface of a Navier-Stokes Fluid, arXiv: 1505.01929v1 4. E. Di Iorio, P. Marcati, S. Spirito, Splash Singularity for a Free-Boundary Incompressible Viscoelastic Fluid Model, (submitted) 5. M.E. Gurtin, An Introduction to Continuum Mechanics. Mathematics in Science and Engineering, vol. 158 (Academic Press, 1981)
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6. R. Keunings, On the high Weissenberg number problem. J. Non-Newtonian Fluid Mech. 20, 209–226 (1986) 7. F. Lin, C. Liu, P. Zhang, On hydrodynamics of viscoelastic fluids. Commun. Pure Appl. Math. 65, 1437–1471 (2005) 8. J.L. Lions, E. Magenes, Non-homogeneous Boundary Value Problems and Applications (Springer, 1972) 9. C. Liu, N.J. Walkington, An eulerian description of fluids containing viscohyperelastic particles. Arch. Rat. Mech. Ana. 159, 229252 (2001) 10. R.G. Owens, T.N. Phillips, Computational Rheology (Imperial College Press, 2002) 11. M. Renardy, Mathematical Analysis of Viscoelastic Flows, vol. 73 (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2000)
An Asymptotic Preserving Mixed Finite Element Method for Wave Propagation in Pipelines Herbert Egger and Thomas Kugler
Abstract We consider a parameter-dependent family of damped hyperbolic equations with interesting limit behavior: the system approaches steady states exponentially fast and for parameter to zero, the solutions converge to that of a parabolic limit problem. We establish sharp estimates and elaborate their dependence on the model parameters. For the numerical approximation, we then consider a mixed finite element method in space together with a Runge–Kutta method in time. Due to the variational and dissipative nature of this approximation, the limit behavior of the infinite-dimensional level is inherited almost automatically by the discrete problems. The resulting numerical method thus is asymptotic preserving in the parabolic limit and uniformly exponentially stable. These results are further shown to be independent of the discretization parameters. Numerical tests are presented for a simple model problem which illustrate that the derived estimates are sharp in general. Keywords Asymptotic preserving · Well-balanced · Parabolic limit Galerkin approximation · Damped wave equation · Exponential stability
1 Introduction Pipeline networks in gas or water supply systems are usually made up of rather long pipes and the time scales of interest are typically large as well. The propagation of pressure waves in such long pipes may then be described by a hyperbolic system ∂t p ε + ∂ x m ε = 0
(1)
ε ∂t m + ∂x p ε + am ε = 0
(2)
2
ε
H. Egger (B) · T. Kugler Technische Universität Darmstadt, Darmstadt, Germany e-mail: [email protected] T. Kugler e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_39
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together with appropriate initial and boundary conditions. Here, p ε corresponds to the pressure, m ε to the momentum or mass flux, and a is a generalized friction coefficient which encodes information about the pipe diameter and roughness. This ˜ of the physical space system can be derived by a parabolic rescaling t = t˜ε2 , x = xε and time variables x, ˜ t˜ from the Euler equations or the shallow water equations under some simplifying assumptions [1, 14] and ε can be assumed to be small. The parameter-dependent hyperbolic problem (1)–(2) has an interesting limit behavior for long time t → ∞ and in the parabolic limit ε → 0 which has been studied intensively in the literature [1, 10–12, 15, 16]. Many interesting results are available even for more general problems including the isentropic Euler equations with damping and rather general hyperbolic systems [3, 13]. In this note, we contribute to this active research field by establishing the following theoretical results: (R1)
For ε → 0, the solutions ( p ε , m ε ) of (1)–(2) converge to the solution ( p 0 , m 0 ) of the corresponding parabolic limit problem and p ε (t) − p 0 (t)2 +
t
m ε (s) − m 0 (s)2 ds ≤ Cε2
0
(R2)
with a constant C that is uniform in ε and independent of time t ≥ 0. Assume that the boundary values are kept constant. Then for any 0 ≤ ε ≤ 1, ¯ m) ¯ and the solutions ( p ε , m ε ) converge to the same steady state ( p, ¯ 2 + ε2 m ε (t) − m ¯ 2 ≤ Ce−γ t p ε (t) − p with constants C and γ > 0 that are independent of t ≥ 0 and ε.
Our proofs are based on careful energy estimates that explicitly take into account the dependence on the parameter ε. As a consequence, the results not only hold for single pipes but can be extended without much difficulty to pipeline networks. Due to the many important applications, the systematic approximation of parameter-dependent hyperbolic problems and, in particular, the preservation of asymptotic stability have been investigated intensively as well [2, 4, 7–9]. For the discretization of the model problem (1)–(2), we here consider a mixed finite element method in space combined with an implicit Runge–Kutta time-stepping scheme. The resulting method can be shown to exactly conserve mass and to be slightly dissipative in energy, thus capturing the relevant physical behavior [5]. In this paper, we additionally establish the following properties: (R3)
ε , m εh,τ ) converge The scheme is asymptotic preserving, i.e., the solutions ( ph,τ 0 with ε → 0 to the solution ( ph,τ , m 0h,τ ) of the parabolic limit problem, and ε ph,τ (t)
−
0 ph,τ (t)2
+ 0
t
m εh,τ (s) − m 0h,τ (s)2 ds ≤ Cε2
with C independent of ε and of the discretization parameters h and τ .
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(R4)
517
The method is uniformly exponentially stable, i.e., for constant boundary ε , m εh,τ ) converge toward steady state ( p¯ h , m¯ h ) and data the solutions ( ph,τ ε (t) − p¯ h 2 + ε2 m 2h,τ (t) − m¯ h 2 ≤ Ce−γ t ph,τ
with C and γ > 0 independent of ε and the discretization parameters h, τ . The numerical method is also well balanced in the sense that it automatically provides a stable approximation ( p¯ h , m¯ h ) for the corresponding stationary problem. Since the proposed discretization strategy is of variational and dissipative nature, the above assertions can be proven with only slight modification of the energy arguments used on the continuous level. In summary, we thus obtain uniformly stable and accurate approximations for the parameter-dependent problem (1)–(2) that capture all relevant physical and mathematical properties of the underlying system. The remainder of this note is organized as follows: In Sect. 2, we prove the assertions (R1) and (R2) for the case of a single pipe. Section 3 is then concerned with the numerical approximation and the proof of assertions (R3) and (R4) for a single pipe. In Sect. 4, we briefly indicate how the results can be generalized with minor modifications to pipe networks. In Sect. 5, we discuss in detail a specific test problem and present numerical results that illustrate the sharpness of our estimates and also indicate directions for possible improvements.
2 Analysis of a Single Pipe Let us start with describing in more detail the model problem under investigation. The pipe shall be represented by the unit interval, and we consider ∂t p ε (x, t) + ∂x m ε (x, t) = 0, ε2 ∂t m ε (x, t) + ∂x p ε (x, t) + a(x)m ε (x, t) = 0,
x ∈ (0, 1), t > 0 x ∈ (0, 1), t > 0.
(3) (4)
We assume that 0 < a ≤ a(x) ≤ a and that the pressure at the boundary is given by p ε (0, t) = g0 ,
p ε (x, t) = g1 ,
x ∈ {0, 1}, t > 0.
(5)
For ease of presentation, g0 , g1 are assumed to be independent of time here. Other boundary conditions could be considered with obvious modifications. From standard results of semigroup theory, one can easily deduce the following. Lemma 1. Let p0 , m 0 ∈ H 1 (0, 1) be given with p0 (0) = g0 and p1 (1) = g1 . Then for any ε > 0 problem (3)–(5) has a unique classical solution ( p, m) ∈ C 1 (R+ ; L 2 (0, 1) × L 2 (0, 1)) × C(R+ ; H 1 (0, 1) × H 1 (0, 1))
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satisfying initial conditions p ε (x, 0) = p0 (x) and m ε (x, 0) = m 0 (x) for all x ∈ (0, 1). The parabolic problem (3)–(5) with ε = 0 also has a unique solution p 0 ∈ C 1 (R+ ; L 2 (0, 1)) × C(R+ ; H 1 (0, 1)), m 0 ∈ C(R+ ; L 2 (0, 1)) satisfying the initial condition p 0 (x, 0) = p0 (x) for all x ∈ (0, 1). Note that only one single initial condition is required in the parabolic limit. By elementary arguments, one can verify that the corresponding stationary problem ¯ = 0, ∂x m(x) ∂x p(x) ¯ + a(x)m(x) ¯ = 0, p(0) ¯ = g0 ,
x ∈ (0, 1) x ∈ (0, 1)
p(1) ¯ = g1
(6) (7) (8)
is independent of ε and has a unique solution ( p, ¯ m) ¯ ∈ H 1 (0, 1) × H 1 (0, 1) as well. Using standard energy arguments and the linearity of the time dependent and of the stationary problem, one can then establish the following assertions. ¯ m) ¯ denote solutions of (3)–(5) and (6)–(8), respecLemma 2. Let ( p ε , m ε ) and ( p, tively. Then for any ε ≥ 0 and any t ≥ 0, there holds
¯ 2 + ε2 m ε (t) − m ¯ 2+2 p ε (t) − p
t
am ε (s) − m ¯ 2 ds
0
≤ p0 − p ¯ 2 + ε2 m 0 − m ¯ 2. For ε > 0, one can additionally bound the time derivatives of ( p ε , m ε ) by ∂t p ε (t)2 + ε2 ∂t m ε (t)2 + 2
t
a∂t m ε (s)2 ds
0
≤ ∂x m 0 2 +
1 ∂x p0 + am 0 2 . ε2
Here and below, · and (·, ·) denote the norm and the scalar product on L 2 (0, 1). In addition, the functions p ε , m ε are understood as functions of time with values in Hilbert spaces. The fact that the second estimate degenerates as ε → 0 resembles the fact that the second initial condition becomes superfluous in the parabolic limit. Proof. Due to linearity of the problem, we may assume without loss of generality that g0 = g1 = 0 and hence p¯ ≡ m¯ ≡ 0. From (3)–(4), we then get ε2 d 1 d p ε 2 + m ε 2 2 dt 2 dt = (∂t p ε , p ε ) + ε2 (∂t m ε , m ε ) = −(∂x m ε , p ε ) − (∂x p ε , m ε ) − (am ε , m ε ).
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519
Using integration by parts for the second term in the last line, the homogeneous boundary conditions for p ε , and the lower bound for the parameter a, we get d d p ε 2 + ε2 m ε 2 ≤ −2am ε 2 . dt dt The first estimate now follows by integration with respect to time. Next assume that ( p ε , m ε ) ∈ C 2 (R+ ; L 2 (0, 1) × L 2 (0, 1)). Then by formal differentiation of the problem, one can see that the time derivative (∂t p ε , ∂t m ε ) also solves (3)–(5) with homogeneous boundary conditions. The previous estimate thus yields ∂t p ε (t)2 + ε2 ∂t m ε (t)2 + 2
t
a∂t m ε (t)2
0
≤ ∂t p ε (0)2 + ε2 ∂t m ε (0)2 . The differential equations (3) and (4) can be used to replace the terms on the righthand side which proves the second estimate for the case of smooth solutions. The general case finally follows by a density argument. A combination of these energy estimates allows us to provide a precise formulation and to prove the first assertion about solutions of the continuous problem. Theorem 1. Let ε > 0 and let ( p ε , m ε ) and ( p 0 , m 0 ) denote the unique solutions of problem (3)–(5) with initial values p ε (0) = p 0 (0) = p 0 and m ε (0) = m 0 . Then
ε
p (t) − p (t) + 0
2
t
am ε (s) − m 0 (s)2 ds
0
≤
ε4 1 (∂x m 0 2 + 2 ∂x p0 + am 0 2 ). 2 2a ε
Proof. Let r ε = p ε − p 0 and wε = m ε − m 0 denote the differences between the solutions of the hyperbolic and the parabolic problem. Then by linearity of the equations, one can deduce that r ε = 0 at the boundary and that ∂t r ε + ∂x wε = 0, ∂x r ε + awε = −ε2 ∂t m ε . Applying similar arguments as in the proof of the previous lemma then leads to 1 d ε r (t)2 + awε (t)2 ≤ ε2 ∂t m ε (t)wε (t) 2 dt ε4 a ≤ ∂t m ε (t)2 + wε (t)2 . 2a 2
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Multiplication by two and integration with respect to time further yields ε
t
r (t) + 2
0
ε4 aw (s) ds ≤ r (0) + a ε
2
ε
t
2
∂t m ε (s)2 ds.
0
Since p ε and p 0 satisfy the same initial conditions, we have r ε (0) = 0, and the remaining integral on the right-hand side can be estimated by Lemma 2. The estimates of Lemma 2 provide uniform bounds for the distance to steady state. A refined analysis reveals that in fact, exponential convergence takes place. Theorem 2. Let ( p ε , m ε ) denote a solution of (3)–(5) for some 0 ≤ ε ≤ 1. Further let ( p, ¯ m) ¯ be the unique solution of the corresponding stationary problem. Then ¯ 2 + ε2 m ε (t) − m ¯ 2 ≤ Ce−γ (t−s) ( p ε (s) − p ¯ 2 + ε2 m ε (s) − m ¯ 2) p ε (t) − p which holds for all 0 ≤ s ≤ t and with some constants C, γ > 0 independent of ε. Proof. Set τ = t/ε and σ = s/ε and define π ε (τ ) = p ε (t) and με (τ ) = εm ε (t). Then by elementary calculations, one can see that ∂t π ε + ∂x με = 0 a ∂t με + ∂x π ε + με = 0. ε The exponential convergence for this problem has been established in [5], and a direct application of Theorem 3.3 in [5] yields ¯ 2 ≤ Ce−cε(τ −σ ) (π ε (σ ) − π¯ 2 + με (σ ) − μ ¯ 2 ). π ε (τ ) − π¯ 2 + με (τ ) − μ Using τ = t/ε and σ = s/ε and the definition of π ε and με then directly yields the estimate for ε > 0. The result for ε = 0 follows directly but also from the uniformity of those for ε > 0 and the convergence to the parabolic limit.
3 A Mixed Finite Element Runge–Kutta Scheme For the discretization of problem (3)–(5), we now consider a mixed finite element method in space and the implicit Euler method in time. More general Galerkin and time-integration schemes could be analyzed in a similar manner. Let Th = {e} denote a uniform mesh of the interval (0, 1) into elements e of size h and denote by Q h = {q ∈ L 2 (0, 1) : q|e ∈ P0 (e)} and Vh = {v ∈ C[0, 1] : v|e ∈ P1 (e)}
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the spaces of piecewise constant and piecewise linear and continuous functions, respectively. Furthermore, let τ > 0 be the time step size, define t k = kτ , and denote by ∂¯τ u(t k ) = τ1 [u(t k ) − u(t k−1 )] the backward difference quotient. We then consider ε Problem 1. Let ph,τ (0) and m εh,τ (0) be the L 2 projections of the initial data onto ε the finite element spaces. For k ≥ 1 find ( ph,τ (t k ), m εh,τ (t k )) ∈ Q h × Vh , such that ε (t k ), qh ) + (∂x m εh,τ (t k ), qh ) = 0 (∂¯τ ph,τ ε (t k ), ∂x vh ) + (am εh,τ (t k ), vh ) = g0 vh (0) − g1 vh (1) ε2 (∂¯τ m εh,τ (t k ), vh ) − ( ph,τ
holds for all test functions qh ∈ Q h and all vh ∈ Vh . Recall that (·, ·) denotes the scalar product of L 2 (0, 1). Existence of a unique disε , m εh,τ ) to Problem 1 and of a unique solution ( p¯ h , m¯ h ) of the crete solution ( ph,τ corresponding stationary problem can be deduced from the results in [5]. ε , m εh,τ ) and Lemma 3. For any ε ≥ 0, Problem 1 admits a unique solution ( ph,τ
ε (t k ) − p¯ h 2 + ε2 m εh,τ (t k ) − m¯ h 2 + 2a ph,τ
k
τ m εh,τ (t j ) − m¯ h 2
j=1
≤ p0 − p¯ h 2 + ε2 m 0 − m¯ h 2 for all k ≥ 0, where ( p¯ h , m¯ h ) ∈ Q h × Vh denotes the unique solution of the corresponding stationary problem. For ε > 0, we additionally have ε ∂¯τ ph,τ (t k )2 + ε2 ∂¯τ m εh,τ (t k )2 + 2a
k
τ ∂¯τ m εh,τ (t j )2
j=1
≤ C(∂x m 0 2 +
a2 1 2 ∂ p + am + m 0 2 ) x 0 0 ε2 ε2
with constant C that is independent of ε and the discretization parameters h and τ . Proof. Without loss of generality, we may set g0 = g1 = 0 and hence p¯ h ≡ m¯ h ≡ 0. ε (t k ) and m k := m εh,τ (t k ). Then by For ease of notation, let us abbreviate p k := ph,τ elementary calculations, one can verify that p k 2 + ε2 m k 2 + p k − p k−1 2 + ε2 m k − m k−1 2 = p k−1 2 + ε2 m k−1 2 + 2τ [(∂¯τ p k , p k ) + ε2 (∂¯τ m k , m k )]. Using the discrete problem and the lower bounds for the parameter, we thus obtain p k 2 + ε2 m k 2 ≤ p k−1 2 + ε2 m k−1 2 − 2aτ m k 2 .
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The first estimate now follows by recursion and by noting that p 0 ≤ p0 and m 0 ≤ m 0 , since the initial iterates were defined as L 2 orthogonal projections of the initial values onto the respective subspaces. By linearity of the problem, one can then deduce in a similar manner that ∂¯τ p k 2 + ε2 ∂¯τ m k 2 + 2a
k
τ ∂¯τ m j 2 ≤ ∂¯τ p 1 2 + ε2 ∂¯τ m 1 2 .
j=2
Using the discrete problem for k = 1, we further get τ (∂¯τ p 1 2 + ε2 ∂¯τ m 1 2 ) = −(∂x m 1 , p 1 − p 0 ) + ( p 1 , ∂x m 1 − ∂x m 0 ) − (am 1 , m 1 − m 0 ) ≤ −(m 1 − m 0 , ∂x p0 + am 0 ) − ( p 1 − p 0 , ∂x m 0 ) ≤ τ ∂¯τ m 1 ∂x p0 + am 0 + τ ∂¯τ p 1 ∂x m 0 . Using Young’s inequality, the bounds for the parameter a, and the stability of the L 2 projection in the H 1 norm, we may conclude that 1 ∂¯τ p 1 2 + ε2 ∂¯τ m 1 2 ≤ C ∂x m 0 2 + 2 (2∂x p0 + am 0 2 + 2a 2 m 0 − m 0 2 ), ε which together with the energy estimate from above completes the proof.
Similarly as on the continuous level, a combination of the previous estimates now ε immediately allows to show convergence of the solutions ( ph,τ , m εh,τ ) of the discrete hyperbolic problem to that of the discrete parabolic problem when ε → 0. ε 0 , m εh,τ ) and ( ph,τ , m 0h,τ ) denote solutions of Problem 1 for Theorem 3. Let ( ph,τ ε 0 ε > 0 and ε = 0, respectively. Further assume that ph,τ (0) = ph,τ (0). Then
ε (t k ) ph,τ
−
0 ph,τ (t k )2
+ 2a
k
τ m εh,τ (t j ) − m 0h,τ (t j )2
j=1
a2 1 ≤ Cε4 (∂x m 0 2 + 2 ∂x p0 + am 0 2 + 2 m 0 2 ) ε ε with constant C independent of ε and of the discretization parameters h and τ . ε 0 Proof. Define r k = ph,τ (t k ) − ph,τ (t k ) and wk = m εh,τ (t k ) − m 0h,τ (t k ). Then by linearity of the discrete problem, one can see that
(∂¯τ r k , qh ) + (∂x wk , qh ) = 0
−(r k , ∂x vh ) + (awk , vh ) = −ε2 (∂¯τ m εh,τ (t k ), vh )
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for all qh ∈ Q h and vh ∈ Vh and for all k ≥ 0. Testing with qh = wk and vh = m k and proceeding similarly as in the previous lemmas leads to the energy estimate r + 2a k 2
k
τ w ≤ r + ε k 2
0 2
2
j=1
k
τ ∂¯τ m εh,τ (t j )wk
j=1
≤ r 0 2 + a
k
τ wk 2 +
j=1
k ε4 ¯ ε τ ∂τ m h,τ (t j )2 . a j=1
The assertion now follows by noting that r 0 ≡ 0 and application of the second estimate of the previous lemma to estimate the last term in this expression. Similarly as on the continuous level, one can again prove uniform exponential convergence of discrete solutions to steady states. ε Theorem 4. Let ( ph,τ , m εh,τ ) denote a solution of Problem 1 and let ( p¯ h , m¯ h ) be the unique solution of the corresponding stationary problem. Then ε (t k ) − p¯ h 2 +ε2 m εh,τ (t k ) − m¯ h 2 ph,τ ε ≤ Ce−γ (k− j)τ ph,τ (t j ) − p¯ h 2 + ε2 m εh,τ (t j ) − m¯ h 2
for all 0 ≤ j ≤ k with constants C, γ > 0 that are independent of ε, h, and τ . Proof. Using a rescaling like in the proof of Theorem 2, the result for ε > 0 can be deduced directly from Theorem 7.4 in [5]. The estimate for ε = 0 follows from the uniformity of the estimates and convergence to the parabolic limit.
4 Extension to Pipe Networks The results of the previous sections can be extended to the following class of hyperbolic problems on networks: Let G = (V , E ) be a finite directed graph representing the topology of the network. On every single pipe e, the dynamics shall again be described by the linear damped hyperbolic system
ε
2
∂t m εe
∂t peε + ∂x m εe = 0 + ∂x peε + ae m εe = 0.
(9) (10)
At any junction v of several pipes e ∈ E (v) of the network, we require that
n e (v)m εe (v) = 0
(11)
e∈E (v)
peε (v) = pv
∀e ∈ E (v).
(12)
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Here, n e (v) takes the value minus or plus one, depending on whether the pipe e start or ends at the junction v. At the boundary vertices v of the network, we require peε (v) = gv .
(13)
Using the arguments developed in [6], all results stated in Theorems 1–4 hold verbatim also for the system (9)–(13). Details are left to the interested reader.
5 Numerical Validation We now illustrate our theoretical results by considering in detail a particular model problem. For constant damping parameter a ≡ 1, initial data p0 = sin(π x), m 0 ≡ 0, and boundary values g0 = g1 ≡ 0, the solution of problem (3)–(5) is given by
2π 2 ε2 1 1 p (x, t) = exp − 2 (1 − s(ε))t 1 − s(ε) s(ε) 2ε 2 2 2π ε 1 1 − exp − 2 (1 + s(ε))t sin(π x) 1 + s(ε) s(ε) 2ε ε
and 1 π exp − 2 (1 − s(ε))t m (x, t) = s(ε) 2ε 1 π exp − 2 (1 + s(ε))t cos(π x). − s(ε) 2ε ε
with parameter s(ε) =
√ 1 − 4π 2 ε2 . By Taylor expansion w.r.t. ε, we deduce that
p ε (x, t) = (1 + O(ε2 )) exp (−π 2 − O(ε2 ))t 1 sin(π x) − O(ε2 ) exp (− 2 + O(1))t ε and m ε (x, t) = (π + O(ε2 )) exp (−π 2 − O(ε2 ))t 1 cos(π x). − (π + O(ε2 )) exp (− 2 + O(1))t) ε
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For ε = 0, we simply obtain p 0 (x, t) = e−π t sin(π x) and m 0 (x, t) = π e−π t cos(π x) and the steady state for this problem is given by p, ¯ m¯ ≡ 0. From the explicit solution formulas, one can then immediately see that exponential convergence toward the steady state takes place with t → ∞ for all 0 ≤ ε ≤ 1 with a rate that is independent of ε which was the assertion of Theorem 2. In Table 1, we depict numerical results obtained with the numerical scheme discussed in Sect. 3. As predicted by Theorem 4, the exponential convergence toward steady state with t → ∞ is uniform in ε also for the discrete schemes. Mesh independence of the exponential decay rate was already demonstrated in [6]. Let us next have a closer look at the convergence to the parabolic limit. Using the analytical solution formulas and Taylor expansion w.r.t. ε, one can deduce that 2
2
p ε − p 0 = O(ε2 )(t + 1) exp −π 2 t + O(1)t 1 sin(π x) + O(ε2 ) exp − 2 t + O(1)t ε and 2 m − m = O(ε )(t + 1) exp −π t + O(1)t 1 2 cos(π x). − (π + O(ε )) exp − 2 t + O(1)t ε ε
0
2
t This shows that p ε − p 0 2 = O(ε4 ) and 0 m ε − m 0 2 = O(ε2 ) which yields exactly the asymptotic behavior predicted in Theorem 1. In Table 2, we display the corresponding results obtained with the proposed discretization scheme. Also here, we can exactly observe the convergence rate predicted by Theorem 3. Note that the second term in the error measure is strictly increasing w.r.t. time, which together with the exponential convergence to steady states explains that the error is almost independent of t here. ε (t) − p¯ 2 + ε 2 m ε − m Table 1 Distance ph,τ ¯ h 2 ≤ Ce−γ t of the numerical solution to the h h,τ discrete steady state for different values of ε and times t = 0, 0.1, 0.5, 1.0 and estimated exponential convergence rate γ . Discretization parameters were set to h = 0.01 and τ = 10−5 t\ε 1/4 1/8 1/16 1/32 1/64 1/128
0.0 0.1 0.5 1.0 γ
5.00e-01 2.72e-01 3.56e-04 8.51e-08 15.59
5.00e-01 9.09e-02 5.35e-06 2.71e-11 23.64
5.00e-01 7.26e-02 1.94e-05 6.64e-10 20.44
5.00e-01 7.02e-02 2.42e-05 1.13e-09 19.90
5.00e-01 6.96e-02 2.54e-05 1.28e-09 19.78
5.00e-01 6.95e-02 2.57e-05 1.32e-09 19.75
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ε (t k ) − p 0 (t k )2 + k ε 0 j j 2 α Table 2 Error ph,τ j=1 am h,τ (t ) − m h,τ (t ) = O(ε ) between the h,τ discrete approximations for the hyperbolic problem and the parabolic limit problem for different values of ε and time steps t k and observed convergence rate α. Discretization with h = 0.01 and τ = 10−5 t k \ε 1/4 1/8 1/16 1/32 1/64 1/128 α
0.1 0.5 1.0
9.81e-02 1.18e-01 1.18e-01
3.47e-02 3.58e-02 3.58e-02
9.41e-03 9.44e-03 9.44e-03
2.38e-03 2.39e-03 2.39e-03
5.89e-04 5.89e-04 5.89e-04
1.39e-04 1.39e-04 1.39e-04
1.87 1.93 1.93
ε (t k ) − p 0 (t k )2 + k ε 0 j j 2 α Table 3 Error ph,τ j=1 am h,τ (t ) − m h,τ (t ) = O(ε ) between the h,τ discrete approximations for the hyperbolic problem and the parabolic limit problem for time t k = 1 and different values of ε and the discretization parameters h and τ ε 1/4 1/8 1/16 1/32 1/64 1/128 α
h h h h
= 0.010, τ = 0.002, τ = 0.010, τ = 0.002, τ
= 10−5 = 10−5 = 10−6 = 10−6
1.18e-01 1.18e-01 1.18e-01 1.18e-01
3.58e-02 3.58e-02 3.58e-02 3.58e-02
9.44e-03 9.44e-03 9.46e-03 9.46e-03
2.39e-03 2.39e-03 2.40e-03 2.40e-03
5.89e-04 5.89e-04 6.00e-04 6.00e-04
1.39e-04 1.39e-04 1.49e-04 1.49e-04
1.93 1.99 1.90 1.90
In Table 3, we report about further numerical tests to illustrate the independence of the results on the discretization parameters. Again, the observations are in perfect agreement with the theoretical predictions made in Theorem 3. Let us finally note that the previous formulas reveal that the error between the solutions of the hyperbolic and the parabolic problem actually behaves like p ε (t) − p 0 (t)2 + u ε (t) − u 0 (t)2 = O(ε4 )
for t ε.
This shows that the estimate of Theorem 3 is dominated by the error in the mass flux within the initial layer 0 ≤ t ε which again resembles the fact that the second initial condition gets superfluous in the parabolic limit. This behavior can also be observed for the numerical approximations obtained with the method discussed in Sect. 3. A theoretical explanation of this fact would require a refined analysis which is left for future research. Acknowledgements The authors would like to gratefully acknowledge financial support by the German Research Foundation (DFG) via grants IRTG 1529, GSC 233, and TRR 154.
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References 1. J. Brouwer, I. Gasser, M. Herty, Gas pipeline models revisited: Model hierarchies, nonisothermal models and simulations of networks. Multiscale Model. Simul. 9, 601–623 (2011) 2. C. Buet, B. Després, E. Franck, Design of asymptotic preserving finite volume schemes for the hyperbolic heat equation on unstructured meshes. Numer. Math. 122, 227–278 (2012) 3. C.M. Dafermos, R.H. Pan, Global BV solutions for the p-system with frictional damping. SIAM J. Math. Anal. 41, 1190–1205 (2009) 4. A. Duran, F. Marche, R. Turpault, C. Berthon, Asymptotic preserving scheme for the shallow water equations with source terms on unstructured meshes. J. Comput. Phys. 287, 184–206 (2015) 5. H. Egger, T. Kugler, Uniform exponential stability of Galerkin approximations for damped wave systems. arXiv:1511.08341 (2015) 6. H. Egger, T. Kugler, Damped wave systems on networks: Exponential stability and uniform approximations. Numer. Math. 138, 839–867 (2018) 7. S. Ervedoza, E. Zuazua, Uniformly exponentially stable approximations for a class of damped systems. J. Math. Pures Appl. 91, 20–48 (2009) 8. T. Gallouët, J.-M. Hérard, O. Hurisse, A.-Y. LeRoux, Well balanced schemes versus fractional step method for hyperbolic systems with source terms. Calcolo 43, 217–251 (2006) 9. L. Gosse, G. Toscani, An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations. Comptes Rendus Mathematique 334, 337–342 (2002) 10. L. Hsiao, T.P. Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping. Commun. Math. Phys. 143, 599–605 (1992) 11. S. Gatti, V. Pata, A one-dimensional wave equation with nonlinear damping. Glasgow Math. J. 48, 419–430 (2000) 12. J. Lopez-Gomez, On the linear damped wave equation. J. Diff. Equ. 134, 26–45 (1997) 13. P. Marcati, B. Rubino, Hyperbolic to parabolic relaxation theory for quasilinear first order systems. J. Diff. Equ. 162, 359–399 (2000) 14. A.J. Osiadacz, Different transient models—limitations, advantages, and disadvantages. Technical Report 9606, PSIG, Pipeline Interest Group (1996) 15. J. Rauch, M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains. Ind. Univ. Math. J. 24, 79–86 (1974) 16. E. Zuazua, Stability and decay for a class of nonlinear hyperbolic problems. Asymptotic Anal. 1, 161–185 (1988)
Non-existence of Irrotational Flow Around Solids with Protruding Corners Volker Elling
Abstract We motivate and discuss several recent results on non-existence of irrotational inviscid flow around bounded solids that have two or more protruding corners, complementing classical results for the case of a single protruding corner. For a class of two-corner bodies including non-horizontal flat plates, compressible subsonic flows do not exist. Regarding three or more corners, bounded simple polygons do not admit compressible flows with arbitrarily small Mach number, and any incompressible flow has unbounded velocity at at least one corner. Finally, irrotational flow around smooth protruding corners with non-vanishing velocity at infinity does not exist. This can be considered vorticity generation by a slip-condition solid in absence of viscosity. Keywords Kutta–Joukowski · Lift · Protruding corner MSC2010 76G25 · 76N99
1 Equations Consider inviscid flow in the 2d plane around a solid body whose boundary is smooth except for one or more corners (see Fig. 1). This setup is an old and important problem in fluid dynamics. The case of a single corner has received particular attention, since it idealizes cross sections of aircraft wings; the corresponding Kutta–Joukowski theory is the basis for the understanding of lift, the upward force on a horizontally moving body. To explain the problem of corners, we first recall some properties of compressible flow: 0 = ∂t ρ + ∇ · (ρv) , 0 = ∂t v + v · ∇v + ∇ p
(1)
V. Elling (B) Department of Mathematics, University of Michigan, 530 Church St, Ann Arbor, MI 48109, USA e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_40
529
530
V. Elling receding corner
v∞
v∞
Protruding corner
lift
Ω
drag
Fig. 1 Left: flow around a body that is smooth except for one protruding corner; right: flow around a polygon
where v is velocity, ρ is density, and p = p(ρ) ˆ is defined up to an additive constant by pˆ ρ = ρ −1 Pˆρ
(2)
ˆ where P = P(ρ) is pressure. We consider the polytropic pressure law ˆ P = P(ρ) = ργ
(3)
with isentropic coefficient γ > 1, so that p(ρ) ˆ =
γ ρ γ −1 . γ −1
(4)
Linearizing (1) around a v = 0 and ρ = ρ = const > 0 background yields 0 = ∂t ρ + ρ ∇ · v , 0 = ∂t v + ρ −1 Pˆρ (ρ)∇ρ.
(5)
∂t of the first minus ρ ∇· of the second equation yields 0 = ∂t2 ρ − Pˆρ (ρ)Δρ
(6)
which is the linear wave equation; this motivates defining the speed of sound c=
Pˆρ (ρ).
(7)
For steady flow 0 = ∇ · (ρv),
(8)
0 = v · ∇v + ∇ p,
(9)
the dot product of (9) with v yields 1 0 = v · ∇v · v + v · ∇ p = v · ∇( |v|2 + p). 2
(10)
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531
Hence, the Bernoulli constant B=
1 2 |v| + p(ρ) ˆ 2
(11)
is constant along streamlines, i.e. integral curves of v, this is, the Bernoulli relation. On each streamline, we can solve for 1 ρ = pˆ −1 (B − |v|2 ). 2
(12)
pˆ −1 is generally undefined for negative arguments due to fractional exponents (see (4)), in particular when γ = 53 (helium and other noble gases) or γ = 75 (air). Hence, √ |v| may not exceed the limit speed v∗ = 2B; at the limit speed, the density reaches 0. There is no meaningful way to extend the model to higher speeds, as becomes clear by putting the observation into a more physical form: groups of gas particles cannot acquire arbitrarily high speed by moving to regions of increasingly lower pressure. (A related observation: gas inside a piston expanding to near-vacuum cannot perform unbounded mechanical work, or put differently, it takes only a finite amount of energy to compress gas from near-vacuum to a given density. These and other observations do not hold for some other pressure laws, which is in part why those should be considered ‘exotic’.) There are important reasons to consider flow with non-zero vorticity ω = ∇ × v, which we are going to motivate by assuming irrotationality ω = 0 and exploring the consequences. Now, (9) can be rewritten 1 (13) 0 = ∇v · v + ∇ p = ∇( |v|2 + p) 2 so that B and v∗ are global constants, same on all streamlines. Equation (8) yields ρv = −∇ ⊥ ψ
(14)
for a scalar function ψ called stream function. The Bernoulli relation takes the form ˆ = F(ρ, |∇ψ|) B = 21 ρ −2 |∇ψ|2 + p(ρ)
(15)
We may use the implicit function theorem to solve for ρ as long as Fρ = −ρ −3 |∇ψ|2 + pˆ ρ (ρ) = ρ −3 (c2 − |v|2 )
(16)
is non-zero; it is positive as long as |v| < c (and hence ρ > 0), i.e. for subsonic flow. Hence, we can solve 1 (17) ρ −1 = τˆ ( |∇ψ|2 ) 2 where τˆ is defined on some maximal interval [0, μ].
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Having solved the mass and Bernoulli equations it remains to ensure irrotationality (which is needed to recover the original velocity equation (9) from the Bernoulli equation): 0=∇ ×v =∇ ×
|∇ψ|2 −∇ ⊥ ψ = −∇ · τˆ ( )∇ψ ρ 2
After differentiation, we have vx vx v y vy 0 = 1 − ( )2 ψ x x − 2 ψx y + 1 − ( )2 ψ yy c c c c
(18)
(19)
where c is a function of ρ, hence ∇ψ. The eigenvectors of the coefficient matrix I − (v/c)2 are v and v⊥ , with eigenvalues 1 − M 2 and 1 where M := |v|/c
(20)
is the Mach number. Hence, (19) is elliptic exactly wherever it is subsonic. The limit of decreasing Mach numbers formally (and under various assumptions provably) yields steady irrotational incompressible flow: 0 = Δψ.
(21)
2d harmonic functions are more conveniently represented as holomorphic maps: consider the complex velocity w := v x − iv y
(22)
as a function of z := x + i y. Then ∂z w =
1 (∇ · v − i∇ × v). 2
(23)
Hence, w represents an incompressible and irrotational flow if and only ifw is holoz morphic. If so, it is convenient to use the complex velocity potential Φ = w dz = φ + iψ (which may be multivalued). The Cauchy–Riemann equations ∂z Φ = 0, together with ∂z Φ=w=v x − iv y , yield v= − ∇ ⊥ ψ = ∇φ, justifying the notation. At solid boundaries, we use the standard slip condition 0 = n · v = s · ∇ψ ,
(24)
where n, s are normal and tangent to the solid. Integration along connected components of (say) a piecewise C 1 boundary yields ψ = const ; if the solid boundary has a single connected component, then we may add an arbitrary constant to ψ without changing v = −∇ ⊥ ψ to obtain the convenient zero Dirichlet condition ψ = 0.
(25)
Non-existence of Irrotational Flow Around Solids with Protruding Corners
533
2 Protruding Corners Consider a neighbourhood of a corner of a 2d solid body, as in Fig. 1. For simplicity assume the two sides are locally straight, at angles 0 and Θ ∈ / {0, π, 2π }, with fluid in the sector from 0 to Θ counterclockwise. Whatever the global problem, in many cases, it is natural to seek existence of an incompressible flow by Hilbert space method, typically yielding an ψ that is locally H01 , i.e. velocity ∇ψ square-integrable near the corner with ψ = 0 on the two sides. A coordinate change to ζ = z π/Θ maps to a small half-ball; using ψ = 0 on the straight side Schwarz reflection yields a harmonic function in a small ball, with local Taylor expansion ψ = Im
∞
ak ζ k = Im
k=1
∞ k=1
ak z kπ/Θ
=Φ(z)
(26)
with real ak , so that w(z) = Φ (z) =
∞
kπ π a1 z π/Θ−1 ak z kπ/Θ−1 . +
Θ Θ k=2
(27)
!
Here, we may observe a key distinction: if Θ < π , i.e. for receding corners, the exponent π/Θ − 1 is positive so a square-integrable velocity is always bounded in the corner. This is not true at protruding corners (Θ > π ) unless a single real scalar constraint is satisfied: a1 = 0. If so, then inspection of the remaining terms shows that a non-zero ψ must attain both negative and positive sign arbitrarily close to the corner. (For nonstraight but sufficiently regular sides analogous results can be obtained.) Whereas unbounded velocity is merely undesirable for incompressible flows, it is mathematically impossible in the compressible problem (19). The latter is quasilinear; when linearizing the operator about v = 0 and ρ = const > 0 we obtain the incompressible operator Δ. For solutions of the linearized problem to be helpful in solving or understanding the non-linear one, we need to require bounded velocity. Hence, every additional protruding corner adds another scalar real constraint to the problem. To satisfy these constraints, a corresponding number of free real parameters may be needed! The velocity w is holomorphic as well as bounded at infinity, so a Laurent expansion yields w(z) = c0 + c1 z −1 + c2 z −2 + ...
(28)
for constants ci . w → c0 =: w∞ at infinity (everywhere, which is a consequence of ω = 0; rotational flows can be far more complex). By rotational symmetry, we may assume w∞ is real positive; it is usually a chosen parameter, such as aircraft speed.
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Re c1 must be zero because it is the coefficient of a velocity term x/|x|2 which would otherwise cause net mass flux through a sufficiently large circle, in conflict with conservation of mass since the slip condition implies zero mass flux through the solid boundary. Hence, c1 = Γ /(2πi) with real Γ =
Σ
v · dx
(29)
which is called circulation; Σ is an arbitrary contour passing once counterclockwise around the circle (due to ω = 0, Γ is independent of the choice of Σ, which may be the boundary of the body, or very large (‘around infinity’), or anything in between). Integrating w in (28) yields ψ = C + w∞ y −
Γ 2π
log r + O(z −1 )
(30)
˜ Their with C some integration constant. Consider two distinct stream functions ψ, ψ. ˜ difference d = ψ − ψ satisfies Δd = 0, so
d ∇d · n ds. dΔd dx = |∇d|2 dx − + 0=− |x| 0} and {ψ < 0} are connected: near infinity both are connected, approximately half-spaces, since ψ = w∞ y + o(|z|) with
Fig. 3 Utility graph argument: intersection is unavoidable
xP
ψ >0
x1 B
x2 xB
x3
ψ 0 as |z| → ∞; neither can have a bounded connected component, by the strong maximum principle. After choosing suitable edges through each set we would have a planar embedding of the utility graph. The contradiction shows that any incompressible flow must have unbounded velocity at all but at most two of the protruding corners. The proof suggests that the non-existence is quite topological in nature. Simple polygons or other bodies with three or more protruding corners probably do allow irrotational flow in other settings, for example, if we restrict infinity to three ducts meeting in a junction in which the body is located. Such settings also have more free parameters to permit satisfying constraints at additional protruding corners.
4 Smooth Infinite Angles In [6], we instead consider an unbounded solid: an infinite protruding smoothened or sharp corner (see Fig. 4 left). It is natural to look for solutions whose velocity is bounded and converges near the upstream wall (graph θ1 ) to a prescribed non-zero constant at spatial infinity. Indeed for supersonic velocity, many shapes have wellknown solutions based on simple waves [3, Sect. 111]. However, we prove that there do not exist any uniformly subsonic irrotational flows of this type. If we do permit rotation, then some of the same shapes do allow easily constructed solutions with straight vortex sheet separating from the wall (Fig. 4 right). In cases where irrotational inviscid models have no solution, what models may be more appropriate? Physical observations suggest to permit rotation, with vorticity generated at some of the protruding corners, for example, vortex sheets separated by regions of irrotational flow [5]. (Another option is to consider transonic flows, which generally feature curved shock waves that—again—produce vorticity.) Vorticity is also needed for resolving part of the d’Alembert paradox [4, 9], namely, that irrotational inviscid models predict zero drag for subsonic flows. Although zero is a fair v∞
r graph θ1
Vortex sheet
slip boundary
Upstream wall
graph θ0
slip
v=0
boundary
Fig. 4 Left: Ω covers an angle Θ at infinity, with variable but smooth boundaries. This domain, for P(ρ) = ρ γ with γ > 1, does not allow compressible uniformly subsonic flows whose velocity is bounded but non-vanishing at infinity. Right: if vorticity is allowed, then there are trivial solutions with v = v∞ = const = 0 above the vortex sheet, in particular near infinity at the upstream wall, whereas v = 0 below the sheet
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approximation for some shapes that are observed to have rather low drag relative to their size, many other shapes do have significant drag even at small viscosity. Historically, there has been a lot of debate about how solid boundaries can generate vorticity. It is frequently stated that the explanation requires viscosity. On the one hand, this is correct in the sense that physical observations—as well as Prandtl’s theory [11]—confirm the essential role of thin viscous boundary layers and their instabilities. On the other hand, our results could be interpreted as saying that viscosity is not needed after all, in the sense that even in complete absence of viscosity some shapes and Mach numbers do not allow flows with zero vorticity. Acknowledgements This material is based upon work partially supported by the National Science Foundation under Grant No. NSF DMS-1054115 and by Taiwan MOST grant 105-2115-M-001007-MY3.
References 1. G.K. Batchelor, An Introduction to Fluid Dynamics (Cambridge Mathematical Library, 1967) 2. L. Bers, Existence and uniqueness of a subsonic flow past a given profile. Commun. Pure. Appl. Math. 7, 441–504 (1954) 3. R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves (Interscience Publishers, 1948) 4. J. le Rond d’Alembert, Essai d’une nouvelle théorie de la résistance des fluides (1752) 5. V. Elling, Compressible vortex sheets separating from solid boundaries. Discr. Cont. Dyn. Sys. (ser. A) (To appear) 6. V. Elling, Non-existence of subsonic and incompressible flows in non-straight infinite angles (Submitted) 7. V. Elling, Nonexistence of low-mach irrotational inviscid flows around polygons. J. Differ. Equ. (To appear) 8. V. Elling, Subsonic flow around bodies with two protruding corners including flat plates (Submitted) 9. R. Finn, D. Gilbarg, Asymptotic behaviour and uniqueness of plane subsonic flows. Commun. Pure Appl. Math. 10, 23–63 (1957) 10. F.I. Frankl, M. Keldysh, Die äussere Neumann’sche Aufgabe für nichtlineare elliptische Differentialgleichungen mit Anwendung auf die Theorie der Flügel im kompressiblen Gas. Bull. Acad. Sci. URSS 12, 561–607 (1934) 11. L. Prandtl, Über Flüssigkeitsbewegung bei sehr kleiner Reibung (In Verhandlungen des III. Internationalen Mathematiker-Kongresses, Heidelberg, 1904) 12. M. Shiffman, On the existence of subsonic flows of a compressible fluid. J. Rat. Mech. Anal. 1, 605–652 (1952) 13. C. Thomassen, The Jordan-Schoenflies theorem and the classification of surfaces. Amer. Math. Monthly 99(2), 116–131 (1992)
A Splitting Approach for Freezing Waves Robin Flohr and Jens Rottmann-Matthes
Abstract We present a numerical method which is able to approximate traveling waves (e.g., viscous profiles) in systems with hyperbolic and parabolic parts by a direct longtime forward simulation. A difficulty with longtime simulations of traveling waves is that the solution leaves any bounded computational domain in finite time. To handle this problem, one should go into a suitable co-moving frame. Since the velocity of the wave is typically unknown, we use the method of freezing (Beyn and Thümmler J Appl Dyn Syst 3:85–116, 2004, [2]), see also (Beyn et al. Current challenges in stability issues for numerical differential equations, 2014, [1]), which transforms the original partial differential equation (PDE) into a partial differential algebraic equation (PDAE) and calculates a suitable co-moving frame on the fly. The efficient numerical approximation of this freezing PDAE is a challenging problem, and we introduce a new numerical discretization which is suitable for problems that consist of hyperbolic conservation laws which are (nonlinearly) coupled to parabolic equations. The idea of our scheme is to use the operator splitting approach. The benefit of splitting methods in this context lies in the possibility to solve hyperbolic and parabolic parts with different numerical algorithms. We test our method at the (viscous) Burgers’ equation. Numerical experiments show linear and quadratic convergence rates for the approximation of the numerical steady state obtained by a longtime simulation for Lie and Strang splitting, respectively. Due to these affirmative results, we expect our scheme to be suitable also for freezing waves in hyperbolic–parabolic coupled PDEs. Keywords Traveling waves · Burgers’ equation · Operator splitting · Freezing method · Central scheme · Hyperbolic-parabolic partial differential algebraic equations
R. Flohr (B) · J. Rottmann-Matthes Institute for Analysis, Karlsruhe Institute of Technology, Englerstraße 2, 76131 Karlsruhe, Germany e-mail: [email protected] J. Rottmann-Matthes e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_41
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1 Introduction Many partial differential equations from applications consist of different parts, u t = Au x x + f (u)x + g(u) =: F(u).
(1)
Sometimes, one part is parabolic while another part is hyperbolic, and these parts are nonlinearly coupled. Examples of such hyperbolic–parabolic coupled PDEs are hyperbolic models of chemosensitive movement or reaction–diffusion equations for which not all components diffuse. One is often interested in special solutions, which arise as (time-)asymptotic limits of solutions to the Cauchy problem for (1). An important class of such solutions is traveling waves. They describe how mass (or information) travels through the domain. From this interpretation, it is obvious that one is often interested not only in the shape but also in the velocity of the traveling wave. Traveling waves are solutions of the PDE (1) of the form u(x, t) = v¯ (x − μt), ¯ x ∈ R, t ∈ R, where v¯ : R → R is the nonconstant profile and μ¯ ∈ R the velocity of the wave. For Burgers’ equation, there is a family of traveling wave solutions, ¯ u(x, t) = ϕ(x − μt) ¯ + 21 (b + c) = v¯ (x − μt), 1 − eax ϕ(x) = a , a = 21 (b − c), μ¯ = 21 (b + c), 1 + eax
(2)
parametrized by the asymptotic states lim x→−∞ u(x, t) = b > c = lim x→∞ u(x, t). As a toy example, we consider the Cauchy problem for the viscous Burgers’ equation (3) u t + ( 21 u 2 )x = u x x on R × [0, ∞), u(·, 0) = u 0 on R. For the numerical approximation of (3), one has to truncate the unbounded spatial domain to a finite interval. This leads to the problem that every traveling wave solution with nonzero speed eventually leaves the computational domain. The simplest remedy is to use periodic boundary conditions on a very large domain, but this is only reasonable in the case of pulses. Instead, we use the freezing method from [2]. The idea is to move the spatial frame with the speed of the traveling wave. We make the ansatz that the solution of (1) is of the form u(x, t) = v x − γ (t), t ,
γ (t) ∈ R,
(4)
where γ (t) denotes a time-dependent position. Then, μ(t) := γt (t) can be interpreted as the velocity of the wave at time t. Plugging (4) into (1) yields vt = F(v) + μ(t)vx ,
(5)
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where both v and μ are unknown. Due to the additional unknown μ one has to complement (5) with an addition algebraic equation, called phase condition in [2], to retain well-posedness. In Burgers’ case, this transforms (3) into the PDAE ⎧ 1 2 ⎪ ⎨ vt = vx x − ( 2 v )x + μvx , 0 = Ψ (v, μ), ⎪ ⎩ γt = μ(t),
v(·, 0) = u 0 , γ (0) = 0.
(6)
This is called the freezing method in [2]. We restrict to two standard choices for the phase condition, the orthogonal phase condition given by 0 = Ψ (v, μ) := vt | vx L 2 = vx x − ( 21 v2 )x + μvx | vx L 2
(7)
and the fixed phase condition given by 0 = Ψ (v, μ) := v − vˆ | vˆ x L 2
(8)
with vˆ an appropriately chosen reference function. For the numerical approximation of (6), we use splitting methods, which we briefly recall for convenience. Assume that a solution to an initial value problem of the form (9) u t = A(u) + B(u) is sought. Let Φ tA (u 0 ) and Φ Bt (u 0 ) denote the solution operators for u t = A(u) and u t = B(u) with initial value u 0 , respectively. The Lie–Trotter splitting, n u n+1 = Φ BΔ t ◦ Φ Δ A t (u ),
(10)
typically converges linearly to the exact solution for Δt → 0. A splitting method that typically leads to second-order convergence is Strang splitting, Δt/2
u n+1 = Φ A
Δt/2
◦ Φ BΔ t ◦ Φ A
(u n ).
(11)
For example, in [4], the authors show that this scheme is second-order convergent for the viscous Burgers’ PDE. To apply this approach to the freezing PDAE (6), we split the equation into two parts to separate the hyperbolic and parabolic problem. Then, we solve each part with a method which is particularly adapted to the respective subproblem. Namely, we solve the hyperbolic problem with an explicit scheme from Kurganov and Tadmor [5]. The parabolic subproblem is solved by an implicit second-order finite-difference approximation, due to the restrictive CFL condition. Our main focus in this article is on approximating the limits of the time evolution and, unlike [4], not on the finite time convergence properties of the scheme. In particular, we aim to understand the preservation of steady states and their stability by our schemes.
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In the case of ordinary differential equations, there is a well-established theory for numerical steady states. For example, in [8], there are results which state that one-step methods preserve fixed points and their stability in a Δt r -shrinking neighborhood under Lipschitz assumptions, where r denotes the order of consistency of the one-step method. An analogous result holds for the Strang splitting: Theorem 1. ([3]) Let A, B ∈ C 3 (Rm , Rm ) and assume that uˆ is a hyperbolic fixed point of (9). Let ϕ A , ϕ B be one-step methods approximating Φ A , Φ B , respectively. If ϕ A , ϕb are second-order Runge–Kutta methods, then there exist Δt0 , K > 0, such Δt/2 Δt/2 ◦ ϕ Δ t A ◦ ϕ B (U n ), that the numerical Strang splitting, U n+1 = ϕ Δt (U n ) = ϕ B 2 has a fixed point Uˆ which is unique in the ball B(u; ˆ K Δt ) for all 0 < Δt ≤ Δt0 . Furthermore, Uˆ is a stable (resp. unstable) fixed point of ϕ Δ t if uˆ is a stable (resp. unstable) steady state of (9). For the freezing method, there are several results available, where it is shown that the (continuous) method provides good approximations including preservation of asymptotic stability of traveling waves for certain problem classes in the continuous and semi-discrete case, see [1, 6, 9]. But the time-asymptotic behavior of a discretization with a splitting approach was never considered before. A different approach to apply adapted schemes for different parts of the freezing PDAE appears in [7], where the freezing method is used to capture similarity solutions of the multidimensional Burgers’ equation. There an IMEX-Runge–Kutta approach is used and second-order convergence for the time-dependent problem is shown on finite time intervals.
2 The Splitting Scheme We now explicitly state our numerical scheme. We split (6) into two subproblems as follows: Let Φ tA : (z 0 , γ0 , μ0 ) → z(t), γ (t), μ(t) be the solution operator to the parabolic problem ⎧ z(·, 0) = z 0 , ⎪ ⎨ zt = z x x , ⎪ ⎩
γt = 0, μt = 0,
γ (0) = γ0 , μ(0) = μ0 ,
(A)
let Φ Bt : (w0 , γ0 , μ0 ) → w(t), γ (t), μ(t) be the solution operator to ⎧ 1 2 ⎪ ⎨ wt = −( 2 w )x + μwx , 0 = Ψ (w, μ), ⎪ ⎩ γt = μ(t),
w(·, 0) = w0 , γ (0) = γ0 .
(B)
Here, Ψ is one of the phase conditions (7) or (8). Note that the initial value μ0 is ignored for this operator (B), because it is uniquely determined by the constraints. Since the splitting approach now iterates both solution operators consecutively, the question when and how to solve the algebraic constraint arises. For the orthogonal
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Fig. 1 Diagram of the Lie–Trotter splitting on the left and the Strang splitting on the right
phase condition we chose an explicit and for the fixed phase condition we use a half-explicit approach. Thus, we calculate the speed μ prior to solving the nonlinear PDE, and the μwx part is then discretized by using finite differences. Lie and Strang splitting are illustrated by classical diagrams in Fig. 1. A step in the vertical direction in Fig. 1 amounts in numerically solving the Cauchy problem for the heat equation (A), whereas a step in the horizontal direction amounts to solve the hyperbolic PDAE (B). Only states on the dashed diagonal line might be considered as approximations to solutions to the original problem. In addition, the order of the subproblems (A), (B) in the splitting approach is chosen such that the phase condition is satisfied at the end of a full time step. More details about how to calculate the speed with the algebraic constraint can be found in the description of the schemes, see (12), (13), (16) and (17). A schematic overview of the schemes is given in Fig. 2.
convergence
order 1 ut +
full problem freezing method sub-problem semi-discrete formulation time discretization splitting method
order 2
orthogonal or fixed p.c. wt = −( 12 w2 )x
1 2 2u x
zt = zxx
= uxx fixed p.c.
wt = −( 12 w2 )x
zt = zxx
Rusanov Scheme discrete Laplacian Kurganov-Tadmor discrete Laplacian forward Euler
backward Euler Lie
Heun’s method
Crank-Nicolson
Strang
Fig. 2 Overview of the applied numerical schemes for the presented schemes which offer a numerical steady state
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2.1 First-Order Scheme We first present a first-order scheme. For this, we use a method of lines (MOL) approach for (B): We choose a finite interval [L − , L + ] and choose a spatial grid with uniform step size Δx and spatially discretize with the semi-discrete version of the Rusanov scheme. We use Dirichlet boundary conditions. It is worth mentioning here that, for example, the LxF or NT scheme does not offer a semi-discrete version. The Rusanov scheme (RS) in its semi-discrete form is given by f w j+1 (t) − f w j−1 (t) d w j (t) = − dt 2Δx κ [w j+1 (t) − 2w j (t) + w j−1 (t)] + 2Δx ∂ 2 w(t) j = −∂0 f w(t) j + κ Δx 2 0 =: RSΔx (w(t)), 1 where ∂0 is the central difference quotient, ∂0 w j = 2Δx (w j+1 − w j−1 ), ∂02 the discrete Laplacian, both with Dirichlet boundary conditions and κ = max j u( jΔx, 0) is the maximum over the initial value evaluated at all grid points. This scheme is in a simplified form: Since the local maximal speeds, used in the Kurganov–Tadmor scheme, ensure that all information of the Riemann fans stay in each cell of the discretized problem, they can be replaced by an upper bound. In the case of the Burgers’ nonlinearity, this upper bound is given by the maximal absolute value of the solution, which, in turn, is given by the maximal absolute value κ of the initial function u 0 due to the maximums principle. The time discretization is done with a uniform step size Δt; for the first-order version, we use the forward Euler method. The numerical approximation of Φ BΔ t Δ will be denoted by φ Δ B,RSΔx t and ϕ B,RSΔx t for the two different phase conditions (7) and (8), respectively. The operator φ Δ B,RSΔx t is given as the function which maps 1 1 1 w0 , γ0 , μ0 to the solution w , γ , μ of the system
⎧ 1 w = w0 + ΔtRSΔx (w0 ) + Δtμ∗ ∂0 w0 , ⎪ ⎪ ⎪ ⎪ ⎪ ∂0 w0 ∂02 w0 − w0 ∂0 w0 ⎪ ⎪ ∗ ⎪ , ⎨μ = − ∂0 w0 ∂0 w0 ⎪ γ 1 = γ0 + Δtμ1 , ⎪ ⎪ ⎪ ⎪ ⎪ ∂ w1 ∂02 w1 − w1 ∂0 w1 ⎪ ⎪ ⎩ μ1 = − 0 , ∂0 w1 ∂0 w1
w0 = w0 , γ 0 = γ0 ,
(12)
where we use a discrete version of the orthogonal phase condition (7). For the fixed phase condition (8), the operator ϕ Δ B,RSΔx t is given as the mapping, which maps w0 , γ0 , μ0 to the solution w1 , γ 1 , μ1 of the system
A Splitting Approach for Freezing Waves
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⎧ 1 w = w0 + ΔtRSΔx (w0 ) + Δtμ1 ∂0 w0 , ⎪ ⎪ ⎪ ⎨ ∂0 vˆ w0 + ΔtRSΔx (w0 ) − vˆ 1 , μ =− ⎪ Δt∂0 vˆ ∂0 w ⎪ ⎪ ⎩ 1 γ = γ0 + Δtμ1 ,
w0 = w0 , γ 0 = γ0 .
(13)
Also for the subproblem (A) we use a MOL approach, namely, we spatially discretize (A) by finite differences, i.e., the discrete Laplacian with Dirichlet boundary conditions, ∂02 , is used to approximate the second spatial derivative, d z j = ∂02 z j , z j (0) = z 0j . dt For the time discretization, we use backward Euler, because implicit methods have better stability properties for this type of equation. Using the linearity of ∂02 , this 1 1 1 leads to φ Δ A,BEx t : (z 0 , γ0 , μ0 ) → (z , γ , μ ) where ⎧ 1 2 −1 0 ⎪ ⎨ z = (I − Δt∂0 ) z , γ 1 = γ 0, ⎪ ⎩ 1 μ = μ0 ,
z0 = z0 , γ 0 = γ0 ,
(14)
μ = μ0 , 0
Δ such that φ Δ A,BEx t ≈ Φ t A . By using the Lie splitting (10), the full scheme for the freezing PDAE (6) is given by ⎛ n⎞ ⎛ n+1 ⎞ v v Δ Δ ⎝γ n+1 ⎠ := φ B,RSΔx t ◦ φ A,BEx t ⎝γ n ⎠ (LO) μn+1 μn
for the orthogonal phase condition and by ⎛
⎛ n⎞ ⎞ vn+1 v Δ n⎠ ⎝γ n+1 ⎠ := ϕ Δ ⎝ γ t ◦ φ t B,RSΔx A,BEx n+1 μ μn
(LF)
for the fixed phase condition.
2.2 Second-Order Scheme To construct a scheme with quadratic convergence in time and space, we have to replace our numerical solution operators by suitable second-order schemes and use Strang splitting instead of Lie splitting. For the nonlinear hyperbolic part, we use the second-order semi-discrete scheme from [5]. It is given as
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1 + d u j (t) = − f u j+ 1/2 (t) + f u −j+ 1/2 (t) − f u +j− 1/2 (t) − f u −j− 1/2 (t) dt 2Δx
κ u +j+ 1/2 (t) − u −j+ 1/2 (t) − u +j− 1/2 (t) + u −j− 1/2 (t) + 2Δx =: KTΔx (u(t)),
(15) where u ±j+ 1 (t) := u j+ 21 ± 21 (t) ∓ 2
Δx (u x ) j+ 21 ± 21 (t) 2
for j = −M, . . . , M with u(t) ∈ R2M+1 and u j (t) ∈ R its jth element. The slopes are approximated using the minmod limiter (u x )nj = minmod
u nj − u nj−1 u nj+1 − u nj , Δx Δx
,
where minmod(a, b) := 21 [sgn(a) + sgn(b)] · min(|a|, |b|). For the time integraΔ tion, we use Heun’s method. In the case of (7), φ Δ B,KTΔx t is the mapping φ B,KTΔx t : 1 1 1 (w0 , γ0 , μ0 ) → (w , γ , μ ) given by the solution of ⎧ ∗ w ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ w1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∗ μ ⎪ ⎪ ⎪ ⎪ γ1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ μ1
= w0 + ΔtKTΔ x(w0 ) + Δtμ∗ ∂0 w0 , = 21 w0 + 21 w∗ + ΔtKTΔ x(w∗ ) + Δtμ∗ ∂0 w∗ , ∂0 w0 ∂02 w0 − w0 ∂0 w0 =− , ∂0 w0 ∂0 w0 = γ0 + Δtμ1 , ∂0 w1 ∂02 w1 − w1 ∂0 w1 =− , ∂0 w1 ∂0 w1
w0 = w0 , γ 0 = γ0 .
(16)
For the fixed phase condition (8), we define ϕ Δ B,KTΔx t as the mapping (w0 , γ0 , μ0 ) → (w1 , γ 1 , μ1 ) ⎧ ∗ w = w0 + ΔtKTΔ x(w0 ) + Δtμ1 ∂0 w0 , ⎪ ⎪ ⎪ ∗ ⎪ 1 Δ ∗ 1 ∗ 1 0 1 ⎪ ⎪ ⎨ w = 2 w + 2 w + ΔtKT x(w ) + Δtμ ∂0 w , ∂0 vˆ w0 + ΔtKTΔ x(w0 ) − vˆ 1 ⎪ , μ =− ⎪ ⎪ ⎪ ∂0 vˆ ∂0 w ⎪ ⎪ ⎩ 1 γ = γ0 + Δtμ1 ,
w0 = w0 , γ 0 = γ0 .
(17)
For the heat equation, we use the Crank–Nicolson1 method to discretize in time and, as in the first-order version, the discrete Laplacian with Dirichlet boundary conditions, ∂02 , is used in space. The solution operator φ Δ A,CNx t is given by the mapping 1 The Crank–Nicolson method used here is only the discretization in time by combining the forward
and backward Euler method.
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(z 0 , γ0 , μ0 ) → (z 1 , γ 1 , μ1 ) of ⎧ 1 ⎪ ⎨ z = (I − γ 1 = γ 0, ⎪ ⎩ 1 μ = μ0 ,
Δt 2 −1 ∂ ) (I 2 0
+
Δt 2 0 ∂ )z , 2 0
z0 = z0 , γ 0 = γ0 , μ0 = μ0 .
These methods were chosen because they offer quadratic convergence for the individual problem, and thus, we can hope for quadratic convergence of the full problem with Strang splitting. Strang splitting (11) leads to our second-order scheme given by ⎛
⎞ ⎛ n⎞ vn+1 v Δt/2 Δt/2 n⎠ ⎝γ n+1 ⎠ = φ B,KTΔx ⎝ γ ◦ φΔ t ◦ φ A,CNx B,KTΔx n+1 μ μn
(SO)
for the orthogonal phase condition and by ⎛
⎞ ⎛ n⎞ vn+1 v Δt/2 Δt/2 ⎝γ n+1 ⎠ = ϕ B,KTΔx ◦ φ Δ ⎝γ n ⎠ A,CNx t ◦ ϕ B,KTΔx μn+1 μn
(SF)
for the fixed phase condition.
3 Numerical Results The purpose of our schemes is to calculate viscous profiles by a simple forward simulation, and thus, we are interested in the quality of those profiles obtained at the end of a longtime simulation. Note that we do not consider the initial convergence order on finite intervals. For all following computations, we use the finite interval [−15, 15] and Dirichlet boundary conditions. The reference function is given by (2) using b = 1.5, c = −0.5, which is also used as initial value with t = 0. This leads to a speed of 0.5 for the traveling wave. Since we are looking for numerical steady states in the co-moving frame, we have to check if our numerical schemes yield steady states. A steady state has the property dtd u(t) = 0, which translates in the numerical case to u n+1 = u n . In Fig. 3, we plot the time against the discrete L 2 distance u n+1 − u n L 2 and see that our schemes yield steady states at around t ≈ 100 for (LO), (LF), and (SF) since u n+1 − u n L 2 ≈ machine precision. For the Strang splitting scheme with orthogonal phase condition (SO), we see that u n+1 − u n L 2 does not converge to zero, and the scheme does not offer a steady state. Solutions for this scheme leave the co-moving frame because the approximation of the speed is incorrect in this case. . Next, we consider For these computations, we use 300 steps in space and Δt = Δx 10 the error profiles of the calculated steady states with different step sizes. This result is shown in Fig. 4. Obviously, we get different numerical steady states for different , which approximates the exact steady state better for smaller steps sizes. dt = Δx 10
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1. Order - fixed p.c. 2. Order - fixed p.c. 1. Order - orth p.c. 2. Order - orth p.c. eps
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Fig. 3 Convergence to a numerical steady state except for the second-order scheme with orthogonal phase condition
O1 scheme - orth. p.c.: error profiles 0.1
O2 scheme - fixed p.c.: error profiles 1
100 x steps
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0
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−10
0
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x - space Δx 10 . Note that the errors dominate where
In addition, we observe that the dominant error occurs in the profile, and there is more-or-less no error at the boundary. The most interesting observation in our case is the convergence of our numerical steady states to the exact one. For this, we plot the discrete L 2 error of our states to the exact one for different step sizes in Fig. 5. Here, we see linear convergence of our numerical steady states to the exact one for our first-order scheme. For the second-order version, we see quadratic convergence. Finally, we note that usually the exact solution of the traveling wave is unknown. Therefore, one has to guess some suitable reference function. In Fig. 6, we see that
A Splitting Approach for Freezing Waves Fig. 5 Convergence rates of the numerical steady states to the exact one. The scheme (LF) was omitted because it produces the same results as (LO), whereas the scheme (SO) was ignored because it does not offer steady states
549 discrete L2 -error to exact solution
100 O1 scheme - orth. p.c. O2 scheme - fixed p.c.
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10−2 10−3 10−4 10−5
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Fig. 6 Solution using initial value and reference function which only covers the rough behavior of the solution
Profile sequence of O2 scheme initial value & reference function vˆ
1
0
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space
10
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time
a rough guess is sufficient for the initial value as well as for the reference function vˆ . The forward simulation approximates the traveling wave as before. Acknowledgements We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173.
References 1. W.-J. Beyn, D. Otten, J. Rottmann-Matthes, Stability and computation of dynamic patterns in PDEs, in Current Challenges in Stability Issues for Numerical Differential Equations, 2014, pp. 89–172 2. W.-J. Beyn, V. Thümmler, Freezing solutions of equivariant evolution equations. SIAM J. Appl. Dyn. Syst. 3(2), 85–116 (2004) 3. R. Flohr, Splitting-Verfahren für partielle Differentialgleichungen mit Burgers-Nichtlinearität. Masters thesis, Bielefeld University (2013) 4. H. Holden, Chr. Lubich and N.H. Risebro: Operator splitting for partial differential equations with Burgers nonlinearity. Math. Comput. 82, 173–185 (2013)
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5. A. Kurganov, E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160, 241–282 (2000) 6. J. Rottmann-Matthes, Stability of parabolic-hyperbolic traveling waves. Dyn. Partial Differ. Equ. 9, 29–62 (2012) 7. J. Rottmann-Matthes, Freezing similarity solutions in multidimensional Burgers’ equation. Preprint, 2016. http://www.waves.kit.edu/downloads/CRC1173_Preprint_2016-27.pdf 8. A.M. Stuart, A.R. Humphries, Dynamical Systems and Numerical Analysis (Cambridge University Press, 1996) 9. V. Thümmler, Numerical analysis of the method of freezing traveling waves. Dissertation, Bielefeld University (2005)
Metastability for Hyperbolic Variations of Allen–Cahn Equation Raffaele Folino
Abstract The Allen–Cahn equation is a parabolic reaction–diffusion equation that has been originally proposed in Allen and Cahn (Acta Metall 27:1085–1095, 1979, [1]) to describe the motion of antiphase boundaries in iron alloys. In general, reaction– diffusion equations of parabolic type undergo the same criticisms of the linear diffusion equation, mainly concerning lack of inertia and infinite speed of propagation of disturbances. To avoid these unphysical properties, many authors proposed hyperbolic variations of the classic reaction–diffusion equations. Here, we consider a hyperbolic variation of the Allen–Cahn equation and present some results contained in Folino (J Hyperbolic Differ Equ 14:1–26, 2017, [6]) and Folino et al. (Metastable dynamics for hyperbolic variations of the Allen–Cahn equation, 2016, [8]) concerning the metastable dynamics of solutions. We study the singular limit of the solutions as the diffusion coefficient ε → 0+ and show that the hyperbolic version shares the well-known dynamical metastability valid for the parabolic equation. Keywords Hyperbolic Allen–Cahn equation · Metastability · Singular perturbations · Slow motion Subject Classifications
35B25 · 35K57 · 35L20
1 Introduction In this work, we collect some results contained in [6, 8] concerning the metastable dynamics of solutions to the hyperbolic Allen–Cahn equation in one space dimension. Metastability is a broad term describing the persistence of unsteady structures for a very long time; metastable dynamics is characterized by evolution so slow that (nonstationary) solutions appear to be stable. For the classic Allen–Cahn equation, R. Folino (B) Department of Information Engineering, Computer Science and Mathematics, University of L’Aquila, Via Vetoio, 67100 L’Aquila, Coppito, Italy e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_42
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this phenomenon was first observed in [2–4, 9]. In particular, there are two approaches to study metastable dynamics for Allen–Cahn equation. The “energy approach”, introduced by Bronsard and Kohn [2], is based on the underlying variational structure of the equation. It gives an intuitive explanation for the slow motion of the solutions and shows the persistence of the metastable states for a time Tε > ε−k , where ε 1 is the diffusion coefficient. On the other hand, the “dynamical approach” of Carr and Pego [3], and Fusco and Hale [9] gives more precise results. They construct an N -dimensional manifold M consisting of functions which approximate metastable states with N transition layers. If the initial datum is in a small neighborhood of M , then the solution remains near M for a time proportional to eC/ε . In addition, this approach gives the explicit form of a system of ordinary differential equations describing the motion of the N transition layers. The aim of this paper is to show that each of these approaches can be adapted and extended to a hyperbolic variation of the Allen–Cahn equation. In particular, we present the main result of [6], where the metastable dynamics for the hyperbolic version is studied by using the energy approach, and the main result of [8], where the same problem is studied by using the dynamical approach. Let us consider the hyperbolic Allen–Cahn equation τ u tt + g(u)u t = ε2 u x x + f (u),
x ∈ (a, b), t > 0,
(1)
where τ, ε are positive parameters, g ∈ C 1 (R) is a strictly positive function and f = −F with F ∈ C 3 (R) a double-well potential with wells of equal depth; namely, we assume g(u) ≥ σ > 0 ∀ u ∈ R, F(±1) = F (±1) = 0, F (±1) > 0, F(u) > 0 for u = ±1.
(2) (3)
In other words, F is a smooth, nonnegative function with global minimum equal to 0 reached only at u = −1 and u = +1, and so f is a bistable reaction term. The simplest example is F(u) = 41 (u 2 − 1)2 . Taking g ≡ 1 in (1), one obtains a damped wave equation with bistable nonlinearity, which is the easiest hyperbolic variation of the classic Allen–Cahn equation. For g = 1 − τ f , we have the Allen–Cahn equation with relaxation τ u tt + (1 − τ f (u))u t = ε2 u x x + f (u),
(4)
that is obtained by substituting Fick’s diffusion law with a relaxation relation of Cattaneo–Maxwell type (see [5, 12, 13]). Indeed, consider the nonlinear system
u t + vx = f (u), τ vt + ε2 u x = −v.
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By differentiating with respect to t the first equation and differentiating with respect to x the second one, we obtain equation (4). Note that taking formally the limit for τ → 0, the second equation, that is, the Maxwell–Cattaneo law, becomes the classic Fick’s diffusion law and we have the classic Allen–Cahn equation u t = ε2 u x x + f (u).
(5)
It is well known that, under assumption (3), there exist traveling wave solutions, Φ(x − ct), for equation (5) connecting the stable states −1 and +1, if and only if c = 0. The profile Φ is a stationary solution of (5), satisfying the problem ε2 Φ + f (Φ) = 0,
Φ(±∞) = ±1.
(6)
Since f = −F and F satisfies assumption (3), there exists a unique solution, up to translation, of the√problem (6). In the simplest example f (u) = u − u 3 , we have that Φ(x) = tanh(x/ 2ε) is the unique solution of (6) with Φ(0) = 0. As previously mentioned, in [2, 3] it is proved the persistence of metastable states for a very long time (a time Tε → +∞ as ε → 0+ ) for equation (5) in a bounded interval of the real line with homogeneous Neumann boundary conditions u x (a, t) = u x (b, t) = 0,
t > 0.
(7)
The metastable states have N transitions between −1 and +1, and they can be constructed using the solution of (6). Fix an integer N > 0 and N transition layers a < h 1 < h 2 < · · · < h N < b. If h = (h 1 , h 2 , . . . , h N ) ∈ R N , let us define U h (x) := Φ (x − h i )(−1)i+1
for x ∈ [h i−1/2 , h i+1/2 ],
i = 1, . . . , N , (8)
where Φ is the solution of (6) with Φ(0) = 0 and h i+1/2 :=
h i + h i+1 , 2
i = 1, . . . , N − 1,
h 1/2 = a, h N +1/2 = b.
Note that U h is an H 1 -function with a piecewise continuous first derivative that jumps at h i+1/2 for i = 1, . . . , N − 1 (example in Fig. 1). For equation (5) with boundary conditions (7) and initial datum u(x, 0) = U h (x) as in (8), the solution maintains this transition layer structure (see Fig. 1) for an exponentially long time. In particular, the evolution of the solution depends only on the interaction of the transition layers h i and these points move with exponentially small velocity. This is an example of dynamical metastability. Indeed, it is well known that the solutions of (5)–(7) tend to stationary solutions as t → +∞ (see, for example, Matano [15, 16]). Furthermore, the only possible stable equilibrium solutions of (5)–(7) are the constant solutions −1 and +1. Thus, for t → +∞, the solutions will be approximately constant. But, in some cases, the time taken for the solutions to reach these equilibria is very long and we have an example of dynamical metastability.
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Fig. 1 Example of function U h in the case f (u) = u − u 3 : N = 8, ε = 0.1
We want to show that dynamical metastability is also present in the hyperbolic variation (1) with homogeneous Neumann boundary conditions (7) and initial data u(x, 0) = u 0 (x),
u t (x, 0) = u 1 (x),
x ∈ (a, b).
(9)
First of all, note that, as in the parabolic case (5), under the assumptions (2)–(3) there exist solutions to (1) of the form Φ(x − ct) satisfying Φ(±∞) = ±1, if and only if c = 0. Indeed, if Φ(x − ct) is a solution, then the profile Φ satisfies (τ c2 − ε2 )Φ − cg(Φ)Φ = f (Φ), and thus, multiplying by Φ and integrating over R, we get c R
g(Φ)(Φ )2 dξ = F(+1) − F(−1).
Since F(+1) = F(−1) and g is strictly positive, it follows that c = 0. So, the traveling wave solutions connecting the stable states −1 and +1 are stationary solutions and solve problem (6), that is, the same problem of the parabolic case. This suggests that the assumptions on the initial profile u ε0 are the same of the parabolic case. We recall that stability of traveling wave solutions for equation (1) was studied in [10] for the damping case (i.e., g ≡ 1) and in [14] for the relaxation case (4). Now, we study the metastable dynamics for the solutions of equation (1) subject to boundary conditions (7) and initial data (9). First, in Sect. 2, we present the main result obtained by using the energy approach. Second, Sect. 3 contains the main result of the dynamical approach.
2 Energy Approach The key point in the energy approach is the use of the modified energy functional E ε [u
ε
, u εt ](t)
:= a
b
τ ε ε ε F(u ε (x, t)) 2 2 u (x, t) + u x (x, t) + d x. 2ε t 2 ε
(10)
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Note that, respect to the parabolic case, there is a new term concerning the derivative with respect to t of u ε , that is, a nonnegative term. The energy functional (10) is a nonincreasing function of time t along the solutions of (1)–(7). Indeed, if u ε is a solution of (1) satisfying (7), then ε
−1
T
0
b
a
2 g(u ε ) u εt d xdt = E ε [u ε , u εt ](0) − E ε [u ε , u εt ](T ).
(11)
For the proof of inequality (11) see [6, Appendix A]. It follows that the positivity of g ensures the dissipative character of the equation (1). In particular, using (2) and (11), we obtain −1
ε σ 0
T
a
b
ε 2 u t d xdt ≤ E ε [u ε , u εt ](0) − E ε [u ε , u εt ](T ).
(12)
Inequality (12) is crucial in the use of the energy approach. Let us briefly explain the strategy of this approach. Let a < y1 < y2 < b, from Young’s inequality we deduce E ε [u ε , u εt ] ≥ E ε [u ε , 0] ≥
y2 y1
|u εx | 2F(u ε ) d x =
u ε (y2 ) u ε (y1 )
2F(s) ds .
If we assume that u ε (y1 ) = −1 and u ε (y2 ) = +1, we see that the energy of a transition between −1 and +1 is greater than or equal to c0 :=
1
2F(s) ds.
−1
This justifies the use of the modified energy (10); indeed, the constant c0 is strictly positive and does not depend on ε (it depends only on F). Fix v : [a, b] → {−1, +1} having exactly N jumps located at a < h 1 < h 2 < · · · < h N < b. In this way, we fix the number of the transition layers and their positions in (a, b) as ε → 0. We assume that the initial data u 0 , u 1 depend on ε and lim u ε0 − v L 1 = 0.
ε→0
(13)
In addition, we suppose that there exist ε0 , C > 0 and a positive integer k such that for all ε ∈ (0, ε0 ), at the time t = 0, the modified energy (10) satisfies E ε [u ε0 , u ε1 ] ≤ N c0 + Cεk .
(14)
If u ε0 is a function with N transitions between −1 and +1, the previous computation shows that b ε ε 2 F(u ε0 ) ε (u ) + d x ≥ N c0 . (15) E ε [u 0 , 0] = 2 0 x ε a
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More precisely, it can be shown (see Modica [17], Sternberg [18]) that if {vε } is a sequence that converges to v in L 1 (a, b), then
b
lim inf ε→0
a
ε ε 2 F(vε ) (v ) + d x ≥ N c0 , 2 x ε
with equality if the sequence {vε } is chosen properly (we will show in Sect. 4 an example such that the equality holds). Therefore, the condition (14) demands that the energy at the time t = 0 exceeds at most Cεk the minimum possible to have N transitions. In particular, for (15) we have that τ 2
b
u ε1 (x)2 d x ≤ Cεk+1 ,
(16)
E ε [u ε0 , 0] ≤ N c0 + Cεk .
(17)
a
So, roughly speaking, the assumptions (13)–(14) say that the initial profile u ε0 satisfies the same assumptions of the parabolic case (cfr. [2]) and the L 2 -norm of the initial k+1 velocity u ε1 is bounded by Cε 2 . Under these assumptions, it can be proved that −(k+1) such that there exists a time Tε ≥ C1 ε E ε [u, u t ](Tε ) ≥ N c0 − C2 εk ,
(18)
for some C1 , C2 > 0 independent of ε. Substituting (14) and (18) in (12), we infer 0
C1 ε−(k+1) b a
ε 2 u t d xdt ≤ σ (C + C2 )εk+1 .
(19)
Then, if ε is small, Tε is very large and there is a very little excess of energy to be dissipated by the motion of the transitions layers, and so, the evolution of the solution is very slow. The proofs of inequalities (18), (19) are in [6, Sect. 2]. Thanks to (19) we can prove the main result of this section. Theorem 1. ([6]) Consider the initial-boundary value problem (1)–(9) with f = −F , and g ∈ C 1 (R), F ∈ C 3 (R) satisfying (2)–(3). Suppose that the initial data u ε0 , u ε1 satisfy (13) and (14) for some k > 0. Then for any m > 0 sup 0≤ t ≤mε−k
u ε (·, t) − v L 1 −−→ 0. ε→0
(20)
Proof. Fix m > 0. Triangle inequality gives:
u ε (·, t) − v L 1 ≤ u ε (·, t) − u ε0 L 1 + u ε0 − v L 1 ,
(21)
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for all t ∈ [0, mε−k ]. From hypothesis (13), it follows that the last term of the righthand side of (21) tends to zero as ε → 0. On the other hand, sup 0≤ t ≤mε−k
u ε (·, t) − u ε0 L 1 ≤
mε−k 0
u εt L 1 dt.
(22)
If ε is small enough so that C1 ε−1 ≥ m, using Cauchy–Schwarz inequality and (19), we obtain
mε−k
0
1
u εt L 1 dt ≤ m 2 ε− 2 (b − a) 2 (C3 εk+1 ) 2 ≤ 1
and as a consequence
k
lim
ε→0 0
1
mε−k
(m(b − a)C3 )ε ,
u εt L 1 dt = 0.
By combining (13), (21), (22) and (23), we conclude the proof of (20).
(23)
As previously mentioned, the energy defined in (10) is a nonincreasing function of t and so, the requirement (14) holds for any time t. Thanks to Theorem 1 we can say that also (13) holds for t ∈ [0, mε−k ]. Hence, we can apply the same reasoning of u ε0 , u ε1 to u ε (·, t), u εt (·, t) for any t ∈ [0, mε−k ]. It follows that in this interval the same estimates of (16) and (17) hold. Thus, in a timescale of order ε−k nothing happens and the solution maintains the same transition layer structure of the time t = 0. Theorem 1 also implies that the transition layers move slower than εk (cfr. [6, Sect. 2]), and so, using the energy approach we proved algebraically slow motion. The energy approach can be improved and imposing stronger conditions on the initial data we can obtain exponentially slow motion (see Grant [11]). In [7], this improvement is applied to the case of systems of hyperbolic Allen–Cahn equations, i.e., when u is a vector-valued function in (1).
3 Dynamical Approach In this section, we study the slow motion of the solutions to the initial-boundary value problem (1)–(7)–(9) with f = −F and g, F satisfying assumptions (2), (3) by using the dynamical approach. Rewrite the hyperbolic equation (1) as the system u t = v, τ vt = L (u) − g(u)v,
where L (u) := ε2 u x x + f (u).
(24)
Fix N ∈ N and ε > 0. The idea is to associate to any configuration h = (h 1 , . . . , h N ) of N layer positions (with h j < h j+1 ) a function u h = u h (x) which approximates
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a metastable state with N transition points at h 1 , . . . , h N . We consider the slow evolution of solutions when the transition points are well separated one from the other and bounded away from the boundary points (for simplicity we take [a, b] = [0, 1]). Fixed (small) ρ > 0, the admissible layer positions lie in the set
Ωρ := h ∈ R N : 0 < h 1 < · · · < h N < 1, h j+1 − h j > ε/ρ for j = 0, . . . , N , where h 0 := −h 1 , h N +1 := 2 − h N . In what follows, we fix a minimal distance δ > 0 with δ < 1/N and we consider the parameters ε and ρ such that 0 < ε < ε0
and
δ
0 to be chosen appropriately small. In such a way, the parameters ρ and ε have the same order of magnitude. A possible way to construct the function u h with h ∈ Ωρ is by using the standing wave (6) to define u h as in (8). Fusco and Hale [9] proposed this strategy; they obtained a continuous function with a piecewise continuous first-order derivative that jumps at h j+1/2 , j = 1, . . . , N − 1. In particular, U h belongs to H 1 and not to H 2 (if N > 1). Here, we follow the strategy used by Carr and Pego [3] adapting it to the case of the hyperbolic Allen–Cahn equation (1) (for more details see [8]). Given > 0, let us define φ(x, , ±1) as the solution to L (φ) := ε2 φx x + f (φ) = 0,
φ − 21 = φ 21 = 0,
(26)
with φ > 0 in (− 21 , 21 ), and φ(·, , −1) as the solution to (26) with φ < 0 in (− 21 , 21 ). The functions φ(·, , ±1) are well defined if /ε > L 0 for some constant L 0 > 0 depending on f , and they depend on ε and only through the ratio ε/ . For h ∈ Ωρ with ρ < 1/L 0 , we define the function u h with N transition points at h 1 , . . . , h N by matching together the functions φ(·, , ±1), using smooth cutoff functions. Given χ : R → [0, 1] a C ∞ function with χ (x) = 0 for x ≤ −1 and χ (x) = 1 for x ≥ 1, set χ j (x) := χ
x − hj ε
and
φ j (x) := φ x − h j−1/2 , h j − h j−1 , (−1) j .
Then, the function u h is given by the convex combination u h := 1 − χ j φ j + χ j φ j+1
in I j := [h j−1/2 , h j+1/2 ],
and the base manifold is M := {u h : h ∈ Ωρ }. A detailed study of the manifold M h is in [3] or [8, Sect. 2]; here we recall only some properties of the functions u .j−1 h If ρ > 0 is sufficiently small and h ∈ Ωρ , then u (x) ≈ Φ (x − h j )(−1) for x near h j and u h (x) ≈ ±1 away from h j for j = 1, . . . , N . Therefore, states u h on the base manifold are well approximated near transition layers by the function
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defined in (8). By definition, u h is a smooth function of x and h and enjoys the properties u h (0) = φ(0, 2h 1 , −1) < 0, u h (h j ) = 0,
u h (h j+1/2 ) = φ 0, h j+1 − h j , (−1) j+1 L (u h (x)) = 0 for |x − h j | ≥ ε,
for any j = 1, . . . , N . In what follows, we use the notation u hj := ∂h j u h ,
∇h u h := u 1h , . . . , u hN .
Finally, we recall that the L 2 -norm of L (u h ) is exponentially small
L (u h ) ≤ Cε1/2 exp(−A h /ε). √ Here and in the following, · is the L 2 -norm, A := min{F (−1), F (1)}, and
h := min{h j − h j−1 } is the minimum distance between layers. Since the hyperbolic Allen–Cahn equation (1) corresponds to the system (24), the dynamics is determined by an additional unknown, the time derivative v = u t , and thus the base manifold M has to be embedded in an extended vector space. Here, taking advantage of the fact that we are looking for a manifold that is only approximately invariant, we perform this extension in a trivial way,
considering the extended base manifold M0 := M × {0} = (u h , 0) : u h ∈ M . In order to restrict the attention to a neighborhood of M0 , we introduce the decomposition u = u h + w, where w is orthogonal to some appropriate approximate tangent vectors k hj to M (for more details see [8, Sect. 2]). Namely, if ·, · is the standard inner product in L 2 (0, 1), w, k hj = 0,
for
j = 1, . . . , N .
(27)
Then, setting
HN2 := w ∈ H 2 (0, 1) : wx (0) = wx (1) = 0, w, k hj = 0
for j = 1, . . . , N ,
we choose the following tubular neighborhood of M0 : given Γ, ρ > 0, we set
ZΓ,ρ := (u, v) : u = u h + w, (h, w, v) ∈ Ω ρ × H N2 × L 2 (0, 1), E h [w, v] ≤ Γ Ψ (h) ,
with the energy functional E h and the barrier function Ψ defined by E h [w, v] := Ψ (h) :=
1 2
1
0
N j=1
ε2 wx2 − f (u h )w2 d x + 21 τ v 2 + ετ w, v,
2 L u h , k hj .
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The barrier function Ψ is exponentially small in ε and the requirement (u, v) ∈ ZΓ,ρ implies w and v exponentially small. The main result of this section states that the channel ZΓ,ρ is invariant for an exponentially long time if the parameters Γ and ρ are appropriately chosen: if the initial datum (u 0 , u 1 ) ∈ ZΓ,ρ , then the solution (u, v) of (24) with Neumann boundary conditions (7) remains in ZΓ,ρ for an exponentially long time. In other words, the manifold M0 is approximately invariant for the hyperbolic system (24). Theorem 2. Let g ∈ C 1 (R) and f = −F with F ∈ C 3 (R) be such that (2)–(3) hold. Given N ∈ N and δ ∈ (0, 1/N ), there exist Γ2 > Γ1 > 0 and ε0 > 0 such that if ε, ρ satisfy (25), Γ ∈ [Γ1 , Γ2 ] and the initial datum satisfies ◦
(u 0 , u 1 ) ∈ Z Γ,ρ = (u, v) ∈ ZΓ,ρ : h ∈ Ωρ and E h [w, v] < Γ Ψ (h) ,
then the solution to the initial-boundary value problem (1)–(7)–(9) remains in ZΓ,ρ for a time Tε > 0 and there exists C > 0 such that ε1/2 u − u h L ∞ + u − u h + τ 1/2 u t ≤ C exp(−A h /ε),
|h |∞ ≤ C(ε/τ )
1/2
(28)
exp(−A /ε), h
(29)
where | · |∞ denotes the maximum norm in R N . Moreover, there exists C > 0 such that Tε ≥ C(τ/ε)1/2 ( h(0) − ε/ρ) exp(Aδ/ε). In Theorem 2, the assumptions on the initial data are stronger than Theorem 1, but we obtain exponentially slow motion. For the complete proof of Theorem 2, see [8]. Here, we briefly explain the strategy of the proof. First, plugging the decomposition u = u h + w into system (24) and using conditions (27), we obtain an ODE-PDE coupled system describing the dynamics for (h, w, v): ⎧ h ⎪ ⎨wt = v − ∇h u · h , τ vt = L (u h ) + L h w + f 2 w2 − g(u h + w)v, ⎪ ⎩
ˆ D(h) − D(h, w) h = Y (h, v),
(30)
where · denotes the inner product in R N . The equation for the pair (w, v) follows from (24) and the expansion L (u h + w) = L (u h ) + L h w + f 2 w2 ,
where f 2 :=
1 0
(1 − s) f (u h + sw) ds,
and the operator L h is the linearization of L about u h : L h w := ε2 wx x + f (u h )w. Differentiating with respect to t the orthogonality condition (27), we get the equation for h, where we use the notation
Metastability for Hyperbolic Variations of Allen–Cahn Equation
Di j (h) := u hj , kih ,
Dˆ i j (h, w) := w, kihj , kihj := ∂h j kih ,
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Yi (h, v) := v, kih .
Second, studying the system (30), it can be shown that if the solution (u, v) belongs to ZΓ,ρ , then the estimates (28) and (29) hold. Finally, we need to estimate the time Tε taken for the solution (u, v) to leave the channel ZΓ,ρ . The boundary of ZΓ,ρ is the union of two parts: the “ends” where h ∈ ∂Ωρ , meaning h j − h j−1 = ε/ρ for some j and “sides” where E h [w, v] = Γ Ψ (h). Using energy estimates in the study of (30), in [8] it is proved that the solution can leave ZΓ,ρ only through the ends. Since, for (29), the transition points move with exponentially small velocity, the solution (u, v) stays in the channel for an exponentially long time and we have the lower bound of Tε . As long as the solution (u, v) of (24) remains in the channel ZΓ,ρ , u is a function with N transition layers (see Fig. 1). In [8, Sect. 4], it is derived an approximation of the equation for h that describes the motion of the transition points. This equation is determined formally by the requirement that u(x, t) = u h(t) (x) is an exact solution. Such request is expected to be appropriate in the limit ε → 0. In this way, we obtain a system of ordinary differential equations for h which does not depend on w and v and has the form (31) τ h + γ h = P(h), where γ := g and the (weighted) average g of the continuous function g is given by 1
g := √
F L 1
1
F(s) g(s) ds,
−1
and P is a function, depending on F. Equation (31) has to be compared with the corresponding one for the parabolic case (5), which is h = P(h). For the damped nonlinear wave equation g ≡ 1, we have γ = 1, while for the Allen–Cahn equation with relaxation (4) we obtain γ = 1 − τ f . Since f is positive, the (physical relevant) relaxation case exhibits smaller friction effects with respect to the damped one (details in [8, Sect. 4]).
4 Concluding Remarks In this final section, we do a comparison between the assumptions on the initial data in the two approaches. Consider an initial profile as in (8), i.e., u ε0 = U h , that is a “well approximation” of the function u h constructed in Sect. 3. Let us focus our attention on an interval [h i−1/2 , h i+1/2 ] (for example, i odd). Since Φ satisfies (6), 2 2 we have ε2 Φ = F(Φ) and so,
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h i+1/2 h i−1/2
li ε h 2 F(U h ) (U )x + dx = Φ (x) 2F(Φ(x)) d x 2 ε −li−1 Φ(−li−1 ) 2F(s) ds − = c0 − −1
1
2F(s) ds,
Φ(li )
where li := (h i+1 − h i )/2. The last two integrals are exponentially small, because Φ(x) = W (x/ε) where W satisfies W + f (W ) = 0,
W (0) = 0,
W (±∞) = ±1.
It follows that 1 − W (x) ≤ c1 e−c2 x
as x → +∞,
W (x) + 1 ≤ c1 ec2 x
as x → −∞,
for some constant c1 , c2 > 0. Hence, in the case of u ε0 as in (8), we have lim u ε0 − v L 1 = 0,
ε→0
N c0 − Ce−c/ε ≤ E ε [u ε0 , 0] ≤ N c0 ,
(32)
where the constant c depends on F and on the distance between the transition layers. Roughly speaking, the energy is exponentially close to N c0 . The lower bound in (32) for the energy holds for any function sufficiently close to v in L 1 (this is proved in [7] in the case of vector-valued functions). Then, in the case of initial profile as in (8), the estimate (17) is satisfied for any positive integer k. Let us stress that the initial velocity u 1 plays a crucial role; transitions may be created or eliminated in a short time by choosing opportunely u 1 (see some examples in [6, Sect. 3]). If the L 2 –norm of the initial velocity is exponentially small, we obtain exponentially slow motion by using the dynamical approach or by improving the energy approach (see [7]). If the L 2 –norm of the initial velocity satisfies the weaker condition (16), we obtain algebraically slow motion.
References 1. S. Allen, J. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1085–1095 (1979) 2. L. Bronsard, R. Kohn, On the slowness of phase boundary motion in one space dimension. Commun. Pure Appl. Math. 43, 983–997 (1990) 3. J. Carr, R.L. Pego, Metastable patterns in solutions of u t = ε2 u x x − f (u). Commun. Pure Appl. Math. 42, 523–576 (1989) 4. J. Carr, R.L. Pego, Invariant manifolds for metastable patterns in u t = ε2 u x x − f (u). Proc. R. Soc. Edinburgh Sect. A 116, 133–160 (1990) 5. C. Cattaneo, Sulla conduzione del calore. Atti del Semin. Mat. e Fis. Univ. Modena 3, 83–101 (1948) 6. R. Folino, Slow motion for a hyperbolic variation of Allen-Cahn equation in one space dimension. J. Hyperbolic Differ. Equ. 14, 1–26 (2017)
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7. R. Folino, Slow motion for one-dimensional hyperbolic Allen–Cahn systems. Differ. Integr. Equat, to appear 8. R. Folino, C. Lattanzio, C. Mascia, Metastable dynamics for hyperbolic variations of the Allen– Cahn equation. Commun. Math. Sci. 15, 2055–2085 (2017) 9. G. Fusco, J. Hale, Slow-motion manifolds, dormant instability, and singular perturbations. J. Dyn. Differ. Equ. 1, 75–94 (1989) 10. T. Gallay, R. Joly, Global stability of travelling fronts for a damped wave equation with bistable nonlinearity. Ann. Scient. Ec. Norm. Sup. 42, 103–140 (2009) 11. C.P. Grant, Slow motion in one-dimensional Cahn-Morral systems. SIAM J. Math. Anal. 26, 21–34 (1995) 12. D.D. Joseph, L. Preziosi, Heat waves. Rev. Modern Phys. 61, 41–73 (1989) 13. D.D. Joseph, L. Preziosi, Addendum to the paper: “Heat waves” [Rev. Modern Phys. 61(1), 41–73 (1989)]. Rev. Modern Phys. 62, 375–391 (1990) 14. C. Lattanzio, C. Mascia, R.G. Plaza, C. Simeoni, Analytical and numerical investigation of traveling waves for the Allen-Cahn model with relaxation. Math. Models Methods. Appl. Sci. 26, 931–985 (2016) 15. H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations. J. Math. Kyoto Univ. 18, 221–227 (1978) 16. H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations. Publ. Res. Inst. Math. Sci., Kyoto Univ. 15, 401–454 (1979) 17. L. Modica, The gradient theory of phase transitions and the minimal interface criterion. Arch. Rat. Mech. Anal. 98, 123–142 (1987) 18. P. Sternberg, The effect of a singular perturbation on nonconvex variational problems. Arch. Rat. Mech. Anal. 101, 209–260 (1988)
Cell-Centred Lagrangian Lax–Wendroff HLL Hybrid Schemes in Cylindrical Geometry David Fridrich, Richard Liska and Burton Wendroff
Abstract Lagrangian hydrodynamics described by Euler equations is treated by the Lax–Wendroff method with the dissipative fluxes in the HLL form, including both artificial viscosity and artificial energy flux. In cylindrical geometry, the velocity is discretized using source terms. The proposed method works reasonably well on Noh, Sedov and spherical Sod tests. The scheme preserves exact symmetry of results on initially polar meshes while the symmetry on initially rectangular meshes remains very good. Keywords Lagrangian hydrodynamics · Lax–Wendroff · HLL · Artificial viscosity · Euler equations
1 Introduction In this work, we present cylindrical extension of two-dimensional cell-centred method for system of Euler equations in Lagrangian form LW+n previously proposed in [1]. The scheme is based on Richtmyer’s finite volume formulation of Lax–Wendroff (LW) scheme. A single step of Richtmyer’s method consists of predictor and corrector substeps. In the predictor phase, the nodal estimates of conservative quantities (volume, momentum and total energy) are computed. Predicted nodal velocities are then used to advance the mesh positions. In the corrector phase, the cell values of the conservative quantities are computed using the nodal estimates for fluxes. Richtmyer’s scheme is dispersive; hence, it produces oscillations near solution discontinuities, which makes the scheme in its pure form useless. In order to reduce D. Fridrich (B) · R. Liska Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Bˇrehová 7, 115 19 Prague 1, Prague, Czech Republic e-mail: [email protected] B. Wendroff Los Alamos National Laboratory, Los Alamos NM, USA © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_43
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Fig. 1 Pseudo-Cartesian frame with pseudoradius R
the oscillations, we propose to include additional dissipative terms to the cell quantities after each LW step. Besides the oscillation reduction, the additional terms can also improve symmetry of the solution and reduce the effect of wall heating. Methods in planar geometry will be referenced as LW+n. LW+1 adds an artificial viscosity to the momentum conservation equation, LW+2 in addition adds an artificial energy flux to the total energy conservation equation and LW+3 optionally adds also an artificial density flux to the volume conservation equation. In order to distinguish between Cartesian and cylindrical scheme, the cylindrical variants will be referenced as LWrz+n while the n suffix will have the same meaning. The dissipative terms are modelled by the Lagrangian form of Harten–Lax–van Leer (HLL) [2] fluxes.
2 Governing Equations Let us assume that in cylindrical case, the solution is axially symmetric around y ≡ z axis. After integration in azimuthal direction, the angular component of momentum vanishes and the system reduces into two spatial dimensions. The factor 2π arising from the integration can be omitted, assuming that all variables are defined per unit radian. In similar way as in [3, 4], we define pseudoradius R = 1 − α + αx, where x ≡ r is one spatial direction and α is dimensionless geometry switch. For cylindrical geometry α = 1 and for Cartesian geometry α = 0. Let A(t) be area of either 2D or 3D element enclosed by simple curve L(t). The situation can be seen in Fig. 1. Using the pseudoradius R, one can define volume V :
Rd A,
V (t) = A(t)
A(t) =
d A, A(t)
S(t) =
Rd L L(t)
(1)
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Let us assume an element of inviscid compressible fluid bounded in time-dependent volume V enclosed in surface S. Its mass m V , momentum MV = (MV,x , MV,y ) and total energy EV are defined by
mV =
d V, MV = V
ud V, EV = V
Ed V,
(2)
V
where is density, u = (u, v) velocity and E total specific energy. Using total Lagrangian derivative, Euler equations can be written in the following integral form: d d V = 0, dt V (t) d dV − ∇ · ud V = 0, dt V (t) V (t) d ud V + ∇ pd V = 0, dt V (t) V (t) d Ed V + ∇ · ( pu)d V = 0, dt V (t) V (t)
(3) (4) (5) (6)
where p is pressure. Equation (4) is Geometric Conservation Law (GCL), and it is equivalent to the equation of motion: dX = u, dt
X(0) = x.
(7)
Equation (3) is continuity equation (mass conservation law), Eq. (5) is momentum conservation law and Eq. (6) is total energy conservation law. The system is closed by ideal gas equation of state p(, ε) = (γ − 1)ε,
cs (, ε) =
γ p(, ε) ,
(8)
where ε = E − u2 /2 is specific internal energy, γ is heat capacity ratio and cs is speed of sound. Following derivation in [4], the system (3–6) describing behaviour of arbitrary volume element V (t) can be rewritten into form that can be directly discretized:
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d d V = 0, dt V (t) d d V − u · nRd L = 0, dt V (t) L d ud V + pnRd L = αex pd A, dt V (t) L A d Ed V + pu · nRd L = 0. dt V (t) L
(9) (10) (11) (12)
3 Numerical Method 3.1 Meshes Following pc-notation [5], the computational mesh consists of cells c, points p and edges e. Each n-lateral cell can be divided into n quadrilateral subzones labelled pc. Edge separating cells c and a is denoted by eca and ne(ca) is its normal. The edges of dual cell p inside the cell c are the separators s pc± . The normals to separators s pc± are nspc± . The corner vector n pc is defined as the sum of adjacent separator normals n spc+ and n spc− . The situation is illustrated in Fig. 2. The cell position Xc = (rc , z c ) is defined as the arithmetic average of node positions X p . Assuming quadrilateral mesh, the area A pc and volume V pc of subzone pc is given by
(b)
(a)
Fig. 2 The primary mesh with the edge normals for one cell in black and one dual cell with the separators (its edges) normals in orange for the quadrilateral mesh (a); summation of the edge normals ne(ca) and ne(cb) into n pc (b)
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1 1 2 (z i+1 − z i )(ri+1 + ri ), V pc = (z i+1 − z i )(ri2 + ri+1 + ri ri+1 ), 2 i=1 6 i=1 (13) where the subscript i means any node of subzone pc and X5 ≡ X1 . Nodal volume V p , nodal area A p , cell volume Vc and cell area Ac are given by sums of subzones adjacent to cell centre c or point p: 4
4
A pc =
Vp =
V pc ,
Ap =
c( p)
A pc , Vc =
c( p)
V pc ,
Ac =
p(c)
A pc .
(14)
p(c)
Let us define subzonal mass m pc = c V pc , then the mass of primary cell m c and the mass of dual cell m p , respectively, can be defined similarly to volumes or areas: mc =
p∈ p(c)
m pc , m p =
m pc .
(15)
c∈c( p)
The basic Lagrangian assumption of no mass flux between cells implies that the cell masses remain constant.
3.2 Summary of Method for Cartesian Geometry Let us begin with short recapitulation of the method in Cartesian coordinates. The vector of conservative variables w, vectors of physical fluxes in x- and ydirection, respectively, f and g, and two column matrix of stacked physical fluxes F are defined by w=
1 (ν, u, v, E)t , f = (u, − p, 0, − pu)t , g = (v, 0, − p, − pv)t , F = ( f, g). m
As mentioned before, the Richtmyer’s FVM formulation of Lax–Wendroff method is used as the main ingredient of proposed scheme. Each step of basic scheme consists of predictor and corrector. In predictor, the nodal estimates of conservative quantities at time level n + 1/2 are computed using cellular quantities defined at time level n. These nodal estimates are then used to compute final LW update of values in cells (time level n + 1). Assuming that R = 1 for planar geometry, the 2D Cartesian version of the scheme can be written in the following form. 3.2.1
Predictor n+ 21
wp
=
1 Δt n n n m pc wcn + F · (nspc+ + nspc− ), m p c( p) 2m p c( p) c
(16)
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3.2.2
Corrector
wcn+1 = wcn +
3.2.3
Δt mc
n+ 1
n+ 1
2 Fe(ca) · ne(ca)2 ,
(17)
e(ca)∈e(c)
Mesh Positions Update
Node positions are updated using nodal velocity estimates: = Xnp + Δtun+1/2 , Xn+1 p p 3.2.4
Xn+1/2 = (Xn+1 + Xnp )/2 p p
(18)
Artificial Dissipation
The dissipative part of HLL solver is used as artificial dissipation term, which is added after each LW step. The state vector with added dissipation is denoted by additional superscript d. Proposed artificial dissipation has form: m c wcd,n+1 = m c wcn+1 + Δt
e(ca)∈e(c)
Dτ ·
σca σac |ne(ca) | d,n (wa − wcd,n ) σca + σac
(19)
where σca = ρc (ccs + |(uc − ua ) · ne(ca) |/|ne(ca) |), σca = ρc (ccs + |(uc − ua ) · ne(ca) |/|ne(ca) |) are modified signal velocities and Dτ = diag(τd , τu , τu , τe ) is diagonal matrix of dimensionless coefficients controlling the amount of added artificial dissipation and hence also the scheme variant. The main advantages the scheme are very good symmetry and the reduction of density dip caused by wall heating for 2D Cartesian Noh test which is described in Sect. 4.1. More detailed description of this method can be found in [1].
3.3 Method for Cylindrical Geometry Now, we will focus on extension of method mentioned in previous paragraph into cylindrical geometry. Similarly as in planar geometry, the Richtmyer’s finite volume formulation of Lax–Wendroff scheme is stabilized using dissipative part of Harten– Lax–van Leer approximate Riemann solver. Because density and volume can be computed directly from updated geometry, one has to discretize only the equations for momentum (11) and total energy conservation (12).
Cell-Centred Lagrangian Lax–Wendroff HLL Hybrid Schemes …
3.3.1
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Predictor n+1/2
Nodal velocity estimates u p
remain the same as in the Cartesian version:
Δt n n n n n c Acp uc + , p n = 2 c cp c∈c( p)
np Anp un+1/2 p
n+1/2
nodal estimate of total specific energy e p m p en+1/2 p
=
c∈c( p)
m cp ecn
(20)
is given by following formula:
Δt n n n p u · (r n) pc , + 2 c c
(21)
where n n n n (r n)npc = (rspc+ nspc+ + rspc− nspc− )/2.
3.3.2
Corrector
As mentioned before, the velocity update incorporates source terms arising from geometry: 1 n Δt n n . (22) ne(ca) ( pa − pcn )re(ca) ucn+1 = ucn − m c e(ca)∈e(c) 2 Total specific energy is updated by rz-version of LW: ecn+1 = ecn −
3.3.3
Δt n+ 21 n+ 21 n+ 21 n+ 21
pp up rp · np . m c p(c)
(23)
HLL Artificial Dissipation
The artificial dissipation term is similar to the Cartesian variant, with two differences: Each flux over the edge is weighted by radial pseudoradius (radial component of edge position) re(ca) and geometry correction term CAV is incorporated into the artificial viscosity. Artificial viscosity is given by following formula: ⎡ ucd,n+1 = ucn+1 − τu
n σca σac |re(ca) ne(ca) |
Δt ⎣ m c e(ca)
σca + σac
⎤
(uad,n − ucd,n ) + C AV ⎦ , (24)
where τu is dimensionless quantity controlling the amount of added artificial viscosity.
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C AV = (C AV , 0),
C AV =
uc 1 σac |ne(ac) ||Xa − Xc |2 , 2 a∈ve(c) rc
(25)
where by ve(c) we mean only the vertical edges of cell c. Artificial energy flux has form: ecd,n+1 = ecn+1 − τ E
Δt mc
e(ca)∈e(c)
σac σca |re(ca) nne(ca) | σac + σca
(E ad,n − E cd,n ) ,
(26)
where τ E is dimensionless coefficient controlling the amount of added energy flux.
3.4 Time Step Control The time step Δt is primarily controlled by the Courant–Friedrichs–Lewy (CFL) condition. At time level n, the time step is given by Δt
cf l
= Cc f l min c
Vcn s,n , cc
(27)
where Cc f l is CFL number. In addition to the CFL condition, the time step is not allowed to grow faster than by 10%, i.e. Δt n = min(Δt c f l , 1.1Δt n−1 ) and finally, if / (0.9, 1.1) for the volume of any cell changes by more than 10%, i.e. if Vcn+1 /Vcn ∈ some cell c, then the time step is reduced by half Δt n,r ed = Δt n /2 and the whole step is recomputed using reduced times step Δt n,r ed .
4 Numerical Results In this section, we advance to presentation of scheme performance on several classical tests for Lagrangian hydrodynamics. All tests except the first one are defined in cylindrical geometry and all except the last one were computed on initially rectangular grids.
4.1 Noh Problem in Cartesian Geometry Noh problem [6] is a classic test describing circularly symmetric implosion of ideal gas with γ = 5/3. The initial density is 1, the initial pressure is 0 (numerically we use 1e-6), and the initial velocities are directed towards the origin with unit magnitude. The solution is an infinite strength circularly symmetric shock reflecting from the origin. The problem has been computed till t = 0.6 with CC F L = 0.15. The initial mesh is square 0; 12 . Reflective wall boundary conditions are applied on axes
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(a)
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(b)
Fig. 3 Density scatter plots for Noh problem in planar geometry by LW+2 scheme with τ = 1.5, cc f l = 0.15 for 50 × 50, 100 × 100 and 200 × 200 meshes (a); Density contours on 50 × 50 mesh (b)
and free BC are used on outer boundaries. The same solution can be obtained when computing on all four quadrants. The solution of Noh by LW+2 can be seen in Fig. 3. Thanks to the artificial energy flux originally suggested in [6], the dip in density near y-axis caused by wall heating is significantly reduced. Symmetry of the scheme on initially rectangular mesh is very good.
4.2 Noh Problem in Cylindrical Geometry The initial and boundary conditions remain the same as for Cartesian variant of this test, but the solution is different. The exact solution of density together with convergence on meshes with different resolutions is plotted in Fig. 4.
4.3 Sedov Problem Sedov test [7] describes self-similar solution of spherical explosion. The square 0; 1.22 filled by ideal gas with γ = 1.4, = 1, p = 10−6 is initially at rest (u = 0). In single cell adjacent to the origin, the blast energy Eblast = 0.851072 is prescribed. The simulations are computed till t = 1. The density convergence for Sedov problem can be seen in Fig. 5. The solution converges to the exact one, and the symmetry of the solution is again well preserved.
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(a)
(b)
Fig. 4 Density scatter plots for Noh problem in cylindrical geometry by LWrz+2 scheme with τ = 1.25, cc f l = 0.2, for 50 × 50, 100 × 100 and 200 × 200 meshes (a); density contours on 50 × 50 mesh (b)
(a)
(b)
(c)
Fig. 5 Convergence of density scatter plots for Sedov problem by LWrz+2 scheme with τ = 1.5, cc f l = 0.2 on the initially rectangular meshes for: a n = 50; b n = 100; c n = 200
4.4 Spherical Sod Problem This test [4] is an extension of classical Sod shock tube into spherical geometry. The initial data of classical Sod shock tube problem are copied on every ray of polar 50 × 100 mesh (50 rays, 100 circles). The boundary conditions are free at the outer boundaries and reflecting on the axes. The initial conditions are
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(a)
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(b)
Fig. 6 Spherical Sod problem at t = 0.2 with τ = 0.125, cc f l = 0.2 and ‘interface fix’ [1] on polar mesh. Density plot (a); density colour map (b)
(1.0, 0, 1.0) r ≤ 0.5 (, u, p) = (0.125, 0, 0.1) r > 0.5
r=
x 2 + y2
(28)
The results for spherical Sod problem can be seen in Fig. 6.
5 Conclusions We have developed an extension of LW+n method into cylindrical geometry. The method has been applied to Euler equations in rz-geometry. The performance of proposed scheme has been tested on three standard tests, namely, Noh, Sedov and spherical Sod problem. The symmetry of results for all presented tests is very good, while other aspects of solutions seem to be reasonable. The presented method is fairly simple to implement and can be applied also on different systems of conservation laws. Acknowledgements D.F. and R.L. have been partially supported by the Czech Science Foundation projects 14-21318S and 18-20962S and by the Czech Ministry of Education project RVO 68407700 and by the Czech Technical University project SGS16/247/OHK4/3T/14.
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References 1. D. Fridrich, R. Liska, B. Wendroff, Some cell-centered Lagrangian Lax-Wendroff HLL hybrid schemes. J. Comput. Phys. 326, 878–892 (2016) 2. A. Harten, P.D. Lax, B. van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25(1), 35–61 (1983) 3. F.L. Addessio, J.R. Baumgardner, J.K. Dukowicz, N.L. Johnson, B.A. Kashiwa, R.M. Rauenzahn, C. Zemach, CAVEAT: A computer code for fluid dynamics problems with large distortion and internal slip (Technical report, Los Alamos National Laboratory, 1992) 4. P-H. Maire, A high-order cell-centered Lagrangian scheme for compressible fluid flows in twodimensional cylindrical geometry. J. Comput. Phys. 228(18), 6882–6915 (2009) 5. R. Loubère, M. Shashkov, A subcell remapping method on staggered polygonal grids for arbitrary-Lagrangian-Eulerian methods. J. Comput. Phys. 209(1), 105–138 (2005) 6. W.F. Noh, Errors for calculations of strong shocks using an artificial viscosity and artificial heat flux. J. Comput. Phys. 72, 78–120 (1987) 7. J.R. Kamm, F. X. Timmes, On efficient generation of numerically robust Sedov solutions. Technical Report LA-UR 07-2849 (Los Alamos National Laboratory, 2007)
Semilinear Shifted Wave Equation in the de Sitter Spacetime with Hyperbolic Spatial Part Anahit Galstian
Abstract In this paper, we discuss the issue of global existence in the Cauchy problem for the semilinear shifted wave equation in the de Sitter spacetime with hyperbolic spatial part. We give an estimate of the lifespan of the solution and obtain Strichartz type estimates. Keywords de Sitter spacetime · Global existence · Semilinear wave equation
1 Introduction In this paper, we discuss the issue of global existence in the Cauchy problem for the semilinear shifted wave equation in the de Sitter spacetime with hyperbolic spatial part. We give an estimate of the lifespan of the solution and obtain Strichartz type estimates. We consider the semilinear wave equation in the spacetime, which is produced by an expanding universe, more exactly, in the de Sitter spacetime. The line element of that spacetime is as follows: ds 2 = − c2 dt 2 + e2H t dr 2 + e2H t r 2 (dθ 2 + sin2 θ dφ 2 ) . Here, H is the Hubble constant. The scale factor e2H t represents an expansion. This spacetime belongs to the family of the Friedmann–Lemaître–Robertson–Walker spacetimes. For simplicity, we set H = 1. The linear wave in the background generated by the metric g obeys the covariant wave equation g ψ = f , where
A. Galstian (B) School of Mathematical and Statistical Sciences, University of Texas RGV, 1201 West University Dr., Edinburg, TX 78539, USA e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_44
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1 ∂ g ψ = √ |g| ∂ x i
|g|g ik
∂ψ ∂xk
.
The fundamental solutions for the covariant wave equation in the de Sitter spacetime with flat spatial part, as a particular case of Klein–Gordon massless equation, are constructed in [10]. The L p − L q estimates for the solutions of the Cauchy problem are obtained in [10] as well. The global-in-time existence in the energy class of solutions of the Cauchy problem for the semilinear Klein–Gordon equation in the Friedmann–Lemaître–Robertson–Walker spacetimes was proved in [2]. In [1], the following Cauchy problem for the covariant non-shifted semilinear wave (massless field) equation in the de Sitter spacetime was considered
ψtt − e−2t ΔH ψ + nψt = F(ψ) , x ∈ Hn , t ∈ [0, ∞), ψ(x, 0) = ψ0 (x) , ψt (x, 0) = ψ1 (x), x ∈ Hn ,
where Hn is a hyperbolic space and ΔH is Laplace–Beltrami operator on L 2 (Hn ). The real hyperbolic spaces Hn are the most simple examples of noncompact Riemannian manifolds with negative curvature. For geometric reasons, one can expect better dispersive properties and, consequently, stronger results than in the Euclidean setting. Henceforth we assume that n = 3. For the Cauchy problem for the semilinear wave equation in Minkowski spacetime in this case, ∂t2 u − Δu = |u|1+α u(0, x) = u 0 (x) , u t (0, x) = u 1 (x) , the surprising answer, which √ is due to John, is that for small data, a global solu√ tion always exists when α > 2, but does not, in general, when α < 2. For the higher dimensional semilinear wave equations, the following conjecture was stated by Strauss [6]: for n ≥ 2 blow-up for all data if p < pn and global existence for all small data, if p > pn . Here, p = α + 1, and pn is the positive root of the equation (n − 1) pn2 − (n + 1) pn − 2 = 0. (For the history of the results which have validated Strauss’s conjecture see, e.g. [3, 5] and the bibliography therein.) In [1], the estimates for the lifespan of the solutions of semilinear wave equation in the de Sitter spacetime with flat and hyperbolic spatial parts under some conditions on the order of the nonlinearity were obtained. In the case of hyperbolic spatial part, the order of nonlinearity was less than the critical value given by Strauss conjecture. Consider now the shifted wave operator H = ∂t2 − A(x, ∂x ) in the hyperbolic space, that is, the conformal Laplace–Beltrami operator on a pseudo-Riemannian manifold of dimension n + 1 = 4: H := −∂t2 + Δ H + ρ 2 , ρ =
n−1 , 2
A(x, ∂x ) = Δ H + ρ 2 , Δ H := ∂r2 + 2ρ(coth(r ))∂r + (sinh(r ))−2 ΔSn−1 .
Semilinear Shifted Wave Equation in the de Sitter Spacetime …
579
The definition of the inhomogeneous Sobolev spaces H s, p (Hn ) in the hyperbolic space Hn one can find in Theorem 2 of [7]. In [7], the Strichartz estimates for the shifted wave equation in the hyperbolic space were derived. For the wave equation on three-dimensional hyperbolic space, global-in-time dispersive estimates and Strichartz estimates were obtained in [4]. In the present paper, we consider the Cauchy problem for the covariant shifted semilinear wave (massless field) equation in the de Sitter spacetime ψtt − e−2t ΔH + ρ 2 ψ + nψt = F(ψ) , x ∈ Hn , t ∈ [0, ∞), ψ(x, 0) = ψ0 (x) , ψt (x, 0) = ψ1 (x), x ∈ Hn .
(1)
To the best of our knowledge, the question of the small data global solution is not examined for this equation. Denote by G the solution operator of the linear problem: ψtt − e−2t ΔH + ρ 2 ψ + nψt = f , x ∈ Hn , t ∈ [0, ∞), ψ(x, 0) = ψ0 (x) , ψt (x, 0) = ψ1 (x), x ∈ Hn .
(2)
with ψ0 (x) = ψ1 (x) = 0, that is, ψ = G[ f ]. The solution operator G will be described in Sect. 3. Let ψ0 is a solution of the problem (2) with f = 0. Any solution ψ of the problem (1) solves also the linear integral equation ψ(x, t) = ψ0 (x, t) + G[F(ψ(·, τ )](x, t), t > 0 . We define the solution of the problem (1) as a solution of the last integral equation. To formulate the results of this paper, we need the following description of the non-linear term. Condition (L ). The function F is said to be Lipschitz continuous with exponent α ≥ 0 in the space H s if there is a constant C ≥ 0 such that F(ψ1 ) − F(ψ2 ) H s ≤ Cψ1 − ψ2 H s ψ1 αH s + ψ2 αH s for all ψ1 , ψ2 ∈ H s .
The following theorem is the main result of this paper. Theorem 1. Consider the Cauchy problem (1) for the covariant shifted semilinear wave equation in the hyperbolic space with F(ψ) satisfying the condition (L ). Assume that n = 3. Then for every ψ0 , ψ1 ∈ H s , s > n/2, there exist T = T (ψ0 , ψ1 ) > 0 and solution ψ ∈ C([0, T (ψ0 , ψ1 )]; H s ) of the problem (1). Denote T = T (δ), where δ > 13 , the non-negative solution of the equation 1 T + e−3T = δ . 3
580
A. Galstian
Then, the lifespan Tls of the solution ψ ∈ C([0, T (ψ0 , ψ1 )]; H s ), s > n/2, with ψ0 , ψ1 ∈ C0∞ (Hn ) can be estimated as follows: Tls ≥ T
−α 1 1 −1−α ψ0 H s + ψ1 H s +3·2 . 3 3
(3)
It is easy to derive from the theorem the following asymptotic for the lifespan of the solution with small initial data: −α 1 −1−α ψ0 H s + ψ1 H s . Tls ∼ 3 · 2 3
2 L p − L q Estimates We use Yagdjian’s integral transform of [9] to prove Theorem 1. That integral transform creates a bridge between solutions of wave equation in the de Sitter spacetime and the wave equation in Minkowski spacetime. Through this transform, we derive, in particular, the estimates for the solutions. Let Ω be
a domain in Rn , Ω ⊆ Rn , while A(x, ∂x ) is a partial differential operator A(x, ∂x ) = |α|≤m aα (x)Dxα with smooth coefficients. For g ∈ C ∞ (Ω × I ), I = [0, T ], 0 < T ≤ ∞, and ϕ0 , ϕ1 ∈ C0∞ (Ω), let the function vg (x, t; b) be a solution to the problem vtt − A(x, ∂x )v = 0 , x ∈ Ω , t ∈ [0, 1 − e−T ] , v(x, 0; b) = g(x, b) , vt (x, 0; b) = 0 , b ∈ I, x ∈ Ω , and let the function vϕ = vϕ (x, t) be a solution of the problem
vtt − A(x, ∂x )v = 0, x ∈ Ω , t ∈ [0, 1 − e−T ] , v(x, 0) = ϕ(x), vt (x, 0) = 0 , x ∈ Ω .
Then, the function u = u(x, t) defined by
φ(t)−φ(b)
1 3 vg (x, r ; b) e 2 (b+t) (e−2b + e−2t ) − r 2 dr 4 0 0 φ(t) 1 t t +e 2 vϕ0 (x, φ(t)) + 2 vϕ0 (x, s) e− 2 3s 2 + 1 e2t − 3 ds 8 0 φ(t) 1 −t +2 vϕ1 (x, s) e 2 1 − e2t s 2 − 1 ds, x ∈ Ω, t ∈ I , 4 0
u(x, t) = 2
t
db
where φ(t) := 1 − e−t , according to Theorem 2.1 [9], solves the problem
(4)
Semilinear Shifted Wave Equation in the de Sitter Spacetime …
581
u tt − e−2t A(x, ∂x )u − 94 u = g, x ∈ Ω , t ∈ I, u(x, 0) = ϕ0 (x) , u t (x, 0) = ϕ1 (x), x ∈ Ω .
Consequently, the function ψ(x, t) = e− 2 t u(x, t) solves the problem for the covariant wave equation: 3
ψtt − e−2t A(x, ∂x )ψ + 3ψt = f, x ∈ Ω , t ∈ I, ψ(x, 0) = ψ0 (x) , ψt (x, 0) = ψ1 (x), x ∈ Ω ,
(5)
3
where g = e 2 t f, ϕ0 = ψ0 , ϕ1 = 23 ψ0 + ψ1 . Then, 1 ψ(x, t) = 2
t
φ(t)−φ(b)
v f (x, r ; b)e3b e−2b + e−2t − r 2 dr
db 0
0
+ e−t vψ0 (x, φ(t)) + +
1 2
φ(t)
vψ0 (x, s)ds
0
φ(t)
vψ1 (x, s) 1 + e−2t − s 2 ds, x ∈ Ω, t ∈ I ,
(6)
0
where φ(t) := 1 − e−t (see [8, 9]). In order to reduce the integration in the above formula, we can appeal to the solutions Vg = Vg (x, t; b) and Vϕ = Vϕ (x, t) of the problems Vtt − A(x, ∂x )V = 0 , x ∈ Ω , t ∈ [0, 1 − e−T ] , (7) V (x, 0; b) = 0 , Vt (x, 0; b) = g(x, b) , b ∈ I, x ∈ Ω , and
Vtt − A(x, ∂x )V = 0, x ∈ Ω , t ∈ [0, 1 − e−T ] , V (x, 0) = 0, Vt (x, 0) = ϕ(x) , x ∈ Ω ,
respectively. Then, vg (x, t; b) = ∂t Vg (x, t; b), vϕ (x, t) = ∂t Vϕ (x, t) and ψ(x, t) = e−t
+
t
V f (x, e−b − e−t ; b)e2b db
0 t
db e
0 −t
3b
e−b −e−t
V f (x, r ; b)r dr
0
+ e vψ0 (x, 1 − e−t ) + Vψ0 (x, 1 − e−t ) 1−e−t −t −t Vψ1 (x, s)s ds . + e Vψ1 (x, 1 − e ) +
(8)
0
Then, according to Theorem 3 [7], for the solution Vψ of the problem (7), and consequently, for vψ of (4), the following estimates hold:
582
A. Galstian
vψ (t) H −s, p ≤ (sinh(t))
−ρ
1 p
− 1p
2 p
Vψ (t) H −s+1, p ≤ (1 + t) (sinh(t))
ψ H s, p , for all t ∈ (0, ∞), −ρ
1 p
−
1 p
(9)
ψ H s, p , for all t ∈ (0, ∞),
(10)
where ρ=
1 n−1 1 , 2 ≤ p < ∞, + = 1 , ρ 2 p p
1 1 − p
p
< 1 , 2s = (ρ + 1)
1 1 − p
p
.
(11)
Note that the representation formulas (6) and (8) are valid for Ω = Hn (see Page 5, [9]) . For the solution ψ = ψ(x, t) of the problem (5), the estimates (9), (10) can be used to prove the following L p − L q estimate. Proposition 1. Assume that the conditions (11) are satisfied. The solution ψ = ψ(x, t) of the problem ψtt − e−2t Δ H + ρ 2 ψ + 3ψt = f , x ∈ H3 , t ∈ [0, ∞), (12) ψ(x, 0) = ψ0 (x) , ψt (x, 0) = ψ1 (x), x ∈ H3 , satisfies the following inequality: ψ(t) H −s, p e−b −e−t
1 t −ρ p1 − 1p −2b e ≤ f (b) H s, p e3b db (sinh(r )) + e−2t − r 2 dr 2 0 0 1−e−t
1 1 −ρ p1 − 1p −t −t −ρ p − p
ψ0 H s, p + ψ0 H s, p (sinh(r )) dr +e (sinh(1 − e ))
1 + ψ1 H s, p
2
1−e−t
(sinh(r ))
−ρ
1 p
− 1p
0
1 + e−2t − r 2 dr
(13)
0
for all t ∈ [0, ∞). Proof. According to the representation formula (6), we have ψ(t) H −s, p φ(t)−φ(b) 1 t ≤ db v f (x, r ; b) H −s, p e3b e−2b + e−2t − r 2 dr 2 0 0 φ(t) + e−t vψ0 (x, φ(t)) H −s, p + vψ0 (x, r ) H −s, p dr 1 + 2
0 φ(t)
vψ1 (x, r ) H −s, p 1 + e−2t − r 2 dr .
0
Applying (9) to the last estimate, we obtain (13). Proposition is proven.
Corollary 1. Assume p = p = 2. For every ∈ R, the solution of the problem (12) satisfies the following inequality:
Semilinear Shifted Wave Equation in the de Sitter Spacetime …
ψ(t) H ≤
1 3
0
t
583
1 f (b) H 1 − e−3(t−b) db + ψ0 H + ψ1 H 3
(14)
for all t ∈ [0, ∞). Proof. The definition of the Sobolev space H s (Hn ) = H s,2 (Hn ) in the hyperbolic space Hn given in [7] and the inequality (13) allow us to write for every s ∈ R the following estimate: e−b −e−t
1 t e−2b + e−2t − r 2 dr f (b) H s e3b db 2 0 0 1−e−t
1 1 + e−2t − r 2 dr + e−t ψ0 H s + (1 − e−t )ψ0 H s + ψ1 H s 2 0
1 t 3b −3b −3t e db f (b) H s e −e ≤ 3 0 2 2e−3t 1 −t −t (15) + e ψ0 H s + (1 − e )ψ0 H s + ψ1 H s − . 2 3 3
ψ(t) H s ≤
The estimate (14) easily follows from (15). Corollary is proven. 3 2t
Corollary 2. Assume that A(D)2 = −(Δ H + ρ 2 ). For u = e ψ define the energy 1 1 9 E (t) := u t (t)2H + e−2t A(D)u(t)2H − u(t)2H 2 2 8 1 1 9 3 3 = (e 2 t ψ)t (t)2H + et A(D)ψ(t)2H − e 2 t ψ(t)2H . 2 2 8 For every ∈ R, the solution of the problem (12) with f = 0 satisfies d E (t) ≤ 0 for all t ∈ [0, ∞) . dt 3
Proof. If ψ is the solution of the problem (12) with f = 0, then function u = e 2 t ψ solves the following problem: u tt − e−2t Δ H + ρ 2 u − 49 u = 0 , x ∈ Hn , t ∈ [0, ∞), u(x, 0) = ψ0 (x) , ψt (x, 0) = 23 ψ0 (x) + ψ1 (x), x ∈ Hn . It is enough to prove corollary for = 0. In that case, we have
584
A. Galstian
d E 0 (t) = (u t (t), u tt (t)) L 2 − e−2t A(D)u(t)2L 2 + e−2t (A(D)u t (t), A(D)u(t)) L 2 dt 9 − (u t (t), u(t)) L 2 4 9 = (u t (t), e−2t (Δ H + ρ 2 )u + u) L 2 − e−2t A(D)u(t)2L 2 4 9 −e−2t (u t (t), (Δ H + ρ 2 )u(t)) L 2 − (u t (t), u(t)) L 2 4 = −e−2t A(D)u(t)2L 2 ≤ 0 .
Corollary is proven.
3
Proof of Theorem 1
From Corollary 1, we obtain ψ(t) H s
1 ≤ 3
t
0
1 1 − e−3(t−b) db + ψ0 H s + ψ1 H s . ψ(b)1+α Hs 3
(16)
Denote the complete metric space X sM := ψ ∈ C([0, T ]; H s ) | ψ X M := max ψ(t) H s ≤ M [0,T ]
with the metric d(ψ, ω) := max ψ(t) − ω(t) H s . [0,T ]
According to [9], the solution operator G = K ◦ C is an integral operator. Here, C is a solution operator from [7], which solves the following Cauchy problem:
wtt − ΔH + ρ 2 w = 0 , x ∈ Hn , t ∈ [0, T1 ) ⊆ R, w(x, 0; b) = f (x, b) , wt (x, 0; b) = 0, x ∈ Hn ,
with parameter b ∈ I = [t0 , T ] ⊆ R, t0 < T ≤ ∞, and with 0 < T1 ≤ ∞. Operator K , which is given by Yagdjian in [9], is
K [w](x, t) = 2
t
φ(t)−φ(b) db
0
0
∈ C. K (t; r, b; M)w(x, r ; b) dr , x ∈ Ω, t ∈ I, M
= 3 . According to Corollary 1, we have Here, Ω = H3 and M 2
Semilinear Shifted Wave Equation in the de Sitter Spacetime …
G[ f ](t) H s ≤
1 3
t
585
f (b) H s 1 − e−3(t−b) db .
0
Consider the map (x, t) + G[F(ψ)](x, t) , (Sψ)(x, t) := ψ ∈ X sM/2 is a given function. Then for ψ ∈ X sM , we have where ψ X s + G[F(ψ)] X s (Sψ) X sM ≤ ψ M M 1 t X s + 1 − e−3(t−b) db ≤ ψ ψ(b)1+α Hs M 3 0 T 1 1+α s 1 − e−3(T −b) db ≤ ψ X M + ψ X M 3 0 1 1+α 1 1 1 T − + e−3T . ≤ M+ M 2 3 3 3 If T > 0 and M > 0 are such that 1 1 1 −3T 1 1+α T− + e < M, M+ M 2 3 3 3 then Sψ ∈ X sM . Similarly, we obtain the contraction property from Sψ − Sω X sM ≤
1 α 1 1 M d(ψ, ω) T − + e−3T . 3 3 3
be a solution of the Cauchy problem Let ψ
ψtt − e−2t Δ H + ρ 2 ψ + nψt = 0 , x ∈ Hn , t ∈ [0, ∞), ψ(x, 0) = ψ0 (x) , ψt (x, 0) = ψ1 (x), x ∈ Hn .
Then from estimate (16), we have 1 X s ≤ ψ0 H s + ψ1 H s . ψ M 3 Banach’s fixed-point theorem completes the proof of existence of a local solution. In order to estimate the lifespan of solution, we denote ε := ψ0 H s + Then
1 ψ1 H s , 3
E(t) := max{ψ(τ ) H s : 0 ≤ τ ≤ t} .
586
A. Galstian
E(t) ≤ ε +
1 3
t
E(t)1+α 1 − e−3(t−b) db .
0
Set Tε := inf{t : E(t) ≥ 2ε} . Then E(Tε ) ≤ ε + implies
1 3
Tε
E(t)1+α 1 − e−3(Tε −b) db
0
1 1 1 −3Tε 1+α Tε − + e 2ε ≤ ε + (2ε) 3 3 3
and
1 1 Tε + e−3Tε ≥ + 3 · 2−1−α ε−α . 3 3
(17)
The estimate for the lifespan (3) follows from (17). Theorem is proven.
Corollary 3. The lifespan Tls → ∞ as ψ0 H s + 13 ψ1 H s → 0.
4 Strichartz Type Estimates The representation formula (8) also can be used to obtain Strichartz type estimates. Proposition
2. Assume that conditions (11) are satisfied and ∈ R. Denote γ = 1 1 ρ p − p . The solution ψ = ψ(x, t) of the problem (12) satisfies the inequality: ψ(t) H −s+1+ , p t e−b −e−t 2 ≤ db e3b f (b) H s+ , p
(1 + r ) p (sinh(r ))−γ r dr 0 0 t 2 (1 + e−b − e−t ) p (sinh(e−b − e−t ))−γ f (b) H s+ , p e2b db + e−t 0 2
2
+ ψ0 H s+1+ , p 21+ p (sinh(1 − e−t ))−γ + ψ1 H s+ , p e−t 2 p (sinh(1 − e−t ))−γ 1−e−t 2
+ψ1 H s+ , p (1 + s) p (sinh(s))−γ s ds for all t ∈ [0, ∞). 0
Proof. It suffices to consider the case of = 0. From the representation formula (8) and estimates (9), (10), we obtain
Semilinear Shifted Wave Equation in the de Sitter Spacetime …
ψ(t) H −s+1, p ≤
t
e−b −e−t
587 2
db e f (b) H s, p
(1 + r ) p (sinh(r ))−γ r dr 0 0 t 2 −t (1 + e−b − e−t ) p (sinh(e−b − e−t ))−γ f (b) H s, p e2b db +e 3b
0 2
+ ψ0 H s+1, p 21+ p (sinh(1 − e−t ))−γ 2
+ ψ1 H s, p e−t 2 p (sinh(1 − e−t ))−γ 1−e−t 2
+ψ1 H s, p (1 + s) p (sinh(s))−γ s ds . 0
Proposition is proven.
Corollary 4. Assume that p = p = 2 and ∈ R. The solution ψ = ψ(x, t) of the problem (12) satisfies the following inequality: ψ(t) H 1+ ≤
2 1 − eb−t 3eb − 2eb−t + 2 f (b) H db 0 t −t e2b (1 + e−b − e−t ) f (b) H db +e 1 6
t
0
+ 4ψ0 H 1+ + 2ψ1 H
for all t ∈ [0, ∞) .
References 1. A. Galstian, Semilinear wave equation in the de sitter spacetime with hyperbolic spatial part, in New Trends in Analysis and Interdisciplinary Applications, Trends in Mathematics (Springer International Publishing, Berlin, 2017). https://doi.org/10.1007/978-3-319-48812-7-62 2. A. Galstian, K. Yagdjian, Global solutions for semilinear Klein-Gordon equations in FLRW spacetimes. Nonlinear Anal. 113, 339–356 (2015) 3. V. Georgiev, H. Lindblad, C.D. Sogge, Weighted Strichartz estimates and global existence for semilinear wave equations. Am. J. Math. 119, 1291–1319 (1997) 4. J. Metcalfe, M. Taylor, Nonlinear waves on 3D hyperbolic space. Trans. Am. Math. Soc. 363, 3489–3529 (2011) 5. C.D. Sogge, Lectures on Non-linear Wave Equations, 2nd edn. (International Press, Boston, 2008) 6. W.A. Strauss, Nonlinear scattering theory at low energy. J. Funct. Anal. 41, 110–133 (1981) 7. D. Tataru, Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation. Trans. Am. Math. Soc. 353, 795–807 (2001) 8. K. Yagdjian, Huygens’ principle for the Klein-Gordon equation in the de Sitter spacetime. J. Math. Phys. 54, 091503 (2013) 9. K. Yagdjian, Integral transform approach to solving Klein-Gordon equation with variable coefficients. Mathematische Nachrichten 288, 1–24 (2015) 10. K. Yagdjian, A. Galstian, Fundamental Solutions for the Klein-Gordon Equation in de Sitter Spacetime. Commun. Math. Phys. 285, 293–344 (2009)
Convergence Rates of a Fully Discrete Galerkin Scheme for the Benjamin–Ono Equation Sondre Tesdal Galtung
Abstract We consider a recently proposed fully discrete Galerkin scheme for the Benjamin–Ono equation which has been found to be locally convergent in finite time for initial data in L 2 (R). By assuming that the initial data is sufficiently regular, we obtain theoretical convergence rates for the scheme both in the full line and periodic versions of the associated initial value problem. These rates are illustrated with some numerical examples. Keywords Benjamin–Ono equation · Finite element method · Convergence rates 2010 Mathematics Subject Classification 65M12 · 65M15 · 65M60 · 35Q53
1 Background We will in the following consider the Benjamin–Ono (BO) equation [2, 7] which serves as a generic model for weakly nonlinear long waves with nonlocal dispersion. Its initial value problem reads
u t + uu x − Hu x x = 0, (t, x) ∈ (0, T ] × R, x ∈ R, u(0, x) = u 0 (x),
(1)
where H denotes the Hilbert transform defined by Hu(·, x) := p.v.
1 π
R
u(·, x − y) dy, y
S. T. Galtung (B) Department of Mathematical Sciences, NTNU Norwegian University of Science and Technology, NO-7491, Trondheim, Norway e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_45
589
590
S. T. Galtung
for which p.v. denotes the Cauchy principal value. We may also consider the 2Lperiodic IVP for the BO equation
u t + uu x − Hper u x x = 0, (t, x) ∈ (0, T ] × T, x ∈ T, u(0, x) = u 0 (x),
(2)
where T := R/2LZ, and Hper denotes the 2L-periodic Hilbert transform defined by Hper u(·, x) := p.v.
1 2L
L −L
u(·, x − y) cot
π y dy. 2L
Based on a method for the Korteweg–de Vries equation due to Dutta and Risebro [4], Galtung [5] proposed a fully discrete Crank–Nicolson Galerkin scheme for (1) where an inherent smoothing effect is used to prove convergence locally for initial data u 0 in L 2 (R) and a finite time T which depends on u 0 L 2 . The scheme for (1) is defined in the following way. First one discretizes a subset of the real line by dividing it in intervals of equal length Δx, I j = [x j−1 , x j ], where x j := jΔx, j ∈ Z. For the temporal discretization, one analogously has tn = nΔt, n ∈ {0, 1, . . . , N }, for a discretization parameter Δt such that T = (N + 1/2)Δt. Let us also for convenience define tn+1/2 := (tn + tn+1 )/2. Consider now the following finite-dimensional subspace of the Sobolev space H 2 (R): SΔx = {v ∈ H 2 (R) | v ∈ Pr (I j ), j ∈ Z},
(3)
where r ≥ 2 is a fixed integer and Pr (I ) denotes the space of polynomials on the interval I of degree less than or equal to r . Given R > 0, we define ϕ ∈ C ∞ (R), for which the derivative is a cutoff function, satisfying the following conditions: 1. 2. 3. 4.
1 ≤ ϕ(x) ≤ 2 + 2R, ϕ (x) = 1 for |x| < R, ϕ (x) = 0 for |x| ≥ R + 1, and 0 ≤ ϕ (x) ≤ 1 for all x.
This function plays a key role in establishing the previously mentioned smoothing effect for the scheme, and it may be chosen to be point symmetric in (0, ϕ(0)). We need a reasonable approximation of u 0 in (1) as initial data u 0 for our scheme, and so we set u 0 = Pu 0 , where P is the L 2 -projection on SΔx . Now, we define a N of the exact solution at each tn by the following sequence of approximations {u n }n=0 n+1 ∈ SΔx such that procedure: find u
Δt n+1/2 2 u n+1 , ϕv − u , (ϕv)x + Δt H u n+1/2 x , (ϕv)x = u n , ϕv , 2
(4)
for all v ∈ SΔx , where u 0 is defined as before and u n+1/2 := (u n + u n+1 )/2. Here, ·, · is the standard L 2 -inner product. Note that the inner product ·, ·ϕ =: ·, · ϕ
Convergence Rates of a Fully Discrete Galerkin Scheme …
591
defines a norm which we denote · 2,ϕ . The nonlinearity appearing in the above implicit scheme calls for some form of iterative method to solve (4) for each time step, and in [5] the following linearized scheme is used: ⎧ ⎨w+1 , ϕv − ⎩
Δt 2
w +u n 2
2
, (ϕv)x + Δt
+1 n H w 2+u , (ϕv)x = w , ϕv , x
w0 = u n ,
(5) which is to hold for all v ∈ SΔx . By assuming a CFL condition of the type Δt = O(Δx 2 ), the above iteration is shown to converge to the solution u n+1 of (4). From this, one can show that there exists T > 0 such that u Δx , which is a piecewise linear 1/2 interpolation of each u n , belongs to the space L 2 (0, T ; Hloc (R)). Then, compactness arguments yield the convergence result. Because a monotone increasing cutoff function is incompatible with the periodicity of (2) one cannot use the same arguments to prove convergence for L 2 -initial data in this case, and so other tools are called for when considering low regularity initial data for the periodic BO equation. However, in this study we will assume the initial data to be as regular as needed, and so we will consider the convergence rate of the method in best-case scenarios. The established well-posedness of the BO equation for these more regular spaces then guarantees that the exact solution at all times is at least as regular as the initial data. This will even make us able to consider the periodic IVP (2) using a slightly adapted scheme where we have simply replaced the cutoff function ϕ with 1 wherever it appears. In the upcoming analysis, we need some preliminary estimates for polynomial approximations in finite element spaces. For a function v ∈ SΔx , we have the following inverse inequalities: C |v| H k (R) , k = 0, 1, (Δx)1/2 C |v| H k (R) , ≤ k = 0, 1, Δx
|v|W k,∞ (R) ≤
(6)
|v| H k+1 (R)
(7)
where the constant C is independent of v and Δx. Both here and in the following, | · |W k, p (R) denotes the seminorm of the Sobolev space W k, p (R) for which H k (R) := W k,2 (R). The reader is referred to [3, p. 142] for a proof of the above inequalities. Let us now consider two projections P : L 2 (R) → SΔx and Pϕ : L 2 (R) → SΔx defined, respectively, by (Pu − u) v dx = 0, v ∈ SΔx ,
(8)
Pϕ u − u ϕv dx = 0, v ∈ SΔx .
(9)
R
and
R
592
S. T. Galtung
For these projections applied to a function u ∈ H 2 (R), we have the bounds P0 u L 2 (R) ≤ Cu L 2 (R) , P0 u H 1 (R) ≤ Cu H 1 (R) , P0 u H 2 (R) ≤ Cu H 2 (R) ,
(10)
where P0 denotes either of the two projections and C is a constant which is independent of Δx. These bounds can be derived from the norm equivalence in a finitedimensional space and the definitions of these projections. We also have the following polynomial approximation error estimate on the discretized domain Ω. Given u ∈ H l+1 (Ω), 0 ≤ m ≤ l and s := min{l, r }, then |P0 u − u| H m (Ω) ≤ CΔx s+1−m |u| H s+1 (Ω) , m = 0, 1, 2,
(11)
where again P0 denotes either of the two projections and C is a constant not depending on Δx. For a proof of (11) for P, we refer to [8, p. 98], and the result for Pϕ follows from an adaption of the same proof. The following properties of the Hilbert transform, which can be found in [6, p. 317], are also useful: Hu, v = − u, Hv for u, v ∈ L 2 (R), (Hu)x = Hu x , Hu L 2 (R) = u L 2 (R) . Note that these properties hold analogously for the 2L-periodic Hilbert transform Hper on T with L 2 (T) = L 2 ([−L , L]), except that Hper u L 2 (T) ≤ u L 2 (T) .
2 Analysis of Convergence Rates In the following, we want to consider the L 2 -norm of the difference u n − u(tn ), and we will do so by decomposing the error as u n − u(tn ) = (u n − P0 u(tn )) + (P0 u(tn ) − u(tn )) =: τ n + ρ n , and we will use the notation wn := P0 u(tn ) for the sake of brevity. Here, P0 = Pϕ in the full line case, and P0 = P for the periodic case. For ρ n , we already have estimates for the L 2 -norm by virtue of (11), and so it remains to estimate the norm of τ n . As the analysis is similar for the full line and periodic problems, we will give detailed estimates for the former case and only indicate the main differences between the two for the latter case. Note that in the following, C will denote a constant which exact value is of no importance. Similarly, C(R) will denote such a constant which depends on R and so on. When we write, e.g., L 2 it is understood from context if we are referring to L 2 (R) or L 2 (T) = L 2 ([−L , L]). For both the full line and periodic case, we have the following result which is proved in the next subsections.
Convergence Rates of a Fully Discrete Galerkin Scheme …
593
Theorem 1. Given sufficiently regular initial data u 0 , say u 0 ∈ H max{r +1,6} , for the IVP of the BO equation, we have the following convergence rate for the fully discrete Galerkin scheme described in the previous section: u n − u(tn ) L 2 = O(Δx r −1 + Δt 2 ), n = 0, . . . , N .
(12)
2.1 Full Line Problem From multiplying (1) by ϕv, where v ∈ H 2 , and integrating by parts, we get u t (t), ϕv −
1 u(t)2 , (ϕv)x + Hu x (t), (ϕv)x = 0, t ∈ (0, T ]. 2
(13)
From (4), (13), and (9), we are able to write
n+1 n+1 τ n+1 − τ n u w − un − wn , ϕv = , ϕv − , ϕv Δt Δt Δt n+1 n −u u , ϕv − u t (tn+1/2 ), ϕv = Δt u(tn+1 ) − u(tn ) , ϕv + u t (tn+1/2 ) − Δt 1
κ n+1/2
= − (u n+1/2 )2 − u(tn+1/2 )2 , (ϕv)x 2 + H(u n+1/2 − u x (tn+1/2 )), (ϕv)x + κ n+1/2 , ϕv , x for v ∈ SΔx . As we are now considering u evaluated at tn+1/2 , we cannot use the previous decomposition of the error directly, but we instead write u n+1/2 − u(tn+1/2 ) = τ n+1/2 + ρ n+1/2 +
u(tn+1 ) + u(tn ) − u(tn+1/2 ) . 2 σ n+1/2
Then, we may rewrite part of the nonlinear term as (u n+1/2 )2 − u(tn+1/2 )2 = (τ n+1/2 )2 + 2τ n+1/2 wn+1/2 + (wn+1/2 )2 − u(tn+1/2 )2 = (τ n+1/2 )2 + 2τ n+1/2 wn+1/2 + (wn+1/2 + u(tn+1/2 ))(ρ n+1/2 + σ n+1/2 ).
594
S. T. Galtung
In the following, we want to use τ n+1/2 ∈ SΔx as test function, and from integrating by parts, we get the following relevant identities: 1 (τ n+1/2 )2 , (ϕτ n+1/2 )x = − (τ n+1/2 )3 , ϕx , 3 2 τ n+1/2 wn+1/2 , (ϕτ n+1/2 )x = − (τ n+1/2 )2 , ϕwxn+1/2 + (τ n+1/2 )2 , ϕx wn+1/2 .
Inserting this in the previous equations, we get 1 1 n 2 1 1 n+1 2 τ 2,ϕ = τ 2,ϕ + Δt − (τ n+1/2 )3 , ϕx − (τ n+1/2 )2 , ϕwxn+1/2 2 2 6 2 1 n+1/2 2 (τ + ) , ϕx wn+1/2 2 1 n+1/2 + (w + u(tn+1/2 ))(ρ n+1/2 + σ n+1/2 ), (ϕτ n+1/2 )x 2 − Hτxn+1/2 , (ϕτ n+1/2 )x − Hρxn+1/2 , (ϕτ n+1/2 )x − Hσxn+1/2 , (ϕτ n+1/2 )x + κ n+1/2 , ϕτ n+1/2 . From the commutator estimates presented in [5], we have the inequalities √ 2 2 Hwx , (ϕw)x ≥ ϕx D 1/2 w L 2 − Cw L2 , and
2 √ w3 , ϕx ≤ ϕx D 1/2 w L 2 + C(1 + w2L 2 )w2L 2
for w ∈ H 2 . By inserting these in the preceding identity and using the L 2 -isometry of the Hilbert transform, we obtain √ 2 1 n+1 2 n+1/2 22,ϕ τ 2,ϕ + Δt ϕx D 1/2 τ n+1/2 L 2 − Δt Cτ 2 1 Δt √ϕx D 1/2 τ n+1/2 2 2 ≤ τ n 22,ϕ + L 2 3 n+1/2 n+1/2 2 + Δt C(1 + u −w L 2 )τ n+1/2 22,ϕ 1 1 + wxn+1/2 L ∞ τ n+1/2 22,ϕ + wn+1/2 L ∞ τ n+1/2 22,ϕ 2 2 1 + wxn+1/2 + u x (tn+1/2 ) L ∞ (ρ n+1/2 2,ϕ + σ n+1/2 2,ϕ )τ n+1/2 2,ϕ 2 1 + wn+1/2 + u(tn+1/2 ) L ∞ (ρxn+1/2 2,ϕ + σxn+1/2 2,ϕ )τ n+1/2 2,ϕ 2 + C R ρxn+1/2 L 2 (τ n+1/2 2,ϕ + C R τxn+1/2 L 2 ) + C R σxn+1/2 L 2 τ n+1/2 2,ϕ + C R κ n+1/2 L 2 τ n+1/2 2,ϕ . x
Convergence Rates of a Fully Discrete Galerkin Scheme …
595
From the Sobolev inequality w L ∞ (R) ≤ w H 1 (R) , the Cauchy–Schwarz inequality, (6) and reordering we then obtain 1 n+1 2 2Δt √ϕx D 1/2 τ n+1/2 2 2 τ 2,ϕ + L 2 3 1 n 2 ≤ τ 2,ϕ + ΔtC R (1 + u n+1/2 2L 2 + wn+1/2 2L 2 )τ n+1/2 22,ϕ 2 1 + wn+1/2 H 2 τ n+1/2 22,ϕ + wn+1/2 H 1 τ n+1/2 22,ϕ 2 + (wn+1/2 H 2 + u(tn+1/2 ) H 2 )(ρ n+1/2 L 2 + σ n+1/2 L 2 )τ n+1/2 2,ϕ n+1/2
n+1/2
+ (wn+1/2 H 1 + u(tn+1/2 ) H 1 )(ρx L 2 + σx 1 n+1/2 τ n+1/2 2,ϕ ) + ρx L 2 (τ n+1/2 2,ϕ + Δx n+1/2
+ σx x
L 2 )τ n+1/2 2,ϕ
L 2 τ n+1/2 2,ϕ + κ n+1/2 L 2 τ n+1/2 2,ϕ .
The following result is a part of Lemma 4.1 in [5] and will be of use. Lemma 1. Let u n be the solution of (4) and assume furthermore that the scheme ˜ where C˜ is a constant depending fulfills a CFL condition of the form Δt 2 /Δx 3 ≤ C, n 2 2 2 on u 0 L . Then, u L ≤ C(u 0 L ) for n = 0, ..., N . Using Lemma 1, (10), Cauchy’s inequality, and dropping the second term on the left-hand side, we get τ n+1 22,ϕ ≤ τ n 22,ϕ + ΔtC(u, R) τ n+1 22,ϕ + τ n 22,ϕ + ρ n+1/2 2L 2 + |ρ n+1/2 |2H 1 1 n+1/2 |2 + σ n+1/2 2 + κ n+1/2 2 , + |ρ H1 H2 L2 Δx 2
which implies (1 − ΔtC(u, R))τ n+1 22,ϕ ≤ (1 + ΔtC(u, R))τ n 22,ϕ + ΔtC(u, R)Sn , where we have the remainder term Sn = ρ n+1/2 2L 2 + |ρ n+1/2 |2H 1 +
1 |ρ n+1/2 |2H 1 + σ n+1/2 2H 2 + κ n+1/2 2L 2 . Δx 2
We will assume Δt small enough that the left-hand side of the previous inequality is strictly positive, say 1 − ΔtC(u, R)) ≥ 1/2. From Taylor’s formula with integral remainder, we can derive the following estimate for the seminorms of σ n+1/2 : |σ n+1/2 |2H k ≤ CΔt 3
tn
tn+1
|u tt (s)|2H k ds,
(14)
596
S. T. Galtung
and the L 2 -norm of κ n+1/2 , κ n+1/2 2L 2 ≤ CΔt 3
tn+1 tn
u ttt (s)2L 2 ds.
(15)
Then, we may estimate the remainder term using (11), (14) and (15), CΔx 2r Sn ≤ CΔx 2(r +1) (|u(tn )| H r +1 + |u(tn+1 )| H r +1 ) + (|u(tn )|2 r +1 + |u(tn+1 )|2 r +1 ) H H Δx 2 t t n+1 n+1 + CΔt 3 u tt (s)2 2 ds + CΔt 3 u ttt (s)2 2 ds H
tn
= CΔx 2(r −1) sup |u(t)|2 0≤t≤T
H r +1
L
tn
+ CΔt 3
t n+1
u tt (s)2
H2
tn
ds +
t n+1
u ttt (s)2 2 ds . L
tn
This yields τ n 22,ϕ ≤
n−1 ! 1 + CΔt n− j 1 + CΔt n 0 2 τ 2,ϕ + ΔtC Sj 1 − CΔt 1 − CΔt j=0
≤ e4C T τ 0 22,ϕ + Δte4C T
n−1 !
Sj
j=0
≤ T C(u, R, T )Δx 2(r −1) + C(T )Δt 4
"
T 0
u tt (s)2
H
2 ds +
T 0
# u ttt (s)2 2 ds L
= C(u, R, T )(Δx 2(r −1) + Δt 4 ).
To ensure that the above norms are bounded, we assume that u 0 ∈ H s (R), s ≥ max{r + 1, 6}, see Theorems 5.3.1 and 9.1 in [1]. Then, we have τ n L 2 ≤ τ n 2,ϕ ≤ C(u, R, T )(Δx r −1 + Δt 2 ), where we have employed (11) to deduce τ 0 L 2 ≤ Pu 0 − u 0 L 2 + u 0 − Pϕ u 0 L 2 ≤ CΔx r +1 , and if one in the original scheme instead had set u 0 := Pϕ u 0 , then one would have τ 0 = 0 directly. From this and (11), we get u n − u(tn ) L 2 ≤ τ n L 2 + ρ n L 2 ≤ C(u, R, T )(Δx r −1 + Δt 2 ), n = 1, . . . , N , which proves Theorem 1 for the full line case.
Convergence Rates of a Fully Discrete Galerkin Scheme …
597
2.2 Periodic Problem For the 2L-periodic case, we follow the steps made for the real line case, but without the cutoff function ϕ involved in the scheme and all inner products now act on [−L , L]. In this case, it is straightforward to check that the L 2 -norm of the fully discrete solution u n is conserved, simply by choosing v = u n+1/2 in the adapted version of (4), integrating by parts and applying the skew-symmetry of the Hilbert transform and the periodicity of u n . The existence and uniqueness of solutions the adapted version of the iterative scheme (5) can be done analogously to the original version. In this case, estimates which were used to we do not have the commutator bound the terms Hτxn , (ϕτ n )x and (τ n )2 , (ϕτ n )x by τ n 22,ϕ , but since these now appear as, respectively, Hper τxn , τxn and (τ n )2 , τxn we use the skew-symmetricity of Hper and the periodicity of τ n to conclude that they both vanish. Apart from this, one proceeds similarly to obtain the estimate (12) for the periodic problem. Note that by obtaining this estimate, we have proved the convergence of the scheme in the periodic case given sufficiently regular initial data using a stability and consistency argument.
3 Numerical Experiments In order to verify the convergence rates numerically, we applied the fully discrete schemes to the problems (1) and (2). Inspired by [4] we define the subspace SΔx as follows. Let f and g be the functions 1 + y 2 (2|y| − 3), |y| ≤ 1, f (y) = 0, |y| > 1, y(1 − |y|)2 , |y| ≤ 1, g(y) = 0, |y| > 1. For j ∈ Z, we define the basis functions v2 j (x) = f
x − xj Δx
, v2 j+1 (x) = g
x − xj Δx
,
M where x j = jΔx. Then, {v j }−M spans a 4M + 2 dimensional subspace of H 2 (R). In the following, we define N := 2M, which is the number of elements used in the approximation. Note that for this choice we have r = 3 in (3), and so we expect convergence rates of order O(Δx 2 + Δt 2 ). To approximate the full line for (1), we have chosen to consider a finite interval with periodic boundary conditions, and we claim that this is a reasonable approximation as long as the approximate and exact solutions are close to zero
598
S. T. Galtung
at the endpoints, simulating the decay at infinity on the real line, which is the case for our examples. We have chosen to set Δt = O(Δx), contrary to the assertion Δt = O(Δx 2 ) from the theory, as smaller time steps did not lead to significant improvement in the accuracy of the approximations. In the iteration (5) to obtain u n+1 , we chose the stopping condition w+1 − w L 2 ≤ 0.002Δxu n L 2 . The integrals involved in the Hilbert transforms were computed with seven and eight point Gauss–Legendre quadrature rules, respectively, for the inner Cauchy principal value integral and the outer integral in the inner product. For t = nΔt, we $ appearing n u v (x). We have measured the relative error set u Δx (x, t) = u n (x, t) = M j j=−M j E := u Δx − u L 2 /u L 2 of the numerical approximation compared to the exact solution u, where the L 2 -norms were computed with the trapezoidal rule in the grid points x j of the finest grid considered.
3.1 Full Line Problem A solution to this problem is the double soliton given by
4c1 c2 c1 λ21 + c2 λ22 + (c1 + c2 )2 c1−1 c2−1 (c1 − c2 )−2 u s2 (x, t) = ,
2 c1 c2 λ1 λ2 − (c1 + c2 )2 (c1 − c2 )−2 + (c1 λ1 + c2 λ2 )2 where λ1 := x − c1 t − d1 and λ2 := x − c2 t − d2 . When c2 > c1 and d1 > d2 , this equation represents a tall soliton overtaking a smaller one while moving to the right. We applied the fully discrete scheme with initial data u 0 (x) = u s2 (x, 0) and parameters c1 = 0.3, c2 = 0.6, d1 = −30, and d2 = −55. The time step was set to Δt = 0.5Δx/u 0 L ∞ and the numerical solutions were computed for t = 90 and t = 180, that is, during and after the taller soliton overtakes the smaller one. To approximate the full line problem, we set the domain to [−100, 100] with the aforementioned periodic boundary condition, and based on this domain we chose the weight function ϕ(x) = 120 + x for all experiments in this setting. The results are presented in Table 1 and a comparison between the approximation for N = 256 and the exact solution is shown in Fig. 1. These results for the full line problem are also presented as numerical examples for this scheme in [5]. The plot shows that the numerical approximation appears to be close to the exact solution and this is confirmed by the errors which are decreasing from N = 256 onwards, but not with a consistent rate. According to our analysis, we should expect a convergence rate of 2, but at t = 180 it varies from slightly below 1 to slightly above 2. As pointed out in [5], this is a complicated numerical example since one has to approximate the nonlinear interaction between two solitons. Moreover, approximating the full line by a periodic finite interval could also be contributing to the error, and thus, we are led to believe that the method applied to the periodic problem will yield results which are in better agreement with theory.
Convergence Rates of a Fully Discrete Galerkin Scheme …
599
Table 1 Relative L 2 -error at t = 90 and t = 180 for full line problem with initial data u s2 and periodic boundary conditions N t = 90 t = 180 E rate E rate 128 256 512 1024 2048 4096
−1.45 1.58 0.68 1.16 0.08
0.01844 0.05021 0.01678 0.01044 0.00467 0.00442
−1.32 1.75 0.74 2.35 0.89
0.11959 0.29755 0.08869 0.05295 0.01040 0.00561
Double soliton
2.5
approximation: N = 256 exact
2
u(x,t)
1.5
1
0.5
0 -100
-80
-60
-40
-20
0
20
40
60
80
100
x
Fig. 1 Numerical approximation for N = 256 and exact solution for t = 0, 90, and 180, respectively, positioned from left to right in the plot, for full line problem with periodic boundary conditions. This figure is reproduced from [5]
3.2 Periodic Problem In our second example, we consider the Cauchy problem for the 2L-periodic BO equation (2). In this case, there exists a 2L-periodic single wave solution that tends to a single soliton as the period goes to infinity, given by u p1 (x, t) =
π 2cδ , δ= . √ 2 cL 1 − 1 − δ cos (cδ(x − ct))
600
S. T. Galtung
We applied the scheme with initial data u 0 (x) = u p1 (x, 0) with parameters c = 0.25 and L = 15. The time step was set to Δt = 0.5Δx and the approximate solution was computed for t = 480, which is four periods for the exact solution. As previously mentioned, we do not have a weight function in this setting, which is equivalent to ϕ = 1. A visualization of the results for N = 16, 32, and 64 is given in Fig. 2. Again, the plot indicates that the numerical approximation closes in on the exact solution, and this is confirmed by the errors in Table 2 which are decreasing with a rate of approximately 2, as predicted by theory. The reason for this better behavior compared to the previous example could also be its somewhat less complicated nature, where the exact solution is simply the translation of a single solitary wave.
Periodic wave
0.8
0.7
approximation: N = 16 approximation: N = 32 approximation: N = 64 exact
u(x,480)
0.6
0.5
0.4
0.3
0.2 -15
-10
-5
0
5
10
15
x
Fig. 2 Exact and numerical solutions of the 2L-periodic problem at t = 480 for element numbers N = 16, 32, and 64, with L = 15 and initial data u p1 Table 2 Relative L 2 -error at t = 480 for 2L-periodic problem with initial data u p1 N
E
rate
16 32 64 128 256 512 1024
0.14960222 0.02807195 0.00577740 0.00129088 0.00030683 0.00007805 0.00002172
2.41 2.28 2.16 2.07 1.97 1.85
Convergence Rates of a Fully Discrete Galerkin Scheme …
601
Acknowledgements The author is grateful to Helge Holden for his encouragement and support to participate in the HYP2016 conference, and to Institut Mittag-Leffler for its hospitality during the Fall of 2016, providing an excellent working environment for this research.
References 1. L. Abdelouhab, J.L. Bona, M. Felland, J.-C. Saut, Nonlocal models for nonlinear, dispersive waves. Phys. D 40(3), 360–392 (1989) 2. T.B. Benjamin, Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29(3), 559–592 (1967) 3. P.G. Ciarlet, The Finite Element Method for Elliptic Problems (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002) 4. R. Dutta, N.H. Risebro, A note on the convergence of a Crank-Nicolson scheme for the KdV equation. Int. J. Numer. Anal. Model. 13(5), 567–575 (2016) 5. S.T. Galtung, A convergent Crank–Nicolson Galerkin scheme for the Benjamin–Ono equation. Discrete Contin. Dyn. Syst. 38(3), 1243–1268 (2018) 6. L. Grafakos, Classical Fourier Analysis, 3rd edn. (Springer, New York, 2014) 7. H. Ono, Algebraic solitary waves in stratified fluids. J. Phys. Soc. Jpn. 39(4), 1082–1091 (1975) 8. A. Quarteroni, A. Valli, Numerical Approximation of Partial Differential Equations (Springer, Heidelberg, 1994)
The Simulation of a Tsunami Run-Up Using Multiwavelet-Based Grid Adaptation Nils Gerhard and Siegfried Müller
Abstract In (Gerhard et al., J Comput Phys 301, 265–288, 2015, [2]) a new class of adaptive discontinuous Galerkin schemes has been introduced for shallow water equations with drying and wetting. Grid adaptation is performed by means of a multiresolution analysis using multiwavelets. In this paper, we focus on the application to more realistic scenarios. To this end, we validate the adaptive scheme with the numerical simulation of a 1:400 scale experiment of a tsunami run-up. Keywords Grid adaptation based on multiwavelets · Tsunami simulations
1 Introduction Discontinuous Galerkin (DG) schemes have been successfully applied to the shallow water equations [8, 13]. The DG method generalizes the concept of the FV method, while relying on the finite element notion of projecting the solution onto a space of trial functions, however, without the restriction of keeping the functions continuous. Therefore, the piecewise discontinuous description of the solution allows to achieve high-order accuracy while keeping locality. In order to improve the efficiency of the DG method, it can be combined with grid adaptation where the resolution is dynamically adapted to the local flow features. In recent years, the concept of dynamical grid adaptation based on multiresolution analysis—quite successful with FV solvers for compressible fluid flow, see [12] and references therein—has been applied to the framework of DG schemes. Originally, the concept was analytically and numerically investigated for one-dimensional scalar conservation laws [7]. Then, the method was extended to multidimensional scalar N. Gerhard (B) · S. Müller Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Aachen, Germany e-mail: [email protected] S. Müller e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_46
603
604
N. Gerhard and S. Müller
conservation laws in [4] and to the compressible Euler equations in [3]. In [2], it was applied to shallow water equations including wetting and drying. In [10], the adaptive scheme from [2] has been validated for numerous academic test cases. In this paper, we continue the work from [2] and focus on the application to real-world problems. Applying DG schemes to shallow water equations requires the projection of the bottom topography to the DG space. In case of adaptive schemes, an efficient projection of the bottom topography to a suitable adaptive grid is needed. In real-world applications, the bottom topography is no longer provided by an analytical function but usually only discrete samples from measurements are available. For that reason, we present a strategy to determine initially an adaptive representation of the bottom topography from discrete data. Finally, we present numerical results of a tsunami run-up onto a complex three-dimensional beach. The simulation is based on a 1:400 scale experiment [14] of a real tsunami that hit the coast of Japan in 1993.
2 Numerical Model 2.1 Discontinuous Galerkin Scheme for Shallow Water Equations In this section, we briefly summarize the main ideas of the Runge–Kutta DG (RKDG) schemes applied to the shallow water equations including the modifications to achieve well-balancing and positivity-preserving. The shallow water equations can be written as (1) ut + ∇ · f(u) = s(x, u) on Ω × (0, T ) with Ω ⊂ Rd , d = 1, 2, the conserved quantities u = (h, hv) ∈ Rm , m = d + 1, flux f = (hv, hv ⊗ v + 0.5gh 2 I). We include Manning’s explicit friction model and hence, the source term s = (0, −gh∇b) + (0, −ghσ ) is used. The friction slope σ is chosen as σ = n 2σ v |v|h −4/3 with the Glaucker–Manning coefficient n σ . Here, h [m] is the water height, v [m/s] is the velocity, b [m] is the bottom topography, and g [m/s2 ] is the gravitational acceleration. These equations are supplemented with initial data u(0, ·) = u0 and appropriate boundary conditions. We discretize the domain Ω by a finite number of cells Vλ denoting by G := {Vλ }λ∈I the computational grid. On this grid, we introduce the DG space of cell-wise polynomials of total degree less than p: S := { f ∈ L 2 (Ω) : f |Vλ ∈ Π p−1 (Vλ ) ∀ λ ∈ I}.
(2)
In order to derive the variational formulation of the DG scheme, we first rewrite (1) in a weak formulation. For this purpose, we multiply (1) by a test function wh ∈ S m and integrate over a cell Vλ . Then, we perform integration by parts and introduce
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numerical fluxes fˆ approximating the flux f · n in direction n. This results in the semi-discrete DG formulation: Find uh (·, t) ∈ S m , t ∈ [0, T ], such that for all wh ∈ S m and λ ∈ I: ∂uh · wh dx − fˆ · wh d S = 0, (3) f(uh ) : ∇wh + s(x, uh ) · wh dx + Vλ ∂t Vλ ∂ Vλ m×d where a · b, a, b ∈ Rm denote we the standard inner product and for A, B ∈ R define the product A : B := i, j Ai j Bi j . The numerical flux fˆ depends on the inner and outer value of uh at the boundary of Vλ and on the outward pointing unit normal vector. The weak formulation of the DG scheme leads to a system of ordinary differential equations solved with an explicit Runge–Kutta scheme. In order to guarantee the stability of the fully discrete scheme, a limiting of the high-order coefficients has to be applied after each stage of the Runge–Kutta scheme. Here, we use the TVB limiter by Cockburn and Shu [1] which is applied to the local characteristic variables to avoid spurious oscillations. In order to achieve the well-balancing property, the local characteristic variables are based on (h + b, hv) instead of (h, hv). For details, we refer to [2]. In order to guarantee well-balancing and positivity-preserving, we follow the strategy of Xing et al. [13]. One of the key ideas from Xing et al. [13] to ensure well-balancing besides the flux modification is to project the bottom topography b to the same DG space as the solution uh . In the adaptive scheme, the grid and the corresponding discretization space are dynamically changing from time step to time step. At the time step tn , we denote the adaptive grid by Gn and the corresponding discretization space by Sn . Since these spaces are changing, we claim that bh is in Sn for each time step tn . For sake of efficiency, we use a static grid for b and make sure that the refinement of the grid for the bottom is always included in the computational grid of the approximative solution [2]. Therefore, the efficiency of the adaptive scheme significantly relies on the choice of the adaptive grid for the bottom topography. We will focus on this in Sect. 3 in detail.
2.2 Review of Multiwavelet-Based Grid Adaptation The DG discretization typically works on an array of coefficients of the polynomial representation of the solution. The underlying idea of our adaptive strategy is to perform a multiresolution analysis of the polynomial coefficients on a hierarchy of nested grids. This provides us with information on the difference between successive refinement levels that may become negligibly small in regions where the solution is locally smooth. Applying hard thresholding, the data are compressed thereby triggering local grid adaptation. In the following, we briefly summarize the idea of multiwavelet-based grid adaptation. For a detailed description, we refer to [2, 4, 7].
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The starting point for a multiresolution analysis is a hierarchy of nested grids G := {Vλ }λ∈I for increasing number of refinement levels . The index set Mλ ⊂ I+1 denotes the refinement set of cell Vλ and for μ ∈ I let λ(μ) ∈ I−1 denote the parent cell on level − 1. Thus, it holds Vλ = μ∈Mλ Vμ . On each level of this hierarchy, we define the DG space S analogously to (2). Obviously, these spaces inherit the nestedness from the underlying grid hierarchy and form a multiresolution sequence: S0 ⊂ S1 ⊂ . . . ⊂ S ⊂ S+1 ⊂ . . . ⊂ L 2 (Ω).
(4)
The nestedness of the discretization spaces implies that there exist orthogonal complement spaces W such that S+1 = S ⊕ W . For a fixed level L, we obtain by recursively applying this two-scale relation: SL = S0 ⊕ W0 ⊕ ... ⊕ W L−1 .
(5)
To perform the multiscale decomposition of the single-scale space SL in (5), we introduce a basis {φi }i∈IS for the single-scale spaces S and a basis {ψi }i∈IW for the complement spaces W . For sake of efficiency, we assume that the support of these basic functions is a single cell. Then, the index i of φi and ψi can be decomposed in a spatial and a polynomial part: i = (λ, i), i.e., IS = I × P and IW = I × P∗ , where P and P∗ characterize the local degrees of freedom of S and W , respectively. In order to exploit the potential of data compression, we make use of the multiscale transformation (5), and determine a local two-scale representation of u ∈ (S )m : uμ,i φμ,i = uλ,i φλ,i + dλ,i ψλ,i , uVλ = μ∈Mλ i∈P
∀ λ ∈ I−1 .
(6)
i∈P∗
i∈P
In Fig. 1, the local two-scale representation (6) is illustrated exemplarily. Applying this local two-scale decomposition recursively, we obtain the multiscale representation of w ∈ (SL )m : w=
i∈I LS
ui φi =
i∈I0S
ui φi +
L−1
di ψi ,
(7)
=0 i∈IW
where we call di the detail coefficients and ui the single-scale coefficients. Furthermore, we denote the arrays with the local degrees of freedom in a cell Vλ by
Fig. 1 Multiscale decomposition for a single refinement level and p = 2
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Uλ (w) := (uλ,i )i∈P and Dλ (w) := (dλ,i )i∈P∗ . For a detailed description of the construction of the basic functions and the efficient realization of the transformations, we refer to [4]. The advantage of the multiscale representation is that the detail coefficients provide information on the difference between two successive refinement levels. In regions where the underlying function is smooth, the detail coefficients may become negligibly small and therefore its contribution in (7) is small. We will use this information to determine an adaptive grid by compressing the multiscale representation (7). For a given threshold value ε ≥ 0, we define level-dependent threshold values ε by ε := 2−L ε. Then, we call a detail coefficient di significant if it is larger than a level-dependent threshold value and define the index set of significant detail coefficients by L−1 W I : max1≤k≤m ( C(ψi , k) |(di )k |) > ε(i) , Dε (w) := i ∈ ∪=0
(8)
where the constant C(ψi , k) depends on the normalization of the multiwavelet function ψi and the magnitude of the kth conserved quantity [2]. Thereby, we can find a sparser representation for w: wε :=
ui φi +
i∈I0S
L−1
di ψi .
=0 i∈Dε (w)
If Dε (w) is a tree, then wε is a projection of w to a coarser grid Gε (w) [2]. This grid can be determined efficiently by Dε (w): proceeding from coarse to fine level a cell is refined whenever there exists a significant detail coefficient. In Fig. 2, this is illustrated exemplarily for L = 2. For details, we refer to [4]. By hard thresholding, we are able to coarsen the grid. Ideally, it would be sufficient to determine the significant details of the next time step to refine the grid. Since this is not possible, we follow Harten’s heuristic idea, which mainly consists of two steps: (i) since
Fig. 2 Example for coarsening via thresholding
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information is transported, details in a local neighborhood may get significant; (ii) due to steepening of gradients and the formation of shocks, additional details on higher levels may become significant. For a detailed description, we refer to [2, 3]. In Fig. 3, the refinement strategy via prediction on the detail coefficients is shown for a one-dimensional example. In the following algorithm, we summarize one time step of the adaptive scheme: Algorithm 1 (Time step of adaptive scheme) (1) Refine grid via prediction on detail coefficients on (h + b, hv)t . (2) Compute time step for evolution with DG scheme on (h, hv)t . (3) Coarsen grid via thresholding on (h + b, hv)t . We have to add some modifications to our adaptation strategy such that coarsening and refinement will maintain well-balancing and positivity-preserving. One of the key ideas from Xing and Shu [13] to ensure well-balancing besides a flux modification is to project the bottom topography b to the same space as the solution uh . In the adaptive scheme, the grid is dynamically changing with each time step. Therefore, we determine initially a fixed adaptive grid for the bottom topography and determine the projection bh . In the computation, we make sure that the adaptive grid of the current solution is at least as refined as the grid for the bottom topography. For a detailed description, we refer to [2].
3 Initial Adaptive Grid: Projection of Initial Data and Bottom Topography To start the computation, we need to generate the initial adaptive grid by means of the initial data u0 and the bottom topography b. The simplest but not very efficient way for the initialization is to start on the highest refinement level L by projecting u0 and b to SL and then applying hard thresholding to determine an initial adaptive representation. The bottleneck of this initialization on a fully refined grid is the high memory requirement. Thus, the use of an alternative bottom-up strategy is recommended. The basic idea of the alternative strategy is to avoid the memory
Fig. 3 Example for refinement via prediction on detail coefficients
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Fig. 4 Alignment of the discrete samples
p= 2 p= 3
requirements by proceeding from coarse to fine and refine whenever a significant detail is present in the multiscale representation. For a detailed description, we refer to [5]. In real-world applications, the bottom topography is typically provided as a data set of discrete points (xi , bi ) instead of an analytical function b(x). From the discrete data, we need to determine the adaptive representation bh ∈ Sinit ⊂ SL . If the bottom topography is given by an analytical function b, then we apply the bottom-up strategy with a cell-wise L 2 -projection. Otherwise, if only discrete samples are available, the representation bh can be obtained by solving local least squares problem resulting from an overdetermined interpolation problem on each cell. Since the DG space S is defined locally and the well-balancing strategy does not require continuity, this can be performed cell by cell. In the following, we consider the projection to Π p−1 on a single cell Vλ . Let Bλ denote the index set of all tuples (xi , bi ) which are located in Vλ or on ∂ Vλ . The local representation bh on Vλ can be defined by the following linear least squares problem: (bi − bh (xi ))i∈Bλ 2 → min .
(9)
In several applications, the discrete data are particularly ordered, e.g., the samples are taken row-wise resulting in a Cartesian alignment as depicted in Fig. 4. Then, this structure should be incorporated into the process of determining the representation bh in order to diminish the interpolation error. For instance, a simple strategy to incorporate the structure for a Cartesian alignment could be to define a (temporary) auxiliary grid for the initialization of the bed which is fitted to the alignment of the discrete samples. For the particular alignment shown in Fig. 4, the auxiliary grid can be constructed in such a way that each cell contains exactly the same number of samples as degrees of freedom in the local polynomial space. Based on this grid, we compute a piecewise polynomial function baux using locally polynomial interpolation. In Fig. 4, such a cell is shown for p = 2 and p = 3 in red and blue, respectively. Here, most of the discrete points are located on the edges connecting two adjacent cells. Thus, one discrete sample is considered in the local interpolation process of two cells. If the data are ordered in a different way, it is also possible to consider triangulations or spline
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approximations in order to determine baux . Finally, the adaptive representation bh is computed using the bottom-up strategy from [5] with the continuous representation baux as underlying function. Note that in several applications the number of discrete samples is much smaller than the degrees of freedom in the fully refined grid on level L. Thus, the initialization using the auxiliary grid does not have the computational complexity of the reference grid.
4 Results—Tsunami Run-Up onto a Complex Three-Dimensional Beach In 1993, Okushiri Island was hit by a tsunami. The Central Research Institute for Electric Power Industry (CRIEPI) in Abiko, Japan, performed a 1:400 scale experiment of this run-up [11]. This scale experiment was a benchmark problem for The Third International Workshop on Long-wave Run-up Models in 2004 [14] providing experimental data for validation. In particular, the maximum run-up height has been measured. Moreover, the time evolution of the water surface has been recorded at three control points between the island and the shore. Their alignment can be depicted from Fig. 5. In the following, we use these information to validate our adaptive method. The computational domain is Ω = [0, 5.488] m × [0, 3.402] m. On this domain, the bottom topography b is provided by a discrete set of 97, 857 points. The data are aligned row-wise as depicted in Fig. 4. Thus, we apply the strategy described in Sect. 3
Fig. 5 Contour of b with control points
Fig. 6 Bottom topography b at y = 2
0.1 0 h+b b
−0.1 0
1
2
3
x [m]
4
5
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·10−2
1 0 −1 −2
h+b 0
5
10
20
15
time t [s] Fig. 7 Surface h + b at inflow boundary x = 0
h + b [cm]
4
4
exp. data L=6
2
2
0
−2
4
exp. data L=6
2
0 0
10
20
time t [s]
(a) (x1 , x2 ) = (4.521, 1.196)
−2
exp. data L=6
0 0
10
time t [s]
20
−2
0
10
20
time t [s]
(b) (x1 , x2 ) = (4.521, 1.696) (c) (x1 , x2 ) = (4.521, 2.196)
Fig. 8 Comparison of the water surface h + b at the control points with experimental data
to compute the adaptive representation bh . In Fig. 9a, the bottom topography and the corresponding adaptive grid are shown. In Fig. 6, the profiles of b and the initial surface h + b in direction normal to the beach at y = 2 are shown. The Glaucker– Manning coefficient is chosen as 0.01. At x = 0, the tsunami wave is entering the computational domain. This is realized by prescribing the recorded water surface h + b (see Fig. 7) from the experiment as inflow boundary condition. For details, we refer to [5]. At all other boundaries, we consider reflecting wall boundary conditions. First, the water retracts from shoreline and then the tsunami wave is flooding the dried up regions and the coast. In our computations, we use quadratic polynomials ( p = 3) and a third-order strong-stability-preserving Runge–Kutta scheme with three stages and a CFL number of 0.1. The Shu constant in the limiter is chosen as M = 1. The threshold value was set to ε L = 1/100 h L . The wet–dry tolerance is chosen as T ol = 10−4 which is two orders of magnitude smaller than the amplitude of the inflow wave. Thereby, the flow is stabilized without introducing too much perturbation. On the coarsest level, we use 8 × 5 cells and consider 6 levels of refinement. The results presented in this section are a summary of the more detailed discussion in this configuration in [5] (Fig. 8). In Fig. 9, the adaptive solution (h + b) and the corresponding grids are shown at different time instances. In these pictures, the dynamical grid adaption can be observed: the grid refinement is tracking the arising waves, whereas in regions with small variations the grid is coarse. For the L = 6 computation, the adaptive grid consists of at most 18, 139 cells, whereas the uniform grid of the reference scheme
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(a) t = 0.0 s
(b) t = 13.5 s
(c) t = 15 s
Fig. 9 Adaptive solutions and the corresponding computational grids at times t = 0 s, t = 13.5 s t = 15 s, t = 16.5 s, t = 18 s and t = 22.5 s
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(d) t = 16.5 s
(e) t = 18 s
(f) t = 22.5 s
Fig. 9 (continued)
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Table 1 Comparison of maximum run-up height, its position, and time of occurrence Maximum run-up Computation Height [m] Position [m] Time [s] L=6 0.088415 Experimental data [11] 0.09
(5.1504, 1.8977) ≈ (5.158, 1.880)
16.676 –
would consist of 163, 840 cells. Thus, the computational cost can be significantly reduced using the grid adaption. In order to validate our adaptive scheme, we compare the numerical results with measurement data from the scale experiment. For that purpose, we first compare the water height at the three control points located between the offshore island and the coast. The specific location of these points can be depicted from Fig. 6. In Fig. 8, the water surface h + b from the L = 6 computation and from the experiment are shown. Here, we observe a slight difference between computational and experimental data. However, our results are similar to results computed with other numerical schemes, cf. [6, 9]. The differences might be caused by measurement inaccuracies in the experiments or modeling errors due to the use of the shallow water equations. However, the essential evolution of the surface h + b is reproduced by the simulation. Next, we focus on the maximum run-up of the tsunami wave. To this end, we have determined the maximum height, its position, and time of occurrence in our adaptive computations. In Table 1, these are listed. Furthermore, the experimental data from [11] are listed for comparison. When comparing our results with the experimental data, we conclude that our adaptive solver can identify the maximum run-up height and its position of occurrence very accurately. The instant of the highest run-up is shown in Fig. 9d.
References 1. B. Cockburn, C.-W. Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems. J. Comput. Phys. 141(2), 199–244 (1998) 2. N. Gerhard, D. Caviedes-Voullième, S. Müller, G. Kesserwani, Multiwavelet-based grid adaptation with discontinuous Galerkin schemes for shallow water equations. J. Comput. Phys. 301, 265–288 (2015) 3. N. Gerhard, F. Iacono, G. May, S. Müller, R. Schäfer, A high-order discontinuous Galerkin discretization with multiwavelet-based grid adaptation for compressible flows. J. Sci. Comput. 62(1), 25–52 (2015) 4. N. Gerhard, S. Müller, Adaptive multiresolution discontinuous Galerkin schemes for conservation laws: multi-dimensional case. Comput. Appl. Math. 35, 312–349 (2016) 5. N. Gerhard, An adaptive multiresolution discontinuous Galerkin scheme for conservation laws. Ph.D. thesis, RWTH Aachen University, submitted 2017 6. J. Hou, Q. Liang, H. Zhang, R. Hinkelmann, An efficient unstructured MUSCL scheme for solving the 2D shallow water equations. Environ. Model. Softw. 66, 131–152 (2015)
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7. N. Hovhannisyan, S. Müller, R. Schäfer, Adaptive multiresolution discontinuous Galerkin schemes for conservation laws. Math. Comput. 83(285), 113–151 (2014) 8. G. Kesserwani, Q. Liang, A conservative high-order discontinuous Galerkin method for the shallow water equations with arbitrary topography. Int. J. Numer. Methods Eng. 86(1), 47–69 (2011) 9. G. Kesserwani, Q. Liang, Dynamically adaptive grid based discontinuous Galerkin shallow water model. Adv. Water Resour. 37, 23–39 (2012) 10. D. Caviedes-Voullième, G. Kesserwani, Benchmarking a multiresolution discontinuous Galerkin shallow water model: implications for computational hydraulics. Adv. Water Resour. 86, 14–31 (2015). Part A 11. M. Matsuyama, H. Tanaka: An experimental study oh the highest run-up height in the, Hokkaido Nansei-oki earthquake tsunami. ITS Proc. 879–889, 2001 (1993) 12. S. Müller, Multiresolution schemes for conservation laws, in Multiscale, Nonlinear and Adaptive Approximation, ed. by R. DeVore (Springer, Berlin, 2009) 13. Y. Xing, X. Zhang, C.-W. Shu, Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations. Adv. Water Resour. 33(12), 1476–1493 (2010) 14. http://isec.nacse.org/workshop/2004_cornell/bmark2.html
Constrained Reconstruction in MUSCL-Type Finite Volume Schemes Christoph Gersbacher and Martin Nolte
Abstract In this paper, we are concerned with the stabilization of MUSCL-type finite volume schemes in arbitrary space dimensions. We consider a number of limited reconstruction techniques which are defined in terms of inequality-constrained linear or quadratic programming problems on individual grid elements. No restrictions on the conformity of the grid or the shape of its elements are made. In the special case of Cartesian meshes, a novel QP reconstruction is shown to coincide with the widely used minmod reconstruction. The accuracy and overall efficiency of the stabilized second-order finite volume schemes are supported by numerical experiments. Keywords Hyperbolic conservation laws · Finite volume scheme · Linear reconstruction · Quadratic programming MSC (2010) 35F25 · 35L65 · 65M12 · 90C20 · 35Q35
1 Introduction First-order finite volume schemes are a class of discontinuous finite element methods widely used in the numerical solution of first-order hyperbolic conservation laws. They are particularly popular for their conservation properties and their robustness. Finite volume methods can be applied to arbitrarily shaped grid elements and locally adapted grids while still being easy to implement. However, due to the large amount of dissipation built in the first-order scheme, discontinuities in the exact solution may be heavily smeared out. C. Gersbacher (B) · M. Nolte Department of Applied Mathematics, University of Freiburg, Hermann-Herder-Str. 10, D-79104 Freiburg, Germany e-mail: [email protected]; [email protected] M. Nolte e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_47
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Higher order schemes in general provide a much better resolution than the firstorder finite volume method. They are, however, prone to developing spurious oscillations and unphysical values that may result in the immediate breakdown of a numerical simulation and require a suitable stabilization. Stabilized methods include MUSCL-type finite volume schemes introduced by van Leer [14] or the essentially non-oscillatory method by Shu and Osher [12]. The former is based on the reconstruction of piecewise linear functions from piecewise constant data, and slope limiters are used to prevent spurious oscillations. Their extension to general unstructured grids in d space dimensions, however, is a challenging task. In this paper, we follow a more recent approach to the design of MUSCL-type finite volume schemes and limited reconstruction operators [5, 6, 11]. We consider a number of reconstruction operators defined in terms of local inequality-constrained linear or quadratic minimization problems. No restrictions on the conformity of the grid or the shape of individual grid elements are imposed. We are able to prove that in the special case of Cartesian meshes our QP reconstruction coincides with the ddimensional minmod reconstruction, which illustrates the reliability of the stabilized finite volume method. The general approach allows for a number of modifications, and we briefly discuss a positivity-preserving stabilization for the Euler equations of gas dynamics.
2 Stabilization of MUSCL-Type Finite Volume Schemes In this section, we want to briefly revisit MUSCL-type finite volume schemes on arbitrary meshes. In the following, let Ω ⊂ Rd , d > 0, be a bounded domain. We consider the first-order system ∂t u + ∇ · F(u) = 0 in Ω × (0, T ), u(·, 0) = u 0 in Ω
(1)
subject to suitable boundary conditions. Here,u : Ω × (0, T ) → U is an unknown function with values in the set of states U ⊂ Rr . The function u 0 : Ω → U denotes some given initial data, and F : U → Rd×r is the so-called convective flux. General notation Now, let G be a suitable partition of the computational domain into closed convex polytopes with nonoverlapping interior. For each element E ∈ G , we denote by N (E) the set of its neighboring elements, i.e., N (E) = E ∈ G \ {E} dim(E ∩ E ) = d − 1 , where for each boundary segment E ∩ ∂Ω, E ∈ G , we assume the existence of an exterior, possibly degenerate ghost cell E ∈ N (E). By X Gk , we denote the piecewise polynomial spaces of order at most k on G ,
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X Gk = u ∈ L ∞ (Ω, Rr ) u E ∈ (P k (E))r for all E ∈ G . Furthermore, for u ∈ X G0 , we write u(x) =
u E χ E (x),
E∈G
where χ E denotes the indicator function for the element E ∈ G and u E denotes the local cell average. The centroid of a convex polytope E ⊂ Rd will be denoted by x E . MUSCL-type finite volume schemes Next, we define the general second-order finite volume scheme. For each element, E ∈ G and E ∈ N (E) denote by G E,E : C ∞ (E ∩ E ) × C ∞ (E ∩ E ) → Rr a conservative numerical flux from E to E that consistent with F, i.e., F(u) · ν E,E d x for all u ∈ C ∞ (E ∩ E ), G E,E (u, u) = E∩E
where ν E,E ∈ Rd denotes the unit outer normal to E on the intersection E ∩ E . After performing a spatial discretization of (1), we seek u : [0, T ] → X G0 such that χE d u(t) = − dt |E| E∈G
G E,E Ru(t)E , Ru(t)E for t ∈ (0, T ),
E ∈N (E)
χE u(0) = ΠG0 u 0 := u 0 dx. |E| E
(2)
E∈G
Here, R : X G0 → X G1 denotes a reconstruction operator mapping piecewise constant to piecewise linear data. The operator is assumed to be locally mass conservative, i.e., for all u ∈ X G0 it holds Ru(x E ) = u E for all E ∈ G . Note that in Eq. (2) we made implicit use of so-called ghost values, which must be determined from the given set of boundary conditions. For the higher order discretization in time we use a second-order accurate Runge–Kutta method. Obviously, the key ingredient to achieving second-order accuracy is the reconstruction operator R. The design of such operators is a delicate matter which ultimately will affect the robustness and accuracy of the overall numerical scheme. The generalization of techniques developed for the one-dimensional case to multiple space dimensions is not always obvious, in particular with respect to arbitrarily shaped grid elements and possibly nonconforming grids. Limited least-squares fitted polynomials Arguably, the most popular class of stabilized reconstruction techniques is due to Barth and Jespersen [2]. It is based on a two-step procedure to be illustrated by a limited least-squares fit.
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For the sake of simplicity, we restrict the following presentation to scalar functions. Let u ∈ X G0 be a piecewise constant function and E ∈ G a fixed grid cell. We fix nonnegative weights ω E,E , E ∈ N (E), and define the quadratic functional JE (v; u) =
E ∈N (E)
2 ω E,E u E − v(x E ) for v ∈ P 1 (Rd ). 2
(3)
We then compute the minimizing polynomial v E = arg min JE (v; u) for all E ∈ G v∈P 1 (Rd )
subject to the local mass conservation property v E (x E ) = u E . It is easy to see that the minimization problem is well-posed, if span{ω E,E (x E − x E ) | E ∈ N (E)} = Rd .
(4)
Next, we introduce a set of locally admissible linear functions (see, e.g., [8]), given by W (E; u) = w ∈ P 1 (Rd ) | w(x E ) = u E and
min{u E , u E } ≤ w(x E ) ≤ max{u E , u E }for all E ∈ N (E) .
(5)
Note that, by definition, the set W (E; u) is convex and non-empty, since the constant function w(x) = u E is always admissible. A function v ∈ X G1 is called admissible, if vE ∈ W (E; ΠG0 v) for all E ∈ G . Having computed the linear polynomial v E , an inexpensive projection onto the set of admissible polynomials W (E; u) is given by a scaling of the candidate gradient ∇v E . We define the mapping R : X G0 → X G1 by Ru E (x) = u E + α E ∇vE · (x − x E ) for all E ∈ G , where α E ∈ [0, 1] is chosen maximal such that the image is admissible. The scalar factor α E can be computed explicitly.
3 Constrained Linear Reconstruction In the previous section, limitation has been considered a separate step in the definition of a stabilized reconstruction operator. Here, we follow a more recent approach of recovering a suitably bounded approximate gradient in a single step by means of local minimization problems. For example, [6, 11] proposed to directly reconstruct an admissible solution through a linear programming (LP) problem.
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Definition 1 (LP reconstruction). Let G be an arbitrary d-dimensional grid. The LP reconstruction operator R : X G0 → X G1 is defined by Ru E = w E = arg min
w∈W (E;u) E ∈N (E)
u E − w(x E )
(6)
where the set of locally admissible functions W (E; u) is given by Eq. (5). This reconstruction has been shown to be equivalent to the following LP problem:
maximize
sign(u E − u E )(x E − x E ) · ∇w,
E ∈N (E)
subject to
0 ≤ sign(u E − u E ) (x E − x E ) · ∇w ≤ |u E − u E |.
In [6, 11], the authors propose to solve this problem by a variant of the classical simplex algorithm. The use of the l 1 -Norm in the objective function in Eq. (6) might seem natural in the context of hyperbolic conservation laws. For the numerical approximation of overdetermined problems, the use of quadratic objective functions, e.g., least-squares fits, is more common. The approach we propose may be summarized as follows: for each grid element E ∈ G , we choose a locally admissible polynomial w E ∈ W (E; u) as the best admissible fit in a least-squares sense to given piecewise constant data. Definition 2 (QP reconstruction). Let G be an arbitrary d-dimensional grid. The QP reconstruction operator R : X G0 → X G1 is defined by Ru E = w E = arg min JE (w; u),
(7)
w∈W (E;u)
where JE is defined as in Eq. (3) and the set of locally admissible functions W (E; u) is given by Eq. (5). First, observe that the optimization problem (7) is equivalent to a standard quadratic programming (QP) problem for the approximate gradient. Indeed, using the notation d E,E = x E − x E and m E,E = u E − u E for E ∈ N (E), a linear function w E is a solution to (7) if and only if ∇w E solves the QP problem minimize subject to
1 ∇w · (H ∇w) − g · ∇w, 2 0 ≤ sign(m E,E ) d E,E · ∇w ≤ |m E,E |,
(8)
where sign(a) ∈ {−1, 1} denotes the sign of a ∈ R. The Hessian H ∈ Rd×d and the gradient g ∈ Rd of the objective function are given by
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H=
ω E,E d E,E ⊗ d E,E ,
E ∈N (E)
g=
E ∈N
ω E,E m E,E d E,E .
(E)
The matrix H is positive definite due to assumption (4) on the choice of weights ω E,E . A QP problem can be solved efficiently in a small number of steps. For a description of the numerical methods, the reader is referred to standard textbooks on constrained optimization. With a different set of linear constraints, the reconstruction in Definition 2 was also proposed in [5]. However, the authors restrict themselves to conforming triangular grids to compute the exact solution to the arising QP. In contrast, we propose the use of an active set strategy to solve the QP problem numerically. As a consequence, we do not have to impose any restrictions on the grid dimension, the conformity of the grid, or the shape of individual grid elements. The main result of this section is given in Theorem 1. In case of Cartesian meshes, the QP reconstruction coincides with the well-known and reliable minmod limiter. In the following, let G denote a d-dimensional Cartesian grid of uniform grid width h = (h 1 , . . . , h d ). For fixed E ∈ G , we will denote its neighbors by E i± , i = 1, . . . , d, defined by E i± = E ± h i ei . Theorem 1. Let G be a d-dimensional Cartesian grid of uniform grid width h = (h 1 , . . . , h d ), and let E ∈ G be a fixed element. Then, the exact solution ∇w E to the QP problem (8) is given by ∇w E =
d i=1
minmod
u Ei+ − u E u E − u Ei− , hi hi
ei .
In particular, ∇w E is independent of the choice of weights ω E,Ei± . Proof. For a Cartesian grid, we have d E,Ei± = ±h i ei and the Hessian H becomes a diagonal matrix with Hii = h i2 (ωi+ + ωi− ), where we denoted ωi± = ω E,Ei± . Similarly, using the notations m i± = m E,Ei± , the inequality constraints simplify to box constraints for ∂i w. Therefore, the optimization problem is equivalent to the d one-dimensional quadratic problems minimize subject to
ωi+ + ωi− (h i ∂i w)2 − (ωi+ si+ |m i+ | + ωi− si− |m i− |) h i ∂i w 2 1 0 ≤ si± ∂i w ≤ |m ± |, hi i Ji (∂i w) =
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where we denote si± = ± sign(m i± ). Now, if si+ = si− or either of m i± vanishes, the constraints require ∂i w = 0, which agrees with the minmod limiter. Otherwise, let si = si+ = si− . The global minimum of each functional Ji is attained for 1 ωi+ |m i+ | + ωi− |m i− | 1 si ∂i w = ≥ min |m i+ |, |m i− | . + − hi hi ωi + ωi The opposite inequality follows directly from the constraints and we conclude ∂i w =
si 1 min |m i+ |, |m i− | = minmod(m i+ , m i− ), hi hi
which proves the statement.
4 Numerical Results In this section, we want to study the accuracy and efficiency of the QP reconstruction, Definition 2. To this end, generic implementations of all reconstruction operators discussed in this paper were written within the Dune framework [3, 4]. For the computations in Sect. 4.3, the parallel grid library Dune- ALUGrid [1] was used.
4.1 Nonlinear Problem Admitting a Smooth Solution The first benchmark problem is taken from [9, Chap. 3.5]. We consider in the unit square Ω = (0, 1)2 a nonlinear balance law 3 ). ∂t u + ∂1 u 2 + ∂2 u 2 = s in Ω × (0, 10
We want to numerically recover a prescribed smooth solution given by u(x, t) =
1 sin (2π(x1 − t)) sin (2π(x2 − t)) . 5
Note that the right-hand side of Eq. (2) must be extended by the discrete source term. The initial data, source term, and Dirichlet boundary values are then determined from the prescribed solution. We consider two different types of domain discretizations: a series of conforming triangular grids generated through refinement of a coarse Delaunay triangulation with 123 elements and a series of nonconforming quadrilateral grids resulting from a checkerboard-like refinement rule, Fig. 1. This latter is particularly challenging as the number of nonconformities grows linearly in the number of elements.
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Fig. 1 Nonconforming quadrilateral meshes with 160 and 640 elements, generated by refining every second element of a Cartesian grid Table 1 Nonlinear benchmark problem admitting a smooth solution, L 1 -errors and convergence 3 rates at final time t = 10 on conforming, unstructured triangular meshes Elements
Limited LSF
u − EOC uG L 1
QP reconstruction
u − EOC uG L 1
(a) Conforming unstructured triangular meshes 123 1.35 ×10−2 — 9.86×10−3 −3 492 5.59×10 1.27 3.09×10−3 −3 1968 2.24×10 1.32 9.09×10−4 −3 7872 1.05×10 1.10 2.61×10−4 31488 5.36×10−4 0.97 7.31×10−5 (b) Nonconforming quadrilateral meshes 640 8.22×10−3 1.19 4.44×10−3 −3 2560 3.18×10 1.37 1.30×10−3 −3 10240 1.45×10 1.13 4.17×10−4 −4 40960 7.27×10 1.00 1.49×10−4
LP reconstruction
u − EOC uG L 1
— 1.68 1.76 1.80 1.84
1.08×10−2 3.75×10−3 1.15×10−3 3.58×10−4 1.07×10−4
— 1.52 1.71 1.68 1.74
1.46 1.78 1.63 1.48
4.44×10−3 1.30×10−3 4.21×10−4 1.51×10−4
1.48 1.77 1.63 1.48
Table 1 shows the L 1 -errors and convergence rates for a second-order Lax– Friedrichs scheme using the three different reconstruction operators. Observe that in all cases the simple limited least-squares fit results in a first-order approximation only. The QP and LP reconstructions give much better results with the QP reconstruction having a slight edge over the LP reconstruction in case of triangular meshes; in case of the highly nonconforming quadrilateral meshes, however, the approximation order of both reconstructions drops to around 1.5.
Constrained Reconstruction in MUSCL-Type Finite Volume Schemes 0.12
0.14
0.10
0.12
0.08
0.10
0.06
0.08
0.04
0.06
0.02
0.04
0.00 −2 10
−1
10
0
10
1
10
2
10
3
10
4
10
0.02 −1 10
0
625
1
10
10
2
10
3
10
4
10
Fig. 2 Efficiency of different reconstructions on a quadrilateral (left) and a triangular (right) sequence of meshes of varying resolution for the solid body rotation benchmark problem
4.2 Linear Problem In this section, we are interested in the efficiency of the QP and LP reconstructions. To this end, we consider the well-known solid body rotation benchmark problem proposed by LeVeque [10]. In the unit square Ω = (0, 1)2 , we consider the counterclockwise rotation about the center with periodicity T = 2π , ∂t u + ∇ · (vu) = 0 in Ω × (0, T ), v(x, y) = 21 − y, x − 21 . The initial data consists of a slotted cylinder, a cone, and a smooth hump, each of which is restricted to a circular domain of radius r = 0.15, ⎧ ⎪ 1 − χ[0.475,0.525] (x1 ) χ[0,0.85] (x2 ) ⎪ ⎪ ⎪ ⎨1 − |x−xc | r u 0 (x) = 1 1 h| ⎪ + 4 cos π |x−x ⎪ 4 r ⎪ ⎪ ⎩ 0
if |x − xs | ≤ r, if |x − xc | ≤ r, if |x − x h | ≤ r, otherwise.
where xs = (0.5, 0.75), xc = (0.5, 0.25), and x h = (0.25, 0.5), respectively. Figure 2 shows two plots of the L 1 -errors over the computation time. We used the exact same series of triangular and quadrilateral meshes as in the previous section. In both cases, the LP and QP reconstructions perform much better than the limited least-squares fit. While the latter is easy to implement and inexpensive to compute, solving a quadratic or linear minimization problem on each cell results in a more efficient scheme even in the presence of strong discontinuities.
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4.3 Euler Equations of Gas Dynamics In this final section, we consider the d-dimensional Euler equations of gas dynamics, ∂t U + ∇ · F(U ) = 0, where U = (ρ, ρv, E) ∈ Rd+2 is the vector of conserved quantities, i.e., the density ρ, the momentum ρv, and the total energy density E. By v ∈ Rd , we denote the primitive particle velocity. The convective flux is given by ⎛
⎞ ρv F(U ) = ⎝ρv ⊗ v + p Id ⎠ ∈ R(d+2)×d , (E + p)v where Id denotes the d-dimensional identity matrix. The pressure p = p(U ) is given by the equation of state p(U ) = (γ − 1) E − ρ2 |v|2 with the adiabatic constant γ = 1.4. A state U is considered physical if the density ρ and the pressure p are strictly positive. It is well known that unphysical values in an approximate solution typically lead to the immediate break down of a numerical simulation. The design of higher order physicality-preserving schemes, however, is a challenging task (see, e.g., the survey article [15]). In contrast to the LP reconstruction presented in Definition 1, the QP approach in Definition 2 allows for a number of modifications to the set of admissible functions. Here, we present a particularly simple variant that in the special case of Cartesian meshes still coincides with the minmod limiter. For all E ∈ G let W (E; u) = W ∈ [P 1 (Rd )]d+2 | W (x E ) = W E and
min{W E , W E,E } ≤ W (x E,E ) ≤ max{W E , W E,E } for all E ∈ N (E) , where for e = E ∩ E with centroid xe we set W E,E =
|x E − xe | |x E − xe | WE + WE . |x E − xe | + |x E − xe | |x E − xe | + |x E − xe |
This ensures that all admissible polynomials are physical at least in the midpoints of inter-element intersections. Next, we are interested in the solution of the three-dimensional shock tube experiment. The computational domain is a cylinder Ω = x ∈ R3 | −1 < x1 < 1 and x22 + x32 < 0.2 . In primitive variables, the initial data is given by
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Table 2 Initial left and right states in primitive variables for the three-dimensional shock tube experiments WL WR ρ v1 v2 v3 p ρ v1 v2 v3 p Sod problem 1 p123 problem 1
0 −2
0 0
0 0
1 0.4
0.125 1
0 2
0 0
0 0
0.1 0.4
Fig. 3 Unstructured tetrahedral mesh for the discretization of the three-dimensional shock tube
W (·, 0) =
W L if x1 < 0, W R otherwise.
The left and right states W L , W R for two different test settings are chosen as in [13, Chap. 4.3], see Table 2. The first test problem is known as Sod test problem; its solution consists of a left rarefaction, a contact discontinuity, and a right shock. The second test problem is the so-called 123 problem; in this case, the solution consists of two rarefactions and a trivial stationary contact discontinuity. The latter benchmark test is particularly challenging as small densities and pressures occur. We solve the Riemann problems for times t ≤ 0.5 in case of the Sod problem, and for times t ≤ 0.15 in case of the p123 problem. At the left and right boundary, we prescribe W = W L and W = W R , respectively. Otherwise, slip boundary conditions are imposed. The solutions can be computed in a quasi-exact manner as in case of the one-dimensional Euler equations, see, e.g., [13]. The computational domain is discretized by a series of unstructured, affine tetrahedral grids. Instead of refining the coarsest grid several times, each grid was created separately using the Gmsh mesh generator [7]. It was ensured that the discontinuity in the initial data is exactly resolved by the grid. An example grid is shown in Fig. 3. For comparison, we computed the solution to a classical first-order finite volume scheme as well; in all cases, a Harten–Lax–van Leer (HLL) numerical flux was used. The L 1 -errors and EOCs against the quasi-exact solution in primitive variables are shown in Table 3. In terms of the convergence rate, the physicality-preserving QP
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Table 3 Three-dimensional shock tube experiment, L 1 -errors, and convergence rates against the quasi-exact solutions in primitive variables to the Riemann problems under consideration Elements
hG
Sod problem
p123 problem
1st-order
W − WG L 1 288
3.48×10−1 5.53×10−2
2nd-order EOC W − WG L 1
1st-order EOC W − WG L 1
2nd-order EOC W − WG L 1
EOC
—
4.05×10−2
—
8.75×10−2
—
6.21×10−2
—
1680
2.13×10−1
4.19×10−2
0.57
2.57×10−2
0.93
6.82×10−2
0.51
4.39×10−2
0.71
9937
1.11×10−1 2.81×10−2
0.61
1.46×10−2
0.86
4.77×10−2
0.54
2.54×10−2
0.83
71788
5.66×10−2 1.75×10−2
0.71
8.49×10−3
0.81
2.97×10−2
0.71
1.48×10−2
0.81
559593
2.88×10−2 1.04×10−2
0.77
4.73×10−3
0.87
1.73×10−2
0.80
8.55×10−3
0.82
4396447
1.50×10−2 6.14×10−3
0.81
2.65×10−3
0.89
1.03×10−2
0.80
5.28×10−3
0.74
reconstruction shows only little advantage over the first-order scheme. We remark that some physicality treatment is required to solve the above problems. In particular, both, the unmodified QP reconstruction as well as the LP reconstruction fails due to the occurrence of unphysical values. For all its simplicity, the modified MUSCL-type scheme is robust and gives considerably smaller errors compared to the first-order method at the exact same number of degrees of freedom.
5 Conclusion In this paper, we proposed a new QP reconstruction method for MUSCL-type finite volume schemes. For each grid element, we computed the best admissible fit in a least-squares sense. We showed that the QP reconstruction generalizes the multidimensional minmod reconstruction for Cartesian meshes. No restrictions to the grid dimension, the conformity of the grid, or the shape of individual grid elements were made. The local cell problems only involve data associated with direct neighbors of a grid element and thus preserve the locality of the numerical method. We compared our reconstruction against similar techniques proposed in the literature. By numerical experiments, we showed that the minimization problems are indeed inexpensive to solve and yield a more accurate and efficient approximation.
References 1. M. Alkämper, A. Dedner, R. Klöfkorn, M. Nolte, The DUNE-ALUGrid Module. Arch. Numer. Softw. 4(1), 1–28 (2016) 2. T.J. Barth, D.C. Jespersen, The Design and Application of Upwind Schemes on Unstructured Meshes (American Institute of Aeronautics and Astronautics, 1989). Preprint AIAA-89-0366
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3. P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Klöfkorn, M. Ohlberger, O. Sander, A generic grid interface for parallel and adaptive scientific computing. part I: Abstract framework. Computing 82(2–3), 103–119 (2008) 4. M. Blatt, A. Burchardt, A. Dedner, C. Engwer, J. Fahlke, B. Flemish, C. Gersbacher, C. Gräser, F. Gruber, C. Grüninger, D. Kempf, R. Klöfkorn, T. Malkmus, S. Müthing, M. Nolte, M. Piatowski, O. Sander, The distributed and unified numerics enviromment,version 2.4. Arch. Numer. Softw. 4(100), 13–29 (2016) 5. T. Buffard, S. Clain, Monoslope and multislope MUSCL methods for unstructured meshes. J. Comput. Phys. 229(10), 3745–3776 (2010) 6. L. Chen, R. Li, An integrated linear reconstruction for finite volume scheme on unstructured grids. J. Sci. Comput. 68(3), 1172–1197 (2016) 7. C. Geuzaine, J.F. Remacle, Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng. 79(11), 1309–1331 (2009) 8. M.E. Hubbard, Multidimensional slope limiters for MUSCL-type finite volume schemes on unstructured grids. J. Comput. Phys. 155(10), 54–74 (1999) 9. D. Kröner, Numerical Schemes for Conservation Laws (Wiley-Teubner, 1997) 10. R.J. LeVeque, High-resolution conservative algorithms for advection in incompressible flow. SIAM J. Numer. Anal. 33(2), 627–665 (1996) 11. S. May, M. Berger, Two-dimensional slope limiters for finite volume schemes on noncoordinate-aligned meshes. SIAM J. Sci. Comput. 35(5), A2163–A2187 (2013) 12. C.W. Shu, S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988) 13. E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics (Springer, Heidelberg, 1997) 14. B. van Leer, Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. J. Comput. Phys. 32(1), 101–136 (1979) 15. X. Zhang, C.W. Shu, Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments. Proc. R. Soc. A 467(2134), 2752–2776 (2011)
A Posteriori Analysis for the Euler–Korteweg Model Jan Giesselmann and Dimitrios Zacharenakis
Abstract Recently, the authors of Giesselmann and Pryer (IMA J Numer Anal, 36(4):1685–1714, 2016, [11]) used a DG scheme in a multiphase problem of elastodynamics to derive a posteriori error estimates for the difference of the numerical and the exact solution. Based on this, our goal is to provide an a posteriori analysis of a DG scheme for the approximation of the one-dimensional isothermal EK system. A fundamental component of our analysis is the reduced relative entropy stability framework (Dafermos, Arch Ration Mech Anal, 70(2):167–179, 1979, [5]), which is employed since the energy density has a multi-well shape. Moreover, this technique requires a certain amount of regularity. Thus, we consider reconstructions (Makridakis and Nochetto, Numer Math, 104(4):489–514, 2006, [18]) of the numerical solution that possess this property. We expect that there exists an a posteriori error estimator using a local DG (LDG) formulation, similar to Karakashian and Pascal (SIAM J Numer Anal, 41(6):2374–2399, 2006, [15]). Keywords Compressible two-phase flow · Liquid–vapor flow · Diffuse interface model · Relative entropy · Discontinuous Galerkin method · A posteriori error analysis
1 Introduction We study a numerical discretization of the Euler–Korteweg (EK) system, which describes the dynamics of a liquid–vapor mixture and undergoes phase transitions. It belongs to the class of phase-field (diffuse interface) models (see [1, 3] and references therein), where the interface is represented by a narrow layer across in which the fields J. Giesselmann · D. Zacharenakis (B) Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany e-mail: [email protected] J. Giesselmann e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_48
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Fig. 1 Double-well shape of the Helmholtz free energy density
W(ρ) W(
ρ 0
vary smoothly. The EK model is obtained from the Navier–Stokes–Korteweg (NSK) equations by neglecting viscosity. The fundamental ideas underlying the NSK model originate in the nineteenth century. It has been studied theoretically by many authors (see [2, 4, 16] and references therein) and has gained popularity in recent years as a tool for numerical simulations of two-phase flows (see [7, 13] and references therein). For a derivation, using modern thermodynamical methods, of NSK one has to consider the conservation of mass, momentum, and energy, and the entropy production equation (second law of thermodynamics) (see [9]). We restrict to the isothermal NSK system in 1d ρt + (ρv)x = 0 2 in × [0, T ), (ρv)t + ρv + p(ρ) x = μvx x + γρρx x x
(NSK)
with the mass density ρ = ρ(x, t) > 0 and the velocity field v = v(x, t) being the unknown quantities. T > 0 and μ ≥ 0 are parameters that describe a final time and viscosity coefficient, respectively. We model capillarity effects by including density gradients in the energy, leading to the term γρρx x x in (NSK), where γ > 0 is a capillarity coefficient. Furthermore, the pressure is denoted by p and is expressed via the Gibbs–Duhem equation p(ρ) = ρW (ρ) − W (ρ) ⇒ p (ρ) = ρW (ρ)
(1)
where W (ρ) is called the Helmholtz free energy density. In order to describe twophase flows, W has a double-well shape, meaning that it is non-convex, which makes the pressure a non-monotone function. We make the following assumptions on W ∈ C 2 ((0, ∞)): (i) ∃ a1 , a2 ∈ (0, ∞) such that a1 < a2 and W > 0 in (0, a1 ) (a2 , ∞) and W < 0 in (a1 , a2 ), (ii) W ≥ 0 in (0, ∞).
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We say a fluid is in vapor phase if ρ ∈ (0, a1 ], in liquid phase if ρ ∈ (a2 , ∞], and in spinodal phase if ρ ∈ (a1 , a2 ), see Fig. 1. We propose a discontinuous Galerkin finite element scheme for the approximation of the EK system and describe an approach for deriving an a posteriori error estimate. In our a posteriori analysis, we consider the use of relative entropy technique, which is the natural stability framework for problems of the form (NSK) and has been studied for hyperbolic conservation laws and related systems by [5, 8]. For a general overview of its development in the last decades, we refer the reader to the references in [6, Sect. 5.7]. The relative entropy technique is based on the fact that systems of hyperbolic conservation laws are usually endowed with a strictly convex entropy/entropy flux pair. This pair can be used to define the notion of relative entropy between two solutions. As we said, in our multiphase model, the energy density has double-well shape. To account for this non-convexity of the energy, we employ the reduced relative entropy technique which is a modification of the relative entropy framework and considers only the convex contributions of the energy [11, 12]. We use it for bounding the difference between the reconstructions and the exact solution of our system. Since reconstructions need to have a certain amount of regularity in order to apply the reduced relative entropy, we have to combine two reconstruction approaches to obtain the desired regularity. We use a posteriori results to bound the difference between the numerical solution and the reconstruction. The latter is based on an elliptic reconstruction operator, which was studied in ([17, 18] and their references). The elliptic reconstructions of our numerical solutions are the exact solutions of elliptic problems. The paper is organized as follows: In Sect. 2, we give some basic definitions and introduce the model problem with its conservative properties. In Sect. 3, we introduce the reduced relative entropy technique which is used to prove a stability result in Theorem 1. In Sect. 4, we define the semi-discrete scheme, some of its properties, and introduce the operators which we require for the a posteriori analysis. Finally, in Sect. 4.4, we derive an a posteriori estimate for solutions using a combination of reconstructions.
2 The Euler–Korteweg (EK) System In this section, we describe the model. At first, we fix some notation. Let ⊆ R be an interval. We introduce the Sobolev spaces H k () := φ ∈ L 2 () : D α φ ∈ L 2 () , for |α| ≤ k which are equipped with norms and semi-norms
u 2k := u 2H k () =
α≤k
D α u 2L 2 () , |u|2k := |u|2H k () =
α=k
D α u 2L 2 ()
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respectively, and D α are derivatives in the weak sense. We also introduce the timedependent Sobolev spaces: C i (0, T ; H 1 ()) := u : [0, T ] → H 1 () : u and i temporal derivatives are continuous .
In the following, we consider the case, where μ = 0 in (NSK): ρt + (ρv)x = 0 2 in × [0, T ). (ρv)t + ρv + p(ρ) x = γρρx x x
(EK)
We couple (EK) with periodic boundary conditions and denote S 1 = to be the unit interval with coinciding end points. Remark 1. We choose periodic boundary conditions because the upcoming calculations are easier since we can remove the boundary terms. Extending our results to other types of boundary conditions is not straightforward. The next Lemma follows from [9]: Lemma 1. (Energy Balance) Let (ρ, ρv) be a smooth solution of (EK). Then it satisfies the energy balance d d η(ρ, ρv) := dt dt
1 γ W (ρ) + ρv2 + (ρx )2 d x 1 2 2 S
= 0.
3 Reduced Relative Entropy The classical relative entropy technique is used for stability analysis of hyperbolic conservation laws and was introduced by Dafermos and Di Perna [5, 8]. Definition 1. (Reduced Relative Entropy) Given states (ρ, ρv) and (ρ, ˆ ρˆ vˆ ), the relative entropy between them is defined as
δη T T ρˆ , ρ − ρˆ η (ρ ρv) (ρˆ ρˆ vˆ ) = η(ρ, ρv) − η(ρ, ˆ ρˆ vˆ ) − δρ δη ρˆ vˆ , ρv − ρˆ vˆ . − δ(ρv) The terms in brackets denote the directional derivatives in the direction ρ − ρˆ and ρv − ρˆ vˆ . Since our energy has non-convex parts, the relative entropy is not suitable to measure the distance between solutions. Therefore, we employ the reduced relative entropy, which only considers the convex parts. It is defined as η R (t) :=
1 2
S1
2 2 ρ v − vˆ + γ ∂x ρ − ρˆ d x.
(2)
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Remark 2. The reduced relative entropy is useful for comparing solutions, since we have the higher order terms in (2). The kinetic term controls the difference v − vˆ and ˆ the second term is equivalent to the squared H 1 semi-norm between ρ and ρ. Remark 3. (L ∞ bound for ρ) Since Lemma 1 holds and the mean value of ρ does not change in time, we notice that ρ L ∞ (S 1 ×(0,∞)) is bounded in terms of the initial data and we call this bound Cρ . Theorem 1. (Reduced Relative Entropy Bound) Let (ρ, ρv) be the solution of (EK) and (ρ, ˆ ρˆ vˆ ) a strong solution to the perturbed system ∂t ρˆ + ∂x ρˆ vˆ = 0 ∂t ρˆ vˆ + ρ∂ ˆ x W (ρ) ˆ x x x ρˆ = R, ˆ + ∂x ρˆ vˆ 2 − γ ρ∂
(3)
where R is some given L 2 (S 1 × (0, t)) function. Let there be a computable lower bound for ρˆ denoted by A. Then, the reduced relative entropy defined in (2) satisfies: −2 η R (t) ≤ η R (0) + R 2L 2 (S 1 ×(0,t)) eCρ A t , ∀t Proof. We explicitly compute the time derivative of η R (t), use the original problem (EK), the perturbed system (3) and obtain
v − vˆ ρ∂ ˆ x W (ρ) ˆ − ρ∂x W (ρ) + v − vˆ ρ − ρˆ ∂x W (ρ) ˆ +
ρ + γ ∂x vˆ ρ − ρˆ ∂x x ρ − ρˆ + v − vˆ − R + ρˆ 1 ρ 2 ρ vˆ 2 ρ + ρˆ vˆ x − (ρv)x v − vˆ d x. + v − vˆ − ∂x v − ρˆ vˆ x 2 ρˆ ρˆ 2
dt η R (t) =
S1
Following the arguments from [14, Lem. 4.4], we obtain dt η R (t) ≤ Cη R (t) +
ρ R d x. − v − vˆ ρˆ S1
We need to derive a bound for the second term. It can be estimated as
√ 2
ρ 2 ρ + R 2 2 ≤ ρ √ρ v − vˆ 2 2 + R 2 2 R d x
≤ v − v ˆ − v − vˆ L L L ρˆ ρˆ ρˆ L ∞ S1 L2 ≤ Cρ A−2 η R (t) + R 2L 2 .
Thus, we have dt η R (t) ≤ Cρ A−2 η R (t) + R 2L 2 (S 1 ) . Finally, using the Gronwall inequality, we obtain the desired result.
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4 Discretization and A Posteriori Setup In this section, we start our numerical analysis with the discretization of our model, where we introduce auxiliary variables in order to reformulate our system. Our scheme is nearly energy consistent and we use two families of reconstructions, one set of discrete reconstructions and one set of elliptic reconstructions. Definition 2. (Finite Element Space) We let the unit interval S 1 = [0, 1] and denote K i = xi , xi+1 the ith subinterval, where we choose some x0 < . . . < x N . K1
S1 = [0, 1] 0= x0
x1
x2
xN−1
1= xN
Since we employ a discontinuous Galerkin scheme our finite element space Vq := : I → R : | K i ∈ Pq (K i ) consists of discontinuous piecewise polynomials of degree less or equal than q ∈ N. We denote by h i := |K i | the length of each N −1 subinterval and by J = {K i }i=0 the set of all cells. By E we denote the set of all common interfaces of J. For xn ∈ E, we define h E (xn ) = 21 (h n−1 + h n ) such that h E ∈ L ∞ (E). In addition, we introduce the broken Sobolev space H 1 (J) = φ : φ| K ∈ H 1 (K ) , ∀K ∈ J and define the jump operator for arbitrary scalar functions v ∈ H 1 (J) as [[v]] := v− − v+ := lim v (· − s) − lim v (· + s) . s0
s0
Definition 3. (Discrete Gradient Operators) We define G ± : H 1 (J) → Vq , such that for any ψ ∈ H 1 (J) we have S1
G ± [ψ] d x =
K ∈J
(∂x ψ) d x − K
E
[[ψ]] ± , ∀ ∈ Vq .
4.1 Mixed Formulation We reformulate (EK) by introducing two auxiliary variables, τ and ω. Furthermore, we use (1) and we search (ρ, v, τ, ω) such that ρt + (ρv)x = 0, ρvt + ρτx = 0, 1 τ − W (ρ) − v2 + γ ωx = 0, ω − ρx = 0. 2
(4)
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4.2 Discrete Scheme We examine the following semi-discrete numerical scheme, where we seek (ρh , vh , τh , ωh ) ∈ C 1 (0, T ; Vq ) × C 1 (0, T ; Vq ) × C 0 (0, T ; Vq ) × C 1 (0, T ; Vq ) such that ∂t ρh + G − [ρh vh ] d x = 0, ∀ ∈ Vq , 1 S ρh ∂t vh + ρh G + [τh ] d x = 0, ∀ ∈ Vq , S1
(5) 1 τh − W (ρh ) − vh2 + γ G + [ωh ] Z d x = 0, ∀Z ∈ Vq , 2 S1 ∂t ωh − G − [∂t ρh ] X d x = 0, ∀X ∈ Vq . S1
Proposition 1. (Near Energy Consistency) Let (ρh , vh , τh , ωh ) be a solution of (5). Then the following holds for 0 < t < T d dt
1 γ W (ρh ) + ρh vh2 + ωh2 d x 1 2 2 S
=
E
[[τh ]]
Pq [ρh vh ]
+
− (ρh vh )+ ,
where Pq : V2q → Vq is a L 2 −orthogonal projection. Proof. Take = τh , = vh , Z = ∂t ρh , X = γ ωh in (5) and use discrete integration by parts. The desired result yields analogously as in [11, Proposition 4.8].
4.3 A Posteriori Setup The fundamental idea of the reconstruction approach [18] is to use the appropriate PDE stability theory, here reduced relative entropy, for bounding the difference between the reconstructions and the exact solution of the PDE. Thus, obviously, the reconstruction needs to have a certain amount of regularity, which is necessary for applying the reduced relative entropy. We combine two reconstruction approaches to obtain the desired regularity. While the discrete reconstruction is computable, the elliptic one is not. Therefore, the “whole” reconstruction is not computable and we need to make use of other (elliptic) a posteriori results, in order to bound the difference between the numerical solutions and the reconstructions and to bound the residual in a computable way. Definition 4. (Discrete Reconstruction Operators) We define D ± : H 1 (J) → Vq+1 by requiring the following equalities for all ∈ H 1 (J):
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± ∂x D ± [] − G ± [] d x = 0, ∀ ∈ Vq and D ± [] = ∓ , on E.
Remark 4. (Notation Convention) We use the convention that a b means a ≤ Cb for a generic constant C, in order to eliminate the excessive use of constants. Lemma 2. The discrete reconstruction operators D ± are constructed such that for all ∈ H 1 (J) the reconstruction D ± [] ∈ Vq+1 ∩ C 0 (S 1 ) are continuous and
S1
D ± [] − d x = 0, ∀ ∈ Vq , ∈ Vq−1 (or thogonalit y).
Furthermore, the following approximation properties are satisfied: 2 − D ± []2 2 1 h ± [[]] E 2 L (S ) L (E) 2
− D ± [] 2 h −1 [[]] . E dG L 2 (E)
Proof. Can be found in [18]. Definition 5. (Continuous Projection Operator) We define PqC : L 2 (J) → Vq ∩ C 0 (S 1 ) as the L 2 (S 1 ) orthogonal projection operator satisfying
S1
PqC [ζ ] d x =
S1
ζ d x, ∀ ∈ Vq ∩ C 0 (S 1 ).
It also has the following properties (see [11]): • PqC [ζ ] L 2 (S 1 ) ≤ ζ L 2 (S 1 ) (L 2 −stability) • P C [ζ ] − ζ 2 1 h q+1 ζ H q+1 (S 1 ) (optimal approximation) q
L (S )
Definition 6. (Elliptic Reconstruction Operators) We define reconstructions R1 [ρh ] ∈ H 3 (S 1 ), R2 [ρh ] ∈ H 2 (S 1 ), R[ρh vh ] ∈ H 2 (S 1 ) as solutions of the following elliptic equations, respectively: C [W (ρh )] − D + [τh ] + γ ∂x x R1 [ρh ] = Pq+1
1 − 2 D [vh ], 2
(6a)
1 γ ∂x x R2 [ρh ] = W (ρh ) − τh + vh2 , 2
(6b)
∂x x R[ρh vh ] = −∂xt R1 [ρh ],
(6c)
where in each case we impose the following normalization condition:
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S1
R1 [ρh ] − ρh d x =
S1
R2 [ρh ] − ρh d x =
S1
R[ρh vh ] − ρh vh d x = 0.
Remark 5. The problems that define the above reconstruction operators are well posed, since we have unique solvability of the elliptic problems. Moreover, elliptic regularity gives the following (see [11]):
R1 [ρh ] H k+1 (S 1 )
R2 [ρh ] H k+1 (S 1 )
1 − 2 1 C + Pq+1 W (ρh ) − D [τh ] + D vh k−1 1 ∀k ∈ {0, 1, 2}, γ 2 H (S ) 1 1 2 W (ρh ) − τh + vh ∀k ∈ {0, 1}, γ 2 k−1 1 H
R[ρh vh ] H k+1 (S 1 ) ∂xt R1 [ρh ] H k−1 (S 1 )
(S )
∀k ∈ {0, 1, 2}.
Remark 6. Note that we discretized the Laplacian in (6b) in a local DG fashion (LDG), cf. (5). There exists an a posteriori error estimator H1 depending only upon ρh and the problem data such that |R2 [ρh ] − ρh |dG
1 1 2 τh − W (ρh ) − vh . H1 ρh , γ 2
The estimator is similar to, and can be derived analogously as, the estimator in [15]. Lemma 3. (Reconstructed PDE System) The reconstructions defined in Definition 6 satisfy a perturbed version of (EK)
∂t R[ρh vh ] + R1 [ρh ]∂x W (R1 [ρh ]) + ∂x
R[ρh vh ] R1 [ρh ]
2
∂t R1 [ρh ] + ∂x R[ρh vh ] = 0 − γ R1 [ρh ]∂x x x R1 [ρh ] = E,
where E := E 1 + E 2 + E 3 + E 4 + E 5 + E 6 R[ρh vh ] ∂x (R[ρh vh ]) − vh ∂x D − [ρh vh ] E 1 = ∂t (R[ρh vh ] − ρh vh ) , E 2 = R1 [ρh ] C [W (ρ )] , E = (R [ρ ] − ρ ) ∂ P C [W (ρ )], E 3 = R1 [ρh ]∂x W (R1 [ρh ]) − Pq+1 4 1 h h h x q+1 h R[ρh vh ]2 1 1 E 5 = γ (ρh − R1 [ρh ]) ∂x x x R1 [ρh ], E 6 = R1 [ρh ]∂x − ρh ∂x D − [vh2 ]. 2 2 2 R1 [ρh ]
Proof. Starting from the second equation in the discrete scheme S1
ρh ∂t vh + ρh G + [τh ] d x = 0, ∀ ∈ Vq ,
we assume ρh > 0 in order to write (7) point-wise:
(7)
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ρh ∂t vh + ρh G + [τh ] = 0 ⇒ ρh ∂t vh + ρh G + [τh ] + vh ∂t ρh − vh ∂t ρh = 0.
(8)
Furthermore, from the first equation in the discrete scheme, we have that vh ∂t ρh = −vh G − [ρh vh ]. Moreover, taking now the x-derivative of (6a) and then multiplying it by −ρh , we obtain C 1 [W (ρh )] − ρh ∂x D + [τh ] + ρh ∂x D − [vh2 ] = 0. −γρh ∂x x x R1 [ρh ] + ρh ∂x Pq+1 2 (9)
Taking the sum of (8) and (9) yields the desired result.
Lemma 4. (Bound on the reconstruction of ρh ) Let (ρh , vh , τh , ωh ) be the solution to the discrete scheme and R1 [ρh ] be the reconstruction given in Definition 6. Then 1 D + [τh ] − τh −1 1 + 1 1 D − [v2 ] − v2 h h −1 1 H (S ) γ γ 2 H (S )
1 1 1 C τh − W (ρh ) − vh2 + Pq+1 +H1 ρh , [W (ρh )] − W (ρh ) −1 1 := M. H (S ) γ 2 γ |R1 [ρh ] − ρh |dG
Proof. Analogous to [11, Lemma 5.3]. As an example, we show how to derive a computable bound for E 3 . Lemma 5. (Computable bound on E 3 ) Let the conditions of Lemma 1 hold, then
2 1 1 1 2q+2 W (ρh )2 q+1
E 3 2L 2 (S 1 ) H1 ρh , τh − W (ρh ) − vh2 + 2 hK (K ) H γ 2 γ K ∈J v2 2 2 h h 2q 2 2 + + 2 [[τh ]] L 2 (E) + [[ρh ]] L 2 (E) + [[ ]] h K W (ρh ) H q+1 (K ) 2 2 γ L (E)
K ∈J
1 − 2 2 1 C + P D · v W τ + − D (ρ ) h h h q+1 2 γ2 L (E) H −1 (S 1 )
2 −1 + h E [[ρh ]] 2
Proof. Using the triangle inequality, we have C [W (ρ )] 2
E 3 2L 2 (S 1 ) = R1 [ρh ]∂x W (R1 [ρh ]) − Pq+1 2 1 h L (S )
2
2
2 C
R1 [ρh ] L ∞ (S 1 ) W (R1 [ρh ]) − W (ρh ) dG + W (ρh ) − Pq+1 [W (ρh )] = dG
R1 [ρh ] 2L ∞ (S 1 ) E 3,1 + E 3,2 .
Now, since we are in 1d we have
R1 [ρh ] 2L ∞ (S 1 ) R1 [ρh ] 2H 1 (S 1 )
2 1 P C W (ρh ) − D + [τh ] + 1 D − v 2 . h q+1 2 γ 2 H −1 (S 1 )
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For E 3,1 we have
2 E 3,1 = W (R1 [ρh ]) − W (ρh ) dG |R1 [ρh ] − ρh |2dG = M12 For E 3,2 we have, similar to [11],
C [W (ρ )] 2 E 3,2 = W (ρh ) − Pq+1 h
dG
2 −1 h [[ρ ]] h E
L 2 (E)
+
2q 2 h K W (ρh ) H q+1 (K )
K ∈J
4.4 A Posteriori Error Estimate Corollary 1. (Reduced Relative Entropy Bound) Let (ρ, ρv) be the solution of (EK) and (ρh , vh , τh , ωh ) be the solution of (5). Then, given the reduced relative entropy η R (t) :=
1 2
R[ρh vh ] 2 ρ v− + γ (∂x (ρ − R1 [ρh ]))2 d x, R1 [ρh ] S1
the following inequality is satisfied η R (t) ≤ η R (0) + E 2L 2 (S 1 ×(0,t)) eCt with E = E 1 + E 2 + E 3 + E 4 + E 5 + E 6 and C = Cρ (min ρh − M)−2 depends on the numerical solution and is computable, where M is defined in Lemma 4. Proof. The proof is a combination of Lemma 3 and Theorem 1.
Remark 7. Using Corollary 1, we can achieve our goal, which is to bound the difference between the exact solutions and the numerical solutions. Remark 8. We expect min ρh − M appearing in Corollary 1 to be positive, provided the mesh size h is sufficiently small. Remark 9. We plan to extend this analysis to the isothermal Navier–Stokes–Korteweg (NSK) system, where the viscosity of the fluid μ > 0, and to several space dimensions. For the latter, the main challenge is the construction of appropriate discrete gradients and discrete reconstructions.
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References 1. D.M. Anderson, G.B. McFadden, A.A. Wheeler, Diffuse interface methods in fluid mechanics. Ann. Rev. Fluid Mech. 30, 139–165 (1998) 2. S. Benzoni-Gavage, R. Danchin, S. Descombes, On the well-posedness for the Euler-Korteweg model in several space dimensions. Indiana Univ. Math. J. 56(4), 1499–1579 (2007) 3. M. Braack, A. Prohl, Stable discretization of a diffuse interface model for liquid-vapor flows with surface tension. ESAIM Math. Model. Numer. Anal. 47(2), 401–420 (2013) 4. D. Bresch, B. Desjardins, C.K. Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems. Comm. Partial Differ. Equ. 28(3–4), 843–868 (2003) 5. C.M. Dafermos, The second law of thermodynamics and stability. Arch. Ration. Mech. Anal. 70(2), 167–179 (1979) 6. C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 3rd edn., Grundlehren Math. Wiss. (Fundamental Principles of Mathematical Sciences), vol 325 (Springer, Berlin, 2010) 7. D. Diehl, J. Kremser, D. Kröner, C. Rohde, Numerical solution of Navier-Stokes-Korteweg systems by local discontinuous Galerkin methods in multiple space dimensions. Appl. Math. Comput. 272(2), 309–335 (2016) 8. R.J. DiPerna, Uniqueness of solutions to hyperbolic conservation laws. Indiana Univ. Math. J. 28(1), 137–188 (1979) 9. J.E. Dunn, J. Serrin, On the thermomechanics of interstitial working. Arch. Ration. Mech. Anal. 88(2), 95–133 (1985) 10. J. Giesselmann, T. Pryer, Reduced relative entropy techniques for a priori analysis of multiphase problems in elastodynamics. BIT Numer. Math. 56, 99–127 (2016) 11. J. Giesselmann, T. Pryer, Reduced relative entropy techniques for a posteriori analysis of multiphase problems in elastodynamics. IMA J. Numer. Anal. 36(4), 1685–1714 (2016) 12. J. Giesselmann, A.E. Tzavaras, Stability properties of the Euler-Korteweg system with nonmonotone pressures. Appl. Anal. 1–19 (2017) 13. J. Giesselmann, C. Makridakis, T. Pryer, Energy consistent dG methods for the Navier-StokesKorteweg system. Math. Comput. 83, 2071–2099 (2014) 14. J. Giesselmann, C. Lattanzio, A.E. Tzavaras, Relative energy for the Korteweg theory and related hamiltonian flows in gas dynamics. Arch. Ration. Mech. Anal. 1–58 (2016) 15. O.A. Karakashian, F. Pascal, A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41(6), 2374–2399 (2006) 16. M. Kotschote, Strong solutions for a compressible fluid model of Korteweg type. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(4), 679–696 (2008) 17. O. Lakkis, C. Makridakis, Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems. Math. Comput. 75, 1627–1658 (2006) 18. C. Makridakis, R.H. Nochetto, A posteriori error analysis for higher order dissipative methods for evolution problems. Numer. Math. 104(4), 489–514 (2006)
Conservation Laws Arising in the Study of Forward–Forward Mean-Field Games Diogo Gomes, Levon Nurbekyan and Marc Sedjro
Abstract We consider forward–forward Mean-Field Game (MFG) models that arise in numerical approximations of stationary MFGs. First, we establish a link between these models and a class of hyperbolic conservation laws as well as certain nonlinear wave equations. Second, we investigate existence and long-time behavior of solutions for such models. Keywords Mean-field games · Conservation laws · Nonlinear wave equations
1 Introduction A few years ago, Lasry and Lions [13] and Caines, Huang and Malhame [11] independently introduced the Mean-Field Game (MFG) framework. These games model competitive interactions in a population of agents with a dynamics given by an optimal control problem. A typical MFG is determined by the system
−u t + H (x, Du) = εΔu + g[m] m t − div(D p H (x, Du(x)m)) = εΔm
Td × [0, T ] Td × [0, T ].
(1)
Here, Td is the d-dimensional torus and the Hamiltonian, H , the coupling, g, and the terminal time, T > 0, are prescribed. The first equation in (1) is a Hamilton–Jacobi equation. This equation states the optimality of the value function, u, associated with the control problem. The second equation is the Fokker–Planck equation that D. Gomes · L. Nurbekyan · M. Sedjro (B) CEMSE Division, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia e-mail: [email protected] L. Nurbekyan e-mail: [email protected] D. Gomes e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_49
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determines the evolution of the density of the agents, m. ε ≥ 0 is a viscosity parameter. If ε = 0, we refer to (1) as a first-order MFG. Otherwise, we refer to (1) as a parabolic MFG. System (1) is typically complemented with initial–terminal conditions: u(x, T ) = u T (x) (2) m(x, 0) = m 0 (x). Extensive research has been conducted in the study of MFGs. For the parabolic problem, strong and weak solutions were, respectively, examined in [8, 9, 12] and [12, 14]. The stationary problem for the parabolic case has also generated great interest— several results on the existence of classical and weak solutions were obtained in [4–7]. The uniqueness of a solution in all these cases relies on the monotonicity of g. Here, we consider a related problem, forward–forward MFG, that is derived from (1) by reversing the time in the Hamilton–Jacobi equation. Accordingly, we consider the system u t + H (x, Du) = εΔu + g[m] (3) m t − div(D p H (x, Du(x))m) = εΔm. Because of the time reversal, we prescribe initial–initial conditions:
u(x, 0) = u 0 (x) m(x, 0) = m 0 (x)
(4)
for (3). Forward–forward models were first introduced in [1] to numerically approximate solutions of stationary MFGs. Before our contributions [10], no rigorous results on the long-time convergence of forward–forward MFGs had been proven. Additionally, the forward–forward problem is interesting on its own right as a learning game. In a standard MFG, agents follow optimal trajectories of a terminal-value optimal control problem. For the forward–forward problem, only initial data is given. Thus, only past optimal trajectories are relevant to the density evolution. Accordingly, the density evolution feeds on past information of the density. This paper complements the results in [10] by examining several cases that can be studied explicitly. In Sect. 2, we consider linear Hamiltonians and show that the wave equation is a special case of the forward–forward model. In Sect. 3, we study quadratic forward–forward MFGs using elementary conservation law techniques. In particular, we compute Riemann invariants and characterize invariant domains. We end the paper by recalling the main result from [10] on the convergence of forward– forward MFGs. The study of forward–forward MFG presents substantial challenges even in dimension one which we consider here. The first-order forward–forward problem can be rewritten as a nonlinear wave equation that inherits the nonlinearity of the Hamiltonian. For quadratic Hamiltonian, the system reduces to elastodynamics equation. In general, the forward–forward problem can be rewritten, formally, as a system
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of one-dimensional conservation law. This reformulation allows us to use methods and ideas from the theory of conservation laws such as hyperbolicity, genuinely nonlinearity, Riemann invariants, and invariant domains. For the parabolic forward–forward problem, standard techniques yield existence and uniqueness of a solution. Here, we investigate the long-time convergence of this solution to the solution of a stationary MFG.
2 First-Order, One-Dimensional, Forward–Forward Mean-Field Games as Nonlinear Wave Equations In this section, we consider the first-order, one-dimensional, forward–forward MFG: u t + H (u x ) = g(m), (5) m t − (m H (u x ))x = 0. To gain insight into the system above, we consider two simple examples: linear and quadratic Hamiltonians. First, we assume that H is linear, that is, H ( p) = p, and that the coupling, g, is smooth invertible with g = 0. In this case, u satisfies the wave equation: (6) u tt − u x x = 0. Thus, for smooth initial data in (4), solutions of (5) are 1 u(x, t) = u 0 (x − t) + 2
x+t
g(m 0 (s))ds,
(7)
x−t
and m(x, t) = m 0 (x − t).
(8)
Next, we assume that H is quadratic, H ( p) = p 2 /2, and that g is logarithmic, g(m) = ln(m). Then, after elementary computations, we obtain that u satisfies the nonlinear wave equation (9) u tt − (1 + u 2x )u x x = 0. This nonlinear equation is known in elastodynamics and, in Lagrangian coordinates, it is a system of hyperbolic conservation laws.
vt − wx = 0 wt − σ (v)x = 0
(10)
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with w = u t , v = u x , and σ (z) = z +
z3 . 3
The system (10) falls within a class of conservation laws investigated in [3], in the whole space, and in [2], in the periodic case.
3 One-Dimensional Forward–Forward Mean-Field Games as Conservation Laws In this section, we discuss how certain one-dimensional forward–forward MFGs can be written as a system of one-dimensional conservation laws. Furthermore, we analyze latter and compute the corresponding Riemann invariants. For simplicity, we consider the forward–forward problem with quadratic Hamiltonian and a quadratic coupling:
u t + u 2x /2 = m 2 /2, m t − (mu x )x = 0.
(11)
We complement (11) with initial–initial condition
u(x, 0) = u 0 (x) m(x, 0) = m 0 (x) > 0.
(12)
We note that the Fokker–Planck equation preserves positivity. As such, the density m(t, ·) is positive for all t > 0. We formally differentiate the first equation with respect to x and then set v = u x . As a result, we obtain
vt + v2 /2 − m 2 /2 x = 0, m t − (mv)x = 0.
(13)
The associated flux function to the system of conservation laws (13) is given by F(v, m) = (v2 /2 − m 2 /2, −vm)
m > 0, v ∈ R.
(14)
Finally, we compute its Jacobian and get
v −m D F(v, m) = . −m −v
(15)
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3.1 Hyperbolicity and Genuine Nonlinearity A simple computation shows that (15) has two distinct eigenvalues, λ1 and λ2 , given by λ1 = − v2 + m 2 and λ2 = v2 + m 2 ,
(16)
with respective eigenvectors given by r1 =
√ √ −v + v2 + m 2 v + v2 + m 2 and r2 = . m −m
(17)
From the discussion above, the system of conservation laws in (13) is strictly hyperbolic. Note that
√ −m 2 + v v − v2 + m 2 ∇λ1 · r1 = (18) √ m v2 + m 2
√ m 2 − v v + v2 + m 2 ∇λ2 · r2 = . √ m v2 + m 2
and
(19)
Observe that
∇λi · ri = 0 ⇐⇒ m 2 − v v + v2 + m 2 = 0
i = 1, 2.
(20)
As a result, (13) is a strictly hyperbolic genuinely nonlinear system outside the set S given by (21) S := {(v, m) : m 2 = 3v2 , m > 0}.
3.2 Riemann Invariants and Invariant Domains In the following proposition, we provide an explicit expression for Riemann invariants for the system of conservation laws in the quadratic case. As a consequence, we obtain invariant sets for the corresponding problem with viscosity. Proposition 1. The system of conservation laws (11) has the following Riemann invariants w1 (v, m) = (m 2 + v2 )3 − v3 + 3vm 2 and w2 (v, m) =
(m 2 + v2 )3 + v3 − 3vm 2 ,
corresponding to the eigenvectors r1 and r2 .
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0.8
0.8
0.6
0.6
m
1.0
m
1.0
0.4
0.4
0.2
0.2
0.0
0.0 1.0
0.5
0.0
0.5
1.0
1.0
0.5
v
0.0
0.5
1.0
v
(a )Level sets of w1 .
(b) Level sets of w2 .
Fig. 1 Invariant domains associated with the hyperbolic conservation laws (13)
Proof. Note that wi is such that ∇wi is parallel to the eigenvector ri . This means that w1 solves (v + m 2 + v2 )∂v w1 − m∂m w1 = 0. (22) In a similar way, for w2 , (−v +
m 2 + v2 )∂v w1 + m∂m w1 = 0.
(23)
Using Riemann invariants, we can identify invariant domains for the viscosity solutions to (13). These are obtained by looking at level curves of w1 and w2 see Fig. 1.
4 Convergence of One-Dimensional, Forward–Forward, Parabolic Conservation Laws Here, we consider (1) in dimension 1. As before, by differentiating the first equation with respect to x and setting v = u x , we get
vt + (v2 /2 − m 2 /2)x = εvx x , m t − (mv)x = εm x x .
(24)
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The system (24) has a unique local smooth solution for bounded initial data. Now, we investigate the long-time convergence of the solution. For that, we, additionally, require v(x, 0)d x = 0, T
m(x, 0)d x = 1,
(25)
T
which are natural assumptions from the perspective of periodic MFGs. The following theorem is proven in [10]. Theorem 1. If v, m ∈ C 2 (T × (0, +∞)) ∩ C(T × [0, +∞)), m > 0, solve (24) and satisfy (25) then, we have that
|v(x, t)|d x = 0,
lim
t→∞ T
|m(x, t) − 1|d x = 0.
lim
t→∞
(26)
T
Acknowledgements D. A. Gomes was partially supported by KAUST baseline funds and KAUST OSR-CRG2017-3452. L. Nurbekyan and M. Sedjro were supported by KAUST baseline and start-up funds.
References 1. Y. Achdou, I. Capuzzo-Dolcetta, Mean field games: numerical methods. SIAM J. Numer. Anal. 48(3), 1136–1162 (2010) 2. S. Demoulini, D.M.A. Stuart, A.E. Tzavaras, Construction of entropy solutions for onedimensional elastodynamics via time discretisation. Ann. Inst. H. Poincaré Anal. Non Linéaire 17(6), 711–731 (2000) 3. R.J. DiPerna, Convergence of approximate solutions to conservation laws. Arch. Ration. Mech. Anal. 82(1), 27–70 (1983) 4. D.A. Gomes, H. Mitake, Existence for stationary mean-field games with congestion and quadratic Hamiltonians. NoDEA Nonlinear Differ. Equ. Appl. 22(6), 1897–1910 (2015) 5. D.A. Gomes, L. Nurbekyan, M. Prazeres, One-dimensional stationary mean-field games with local coupling. Dyn. Games Appl. 8(2), 315–351 (2018) 6. D.A. Gomes, S. Patrizi, Obstacle mean-field game problem. Interfaces Free Bound. 17(1), 55–68 (2015) 7. D.A. Gomes, S. Patrizi, V. Voskanyan, On the existence of classical solutions for stationary extended mean field games. Nonlinear Anal. 99, 49–79 (2014) 8. D.A. Gomes, E. Pimentel, Time-dependent mean-field games with logarithmic nonlinearities. SIAM J. Math. Anal. 47(5), 3798–3812 (2015) 9. D.A. Gomes, E. Pimentel, Local regularity of mean-field games in the whole space. Minmax Theor. Appl 1(1), 65–82 (2016) 10. D.A. Gomes, L. Nurbekyan, M. Sedjro, One-dimensional forward-forward mean-field games. Appl. Math. Optim. 74(3), 619–642 (2016) 11. M. Huang, R.P. Malhamé, P.E. Caines, Large population stochastic dynamic games: closedloop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6(3), 221–251 (2006) 12. J.-M. Lasry, P.-L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal. C. R. Math. Acad. Sci. Paris 343(10), 679–684 (2006) 13. J.-M. Lasry, P.-L. Lions, Mean field games. Jpn. J. Math. 2(1), 229–260 (2007) 14. A. Porretta, Weak solutions to Fokker-Planck equations and mean field games. Arch. Ration. Mech. Anal. 216(1), 1–62 (2015)
On the Relaxation Approximation for 2 × 2 Hyperbolic Balance Laws Martin Gugat, Michael Herty and Hui Yu
Abstract The relaxation approximation for systems of conservation laws has been studied intensively for example by Bianchini (Commun Pure Appl Math, 59:688– 753, 2006, [5]), Jin and Xin (Commun Pure Appl Math, 48:235–276, 1995, [17]), Liu (Arch Ration Mech Anal, 80:1–18, 1982, [19]), Yong (Basic aspects of hyperbolic relaxation systems, Birkhäuser, Boston, pp. 259–305, 2001, [24]). In this paper, the corresponding relaxation approximation for 2 × 2 systems of balance laws is studied. Our driving example is gas flow in pipelines described by the isothermal Euler equations. We are interested in the limiting behavior as the relaxation parameter tends to zero. We give conditions where the relaxation converges to the states of the original system and counterexamples for cases where the steady states depend on the space variable. Keywords Jin–Xin relaxation · Lyapunov function · Stabilization · Steady states
1 Introduction For numerical purposes, the dynamics of hyperbolic conservation laws have been successfully approximated by relaxation schemes as, for example, proposed in [17]. We are interested in the effects of this relaxation on the damping of perturbations M. Gugat (B) Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Department Mathematik, Cauerstr. 11, 91058 Erlangen, Germany e-mail: [email protected] M. Herty · H. Yu RWTH Aachen University, Institut für Geometrie und Praktische Mathematik, Templergraben 55, 52056 Aachen, Germany e-mail: [email protected] H. Yu Yau Mathematical Sciences Center, Tsinghua University, Jin Chun Yuan West Building, 100084 Beijing, China e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_50
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of steady states of hyperbolic balance laws. Our primary example is gas dynamics in pipelines. We use a Lyapunov function approach similar to [9, 15] to obtain exponential decay of the perturbation. The advantage of the relaxation system is its semilinear structure in contrast to the quasilinear structure of the original system, see [8] for an overview. In the case of conservation laws, the limit of the relaxation system toward the original system has been studied by many authors, among others [5, 7, 10, 11, 17, 20, 24]. This has also been exploited intensively for numerical purposes, see among others [1, 2, 12, 18, 22]. In the case of hyperbolic balance laws, we are not aware of a similar analysis for the corresponding relaxation systems with nonzero source term. We derive decay results independent of the relaxation parameter for time toward infinity extending existing results [13]. This requires the treatment of a positive semidefinite operator in the relaxation approximation. However, the convergence toward the nonconstant steady states of the original system does not hold true in general. To illustrate the different situations, we discuss two examples from gas dynamics. Similar systems have been studied recently, e.g., in [10, 11]. In this paper, our focus is on the limit behavior for small relaxation parameter. We study the long-term behavior using an L 2 –Lyapunov function, whereas in [10] BV, estimates are derived. Also, in [23, 24], the relaxation limit for conservation laws has been discussed and sufficient conditions for the existence of the limiting equation as well as boundary conditions have been analyzed. Note that if the stationary states of the original balance law are constants (in space), then similar results as in [23] for conservation laws hold true. However, for a general balance law, it may well be that the stationary states are spatially dependent.
2 Natural Relaxation Approximation for Hyperbolic Balance Laws Let t ∈ R+ and x ∈ R denote the time and spatial variables, respectively. Consider a 2 × 2 hyperbolic system of balance laws in the following form:
ρt + qx = 0 ,
(1a)
qt + f (ρ, q)x = g(ρ, q) ,
(1b)
where ρ(t, x) and q(t, x) are unknown scalar functions. f and g are given functions that depend continuously differentiably on ρ and q. Clearly, the stationary states of (1), denoted by (ρs , qs ) satisfies qs = constant ,
f (ρs , qs )x = g(ρs , qs ) .
(2)
Our driving example is gas flow in pipeline systems where g denotes the pipe wall friction and f is the momentum flux, see e.g., [4]. The equations are presented in Sect. 4.
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With two real parameters Λ and ε > 0, we consider the following natural semilinear relaxation formulation of the balance law (1), where the additional relaxation variables u and w are introduced: ⎧ ⎪ ⎪ ρt + u x = 0 , ⎨ qt + wx = g(ρ, q) , (3) u t + Λ2 ρx = 1ε (q − u) , ⎪ ⎪ ⎩ wt + Λ2 qx = 1ε ( f (ρ, q) − w) . The parameter Λ determines the slopes of the characteristic curves of the semilinear relaxed problem, and hence it defines the domains of influence in the relaxation system. The value of Λ should be chosen sufficiently large such that the domain of dependence contains the one for the original system and satisfies the subcharacteristic condition proposed in [20]. The second real relaxation parameter is ε > 0 and it yields a stiff source term. In the case of conservation laws, convergence with respect to ε → 0+ has been established by various authors, see e.g., [5, 11, 23]. Formally, we expect that (u, w) tend to (q, f (ρ, q)) when ε → 0 + . In contrast to the relaxation for conservation laws, in the relaxed system (3) in the second equation, the source term g(ρ, q) appears. For any fixed (Λ, ε), the steady states (ρ, ¯ q, ¯ u, ¯ w) ¯ of (3) solve the ordinary differential equations ¯ q), ¯ ρ¯x = u¯ x = 0, w¯ x = g(ρ,
1 1 (q¯ − u), ¯ q¯ x = ( f (ρ, ¯ q) ¯ − w). ¯ εΛ2 εΛ2
In general, those states are completely different from the stationary states of (1); see the example of isothermal Euler equations in Sect. 4.1, where ρs is strictly concave and ρ¯ is strictly convex. Next, we rewrite the system (3) in vector matrix notation. Define U= (ρ, q, u, w)T , ⎛
0 ⎜0 A=⎜ ⎝ Λ2 0
0 0 0 Λ2
1 0 0 0
⎞ 0 1⎟ ⎟ 0⎠ 0
⎛
and
⎞ 0 ⎜ ⎟ g(ρ, q) ⎟ . (4) F(ρ, q, u, w) = ⎜ 1 ⎝ (q − u), ⎠ ε 1 ( f (ρ, q) − w) ε
Then, (3) is equivalent to Ut + A Ux = F(ρ, q, u, w).
(5)
Remark 1. The matrix A has the eigenvalues −Λ and Λ. For each eigenvalue, two linear independent eigenvectors exist. A basis of eigenvectors is given by ⎛
⎛ ⎛ ⎞ ⎛ ⎞ ⎞ ⎞ 0 −1 0 1 ⎜ −1 ⎟ ⎜ 0⎟ ⎜1⎟ ⎜0⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ ⎟ v1 = ⎜ ⎝ 0 ⎠ , v2 = ⎝ Λ ⎠ , v3 = ⎝ 0 ⎠ , v4 = ⎝ Λ ⎠ . Λ 0 Λ 0
(6)
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Thus for the relaxed system at each boundary point, we prescribe two boundary conditions instead of one for the original system. The additional boundary conditions should be chosen in such a way that they are compatible with the original system (1), that is, such that the last two components of F vanish. Those conditions are independent of ε. This choice is as suggested in [17]. In order to analyze the relaxed system (3), (5), respectively, we bring it first in a form with a symmetric system matrix. Lemma 1. Let Y ∈ R4×4 be symmetric and such that A Y = Y A T . Then we have
Y =
where Y1 =
yb ye
ya yb
Y2 Λ2 Y1
Y1 Y2
,
,
Y2 =
(7)
yc yd
yd yg
(8)
with ya , yb , yc , yd , ye and yg being real numbers. Moreover, if Y1 is positive definite and Λ is sufficiently large, the matrix Y is positive definite. The matrix Y will be used to symmetrize the system in a similar way as in assumption (H1) in [6]. However, in our case since g = 0, it is not possible to symmetrize the (linearized) source term simultaneously. Since g = 0, the stationary solutions depend on the space variable x. If we linearize the system locally around a stationary state and make the source term at the same time as symmetric as possible, the corresponding symmetrizer Y will also depend on x, that is Y = Y (x), and we need to take this into account in the forthcoming analysis. Let f ρ , gρ denote the partial derivatives of f , g with respect to ρ and f q , gq denote the partial derivatives of f , g with respect to q. Let U¯ = (ρ, ¯ q, ¯ u, ¯ w) ¯ T denote a continuously differentiable stationary solution of (3), that is ¯ q, ¯ u, ¯ w) ¯ . A U¯ x = F(ρ, = (ρ, Introduce the perturbation solution U ˜ q, ˜ u, ˜ w) ˜ T and decompose the solution ¯ of (3) around the stationary state: U = U + U . We obtain the linearized system for as the first-order perturbation U x = −B U t + A U U
(9)
¯ q), ¯ etc., the matrix B is given by where with the notation f¯ρ = f ρ (ρ, ⎛
0 ⎜ g¯ ρ ⎜ B = −⎜ ⎝ 0 1 ¯ fρ ε
0 g¯ q
0 0
1 ε
− 1ε
1 ¯ f ε q
0
⎞ 0 0 ⎟ ⎟ ⎟ , 0 ⎠ − 1ε
(10)
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and we did not indicate the dependence on x for readability. Since it is not possible to have a completely symmetric BY , we construct a symmetrizer Y such that B Y is symmetric up to a 2 × 2 submatrix. Such a symmetrizer Y is presented in the following lemma. Lemma 2. For ε > 0, consider the symmetrizer Y (x, ε) of A given by
Y1 =
−g¯ q g¯ ρ
g¯ ρ ye
,
Y2 =
g¯ ρ yd
yd yg
,
(11)
where f¯ρ g¯ ρ + f¯q ye − ε g¯ ρ yd . yd = f¯q g¯ ρ − f¯ρ g¯ q , ye = yd + ε g¯ q yd + g¯ ρ2 , yg = 1 + ε g¯ q If Y1 is positive definite and Λ is sufficiently large, the matrix Y is positive definite. Note that Y1 positive definite in particular implies g¯ q < 0. Using the above Y , we have ⎞ ⎛ 0 0 0 0 ⎜ 0 −g¯ q ye − g¯ ρ2 −g¯ q yd − g¯ ρ2 −g¯ q yg − g¯ ρ yd ⎟ ⎟ ⎜ ⎟ . ⎜ (12) BY =⎜ Λ2 g¯ ρ −yg −(Λ2 − f¯ρ )g¯ q −g¯ ρ f¯q 2 ⎟ ε ε ⎠ ⎝ 0 −g¯ q yd − g¯ ρ (Λ2 − f¯ρ ) g¯ ρ − f¯q yd Λ2 ye − f¯ρ yd − f¯q yg 0 −g¯ q yg − g¯ ρ yd ε ε Hence, only the (3, 4)−th and (4, 3)−th components are left to be symmetrized. Denote the symmetric part of the 3 × 3 submatrix in the bottom right corner of BY by B3 , i.e., ⎛
−g¯ q ye − g¯ ρ2
⎜ 2 B3 = ⎜ ⎝ −g¯ q yd − g¯ ρ −g¯ q yg − g¯ ρ yd
−g¯ q yd − g¯ ρ2
−g¯ q yg − g¯ ρ yd
−(Λ2 − f¯ρ )g¯ q −g¯ ρ f¯q (2 Λ2 − f¯ρ ) g¯ ρ − f¯q yd −yg ε 2ε Λ2 ye − f¯ρ yd − f¯q yg (2 Λ2 − f¯ρ ) g¯ ρ − f¯q yd −yg 2ε ε
⎞ ⎟ ⎟ . ⎠
(13)
Lemma 3. The matrix B Y given by (12) is positive semidefinite in the sense that z T BY z ≥ 0 for any z ∈ R4 if B3 is positive semidefinite. The matrix B3 is positive definite if all the leading principal minors are positive. The sufficient conditions are given as follows: for ε small enough and Λ large enough, g¯ q < 0,
f¯q g¯ ρ > f¯ρ g¯ q > 0, yd > −
g¯ ρ2 g¯ q
, Λ2 > −
yd , g¯ q
and ε < −
1 . g¯ q (14)
If (14) holds, (9) is symmetrized in such a way that the matrix of the source term has a one-dimensional kernel and is positive semidefinite. Starting from this representation, the system matrix A can be diagonalized by an orthogonal transformation in such a
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way that the eigenvalues of the symmetric part of the source term are not changed. In other words, the source term remains positive semidefinite with a one-dimensional kernel even after diagonalization of A. For our analysis, the limit behavior of the matrices A and Y as functions of ε are essential. Moreover, Y may depend on the spatial variable x if the stationary states (ρ, ¯ q) ¯ are nonconstant. Let Y (x, ε) denote Y defined as in Lemma 2. Note that Y (x, ε) remains uniformly bounded as ε tends to zero. In fact, we have ⎞ g¯ ρ g¯ ρ yd −g¯ q ⎜ g¯ ρ yd yd f¯ρ g¯ ρ + f¯q yd ⎟ ⎟ Y0 (x) := lim Y (x, ε) = ⎜ 2 ⎠ ⎝ g¯ ρ yd −Λ g¯ q Λ2 g¯ ρ ε→0+ yd f¯ρ g¯ ρ + f¯q yd Λ2 g¯ ρ Λ2 yd ⎛
(15)
and Y0 (x) is positive definite. This implies that if the conditions from Lemma 2 hold, 1/2 1/2 the square roots Y 1/2 (x, ε) and Y0 (x) also exist and lim Y (x, ε)1/2 = Y0 (x), see [16]. Since A is a constant matrix, we obtain ⎛
yd g¯ ρ ⎜ yd ¯ρ g¯ ρ + f¯q yd f lim A Y (x, ε) = A Y0 (x) = ⎜ ⎝ −Λ2 g¯q Λ2 g¯ ρ ε→0+ 2 Λ g¯ ρ Λ2 yd
ε→0+
⎞ −Λ2 g¯q Λ2 g¯ ρ 2 2 ⎟ Λ g¯ ρ Λ yd ⎟ . ⎠ Λ2 g¯ ρ Λ2 yd 2 2 ¯ ¯ Λ yd Λ ( f ρ g¯ ρ + f q yd )
Similar to [6, 19], we consider the symmetric matrix ¯ A(x, ε) = Y (x, ε)−1/2 A Y (x, ε)1/2 . ¯ Since the matrix A(x, ε) is similar to A, it has the same eigenvalues ±Λ that do ¯ not depend on the spatial variable x. A basis of eigenvectors of A(x, ε) is given by Y (x,ε)−1/2 vi wi = Y (x,ε)−1/2 vi for i ∈ {1, 2, 3, 4} with vi defined in (6). ¯ Since A(x, ε) is symmetric, it can be diagonalized by a real orthogonal matrix Q(x, ε) to a constant diagonal matrix D with the diagonal (−Λ, −Λ, Λ, Λ), that is ¯ ε) Q(x, ε) = D . Q T (x, ε) A(x, To make sure that Q(x, ε) can be chosen in such a way that it remains uniformly bounded with respect to ε, we assume that the first column of Q(x, ε) that contains an eigenvector for the eigenvalue −Λ is given by w1 and the third column of Q(x, ε) that contains an eigenvector for the eigenvalue Λ is given by w3 . By orthogonalization, we define the second column in the following way: Let w˜ 2 = v2 − (v2T w1 ) w1 , and define the second column of Q(x, ε) as ww˜˜ 22 . Similarly, we obtain the fourth column. Then, Q(x, ε) is uniquely determined and remains uniformly bounded with respect to ε. Moreover, the limit lim Q(x, ε) =: Q 0 (x) exists. Using the transformation ε→0+
matrix Y (x, ε)1/2 Q(x, ε) that is uniformly bounded with respect to ε, we transform
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our system to a diagonal form. Through this transformation, the source term that corresponds to the matrix B(x, ε) Y (x, ε) remains positive semidefinite, but it is not necessarily uniformly bounded with respect to ε.
3 Uniform Exponential Decay Now we consider the problem of boundary stabilization of (9) on a finite space interval [0, L]. We show that for appropriate boundary conditions (for which (17) holds) the solution decays exponentially. An example is z 1 = z 2 = 0 at x = L and z 3 = z 4 = 0 at x = 0. For the proof, we analyze the exponential decay of a suitably defined Lyapunov function. The decay rate is independent of ε. This is related to the results presented in [13]. Define the positive semidefinite matrix ˜ B(x, ε) = Q(x, ε)T Y (x, ε)−1/2 B(x, ε) Y (x, ε)1/2 Q(x, ε) . ˜ Note that B(x, ε) has a one-dimensional kernel. Define the unit vectors e1 , e2 , e3 , e4 as e1 = (1, 0, 0, 0)T , etc. Then, ker B Y (x, ε) = e1 is the vector space spanned ˜ ε) = Q(x, ε)T Y (x, ε)1/2 e1 . We define by e1 . Hence, ker B(x, R(x, ε) = D Q Tx Q + Q T Y −1/2 x Y 1/2 Q , where the right-hand side depends on x and ε. We use the classical transformation z(t, x) = Q T (x, ε) Y (x, ε)−1/2 U˜ (t, x) that yields the system in diagonal form z t + D z x = (− B˜ + R) z .
(16)
Note that z depends on the relaxation parameter ε as well. According to the previous discussion, we see that the limit limε→0+ R(x, ε) =: R0 (x) exists. We use the notation λ1 = λ2 = −Λ, λ3 = λ4 = Λ. We consider the system in a finite space interval [0, L], where L > 0 is some given positive constant. To examine the behavior of solutions z, we start by introducing the Lyapunov functions in terms of z and ∂t z. Let J0 (t) =
1 2
L 0
eμ1 x z 12 (t, x) + eμ2 x z 22 (t, x) + e−μ3 x z 32 (t, x) + e−μ4 x z 42 (t, x) d x,
1 J1 (t) = 2
L 0
4 i=1
e−sgn(λi )μi x (∂t z i (t, x))2 d x
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where the numbers μi are positive constants to be determined to ensure the expo˜ nential decay. Recall that B(x, ε) is positive semidefinite and that R0 (x), the limit of R(x, ε) as ε → 0, exists. Hence, there exists a constant parameter C0 that is independent of ε such that z T R(x, ε)z ≤ C0 |z|2
for all z ∈ R4 .
Theorem 1. Define the positive constant β0 = min |λi | μi . 1≤i≤4
Choose μi sufficiently large such that β0 − C0 > 0. Assume that 4
λi [z i2 (t, 0) − e−sgn(λi )μi L z i2 (t, L)] ≤ 0 .
(17)
i=1
Then, J0 decays exponentially to zero with the rate ν0 := β0 − C0 that is independent of ε, that is for all t ≥ 0 we have the inequality J0 (t) ≤ exp(−ν0 t) J0 (0).
(18)
Further, assume that 4
λi [(∂t z i (t, 0))2 − e−sgn(λi )μi L (∂t z i (t, L))2 ] ≤ 0 .
i=1
Then, J1 decays exponentially to zero with the same rate ν0 . Since the Lyapunov function J0 is equivalent to the L 2 -norm of z, Theorem 1 also yields the exponential L 2 decay of z (and z t , respectively). Proof. The time derivative of J0 is given by J0 (t) = =
L 0
4
4 i=1
e−sgn(λi )μi x z i ∂t z i d x
i=1
⎛
L
e 0
−sgn(λi )μi x
⎞ 4 z i ⎝−λi ∂x z i + (− B˜ + R)i j z j ⎠ d x j=1
4 L 1 L −sgn(λi )μi x e λi ∂x (z i2 ) d x + e−sgn(λi )μi x z i (− B˜ + R)i j z j d x . =− 2 0 0 i=1
1≤i, j≤4
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Integration by parts yields J0 (t)
L 4 4 1 1 −sgn(λi )μi x 2 −sgn(λi )μi x 2 =− |λi |μi e zi d x + λi e z i (t, x)|0L 2 i=1 2 i=1 0 L + e−sgn(λi )μi x z i (− B˜ + R)i j z j d x 1≤i, j≤4 0
1 −sgn(λi )μi x 2 λi e z i (t, x)|0L ≤ −ν0 J0 (t) . 2 i=1 4
≤ −(β0 − C0 )J0 (t) +
Gronwall’s inequality implies that J0 (t) decays with an exponential rate ν0 . Similarly, for J1 , we have 1 −sgn(λi )μi x λi e (∂t z i (t, x))2 |0L ≤ −ν0 J1 (t) . 2 i=1 4
J1 (t) ≤ −(β0 − C0 )J1 (t) +
4 Application to the Isothermal Euler Equations To exemplify the theoretical results, we apply the relaxation model to the isothermal Euler equations describing gas flow in pipelines. The isothermal Euler equations are a well-established model for realistic high-pressure gas flow in pipes; see e.g., [4, 21]. Denote by d > 0 the diameter of the pipe, λ f ric > 0 the friction coefficient, λ and ϕ ∈ (−π, π ] the slope of the pipe. Define ξ = sin(ϕ ) and θ = fdric . Let g denote the gravitational constant. Let ρ(t, x) denote the gas density, p the gas pressure, and q(t, x) the flow rate. We consider ideal gas where p := p(ρ) = Ru T ρ.
(19)
Here, Ru is the gas constant and T is the constant temperature. The isothermal Euler equations for a sloped pipe with friction are then given by
ρt + qx = 0 , 2 qt + p + qρ = − 21 θ q ρ|q| − g ξρ .
(20)
x
The velocity of the gas flow is given by v = ρq and the sound speed c is given by √ √ √ c = p (ρ) = Ru T . For the Mach number M, this yields M = vc = Ru T qp . Stationary states for the case of an ideal gas have been considered in [14]. We consider the case of subsonic flow where the absolute value of the velocity of the gas is strictly less than speed of sound in the gas, that is |M| < 1. Stationary states for the original and relaxation systems for different choices of parameters will be
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investigated. The asymptotic behavior of the relaxation model incorporated with the suitable boundary conditions proposed in Theorem 1 will show the convergence of U˜ to zero in time. We will consider both a horizontal pipe with ξ = 0 and a pipe with nonzero slope.
4.1 Horizontal Pipe with Nonzero Gas Velocity We show that the system (20) and its relaxation system in the form of (3) possess Suppose qs ≥ 0 and a different stationary states by examination of the sign of ρx x . √ horizontal pipe, ξ = 0. Note that |M| < 1 and therefore ρq < Ru T . For the steady states (ρs , qs ) of the system (20), we have ∂x x ρs (x) =
gρ f ρ − g f ρρ θ Ru T qs2 g ∂ ρ (x) = =: m < 0 . x s f ρ2 2ρs2 f ρ3
For the steady states (ρ, ¯ q) ¯ of the corresponding relaxation system (3), we obtain ∂x x ρ¯ =
1 2 ε Λ4
x
f (ρ, ¯ q) ¯ −
g(ρ, ¯ q) ¯ dy
>0.
0
Hence, the relaxation approximation does not give to the correct qualitative behavior of the steady states in this case. In particular, the difference ρ¯ − ρs is strongly convex, since we have (ρ¯ − ρs )x x ≥ |m| > 0 where m is independent of the relaxL L ation parameter ε. For L > 0, we have 0 ((ρ¯ − ρs )x x )2 (x) d x ≥ 0 m 2 (x) d x > 0 (Note that for the case of conservation laws, we have m = 0 and thus ρ¯ − ρs can converge to zero). Numerically, Fig. 1 shows the discrepancy where the blue curve is the steady state ρs for the system (1) and the red curve with crosses is ρ¯ for the system (3). The different behaviors of the steady states are clearly visible. Hence, we obtain a negative result regarding the convergence to equilibrium: in fact, we have that ρ¯ − ρs L 2 (0,L) ≥ C1 > 0 for some constant C1 independent of ε. For the system with appropriate boundary conditions such that (17) holds, Theorem 1 implies that the transient solution ρ(t, x) to system (3) does not converge to ρs (x) for ε → 0+. This can be seen as follows. Theorem 1 implies that there exists a constant C2 > 0 such that for all t ≥ 0 we have ρ(t, ·) − ρ(·) ¯ L 2 (0,L) ≤ C 2 exp(−ν0 t). Provided that t is sufficiently large, we have the inequality C2 exp(−ν0 t) ≤ C21 which implies ¯ − ρs (·) L 2 (0,L) − ρ(t, ·) − ρ(·) ¯ ρ(t, ·) − ρs (·) L 2 (0,L) ≥ ρ(·) L 2 (0,L) C1 >0. ≥ C1 − C2 exp(−ν0 t) ≥ 2 Since ν0 is independent of ε, the previous inequality holds for all ε > 0.
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1.1 Hyperbolic system
1 Relax system
0.9
0.8
0.7
0.6 0
2
4
6
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x
Fig. 1 Consider the domain [0, 10] for x. The number of mesh for the discretization in space is N x = 1000. The steady states of the systems (1) and (3) are solved using a second-order Runge– Kutta method. The blue curve is ρs (x) for (1) and the red curve with crosses is ρ(x) ¯ for (3) 1.1 Hyperbolic system
1.05
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1
(a)
s (x) =
2
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4
5
t
x
(x).
(b)
= 100, = 0.01.
Fig. 2 The domain for x is [0, 10]. The number of meshes for the spatial discretization is N x = 1000. The time step is t = 10−6 . The final time is t = 5. a ρs (x) is given by the blue curve and ρ(x) ¯ is given by the red curve with crosses. b The natural logarithm of the L 2 norm of ρ(x, ˜ t) as a function of t which is decaying to zero in time
4.2 Constant Stationary States Now, we consider the case of constant stationary states (ρs , qs ). This implies that 1 θqs |qs | for ξ = 0. If ξ = 0, then the only constant stationary state is ρs2 = − 2gξ ¯ q) ¯ = 0. Further, qs = 0 and ρs = const. For the relaxation system (3), we have g(ρ, we have u¯ = q¯ and w¯ = f (ρ, ¯ q). ¯ In this case, Y, Q and B˜ do not dependent on x. ˜ . Therefore, R(x, ε) = 0 which yields the simpler diagonal form z t + Dz x = − Bz ˜ Since B is positive semidefinite, the same estimates as in [6, 23] can be applied.
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For the parameters in the system (20), we have Ru T = 400, θ = 1, g = 9.80665, ¯ q) ¯ = (1, 4.4287) for both of the ξ = −1. The constant steady states (ρs , qs ) = (ρ, systems (1) and (3). The profiles of ρs = ρ¯ are given in Fig. 2a. In order to illustrate the convergence result, we perturb the constant steady states (ρs , qs ) = (1, 4.4287) using a sinusoidal function at the initial time t = 0. Consider the zero boundary condition which satisfies the assumption in Theorem 1 and let Λ = 100, ε = 0.01. We expect the perturbation U˜ is decreasing to zero in time, as shown in Fig. 2b. As expected, we also observe the exponential decay over time.
5 Conclusion Nonconstant stationary states appear in many practical applications. Relaxation systems provide a useful numerical method for solving conservation and balance laws with constant stationary states. In the case of nonconstant stationary states, we could show exponential decay of the relaxation solution over time for suitable boundary conditions. The rate of convergence is independent of the relaxation parameter. However, the relaxation solution does not necessarily converge toward the stationary state of the original system in the case of nonconstant stationary states. Acknowledgements This work was supported by DFG in the framework of the Collaborative Research Centre CRC/Transregio 154, Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks, project C03, as well as by DFG Cluster of Excellence ‘Integrative Production in High–Wage Countries’ HE5386/13,14,15-1, and NSF RNMS (KI-Net) grant 11-07444.
References 1. R. Abgrall, S. Karni, Two-layer shallow water system: a relaxation approach. SIAM J. Sci. Comput. 31, 1603–1627 (2009) 2. M.K. Banda, M. Seaid, Higher-order relaxation schemes for hyperbolic systems of conservation laws. J. Numer. Math. 13, 171–196 (2005) 3. G. Bastin, J.-M. Coron, Stability and Boundary Stabilization of 1-d Hyperbolic Systems (Birkhäuser, Switzerland, 2016) 4. M.K. Banda, M. Herty, A. Klar, Gas flow in pipeline networks. Netw. Heterog. Media 1, 41–56 (2006) 5. S. Bianchini, Hyperbolic limit of the Jin-Xin relaxation model. Commun. Pure Appl. Math. 59, 688–753 (2006) 6. S. Bianchini, B. Hanouzet, R. Natalini, Asymptotic behaviour of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Commun. Pure Appl. Math. 60, 1559– 1622 (2007) 7. A. Chalabi, Convergence of relaxation schemes for hyperbolic conservation laws with stiff source terms. Math. Comput. 68, 955–970 (1999) 8. J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, vol. 136 (American Mathematical Society, Providence, 2007)
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9. J. Coron, G. Bastin, B. d Andrea-Novel, Dissipative boundary conditions for one dimensional nonlinear hyperbolic systems. SIAM J. Control Optim. 47, 1460–1498 (2008) 10. C.M. Dafermos, Asymptotic behavior of BV solutions to hyperbolic systems of balance laws with relaxation. J. Hyperbolic Differ. Equ. 12, 277–292 (2015) 11. C.M. Dafermos, Hyperbolic balance laws with relaxation. Discrete Contin. Dyn. Syst. 36, 4271–4285 (2016) 12. A. Delis, I. Papoglou, Relaxation approximation to bed-load sediment transport. J. Comput. Appl. Math. 213, 521–546 (2008) 13. M. Dick, M. Gugat, M. Herty, S. Steffensen, On the relaxation approximation of boundary control of the isothermal Euler equations. Int. J. Control 85, 1766–1778 (2012) 14. M. Gugat, F.M. Hante, M. Hirsch-Dick, G. Leugering, Stationary states in gas networks. Netw. Heterog. Media 10, 295–320 (2015) 15. M. Gugat, G. Leugering, K. Wang, Neumann boundary feedback stabilization for a nonlinear wave equation: A strict H 2 -Lyapunov function. Math. Control Relat. Fields 7(3) (2017) 16. N. Higham, Functions of Matrices: Theory and Computation, Society for Industrial and Applied Mathematics (2008) 17. S. Jin, Z.P. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commun. Pure Appl. Math. 48, 235–276 (1995) 18. S. Liotta, V. Romano, G. Russo, Central schemes for balance laws of relaxation type. SIAM J. Numer. Anal. 38, 1337–1356 (2000) 19. T.P. Liu, Transonic gas flow in a duct of varying area. Arch. Ration. Mech. Anal. 80, 1–18 (1982) 20. T.-P. Liu, Hyperbolic conservation laws with relaxation. Commun. Math. Phys. 108, 153–175 (1987) 21. A. Osiadacz, Dynamic optimization of high pressure gas networks, Technical Report at Warsaw University of Technology (2002) 22. L. Pareschi, G. Russo, Implicit-Explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation. J. Sci. Comput. 25, 129–155 (2005) 23. W.-A. Yong, Singular perturbations of first-order hyperbolic systems with stiff source terms. J. Differ. Equ. 155, 89–132 (1999) 24. W.-A. Yong, Basic aspects of hyperbolic relaxation systems, (Birkhäuser Boston, Boston, 2001), pp. 259–305
Numerical Solutions for a Weakly Hyperbolic Dispersed Two-Phase Flow Model Maren Hantke, Christoph Matern and Gerald Warnecke
Abstract We construct numerical solutions for a dispersed isothermal two-phase flow model. The system is a weakly hyperbolic, isothermal system describing the evolution of mass, momentum as well as volume fraction for the dispersed particles as well as the carrier fluid. The dispersed phase is modeled pressureless. We construct a new HLL-type Riemann solver and perform numerical simulations on the homogeneous part of the model. In each time step, an approximate MUSCL–Hancock finite volume scheme is used in which intercell Riemann problems are solved using the new GHLL solver. Keywords Two-phase flows · Approximate Riemann solver · Delta-shocks Non-classical waves · Vacuum states
1 Introduction Multiphase flows occur in many natural and engineering applications. In atmospheric physics, cloud formation is of considerable interest and involves liquid water and its vapor as well as a mixture of other gases. Ship manufacturers are also interested in knowing the effects of bubbles formed in water due to cavitation. In the beverage industry, coffee beans are sugar coated using a fluidized bed granulator. The process involves a mixture of solid, liquid, and gaseous phases. All such processes are modeled using two-phase or multiphase flow models.
M. Hantke (B) · C. Matern · G. Warnecke Institute for Analysis and Numerics, Otto-von-Guericke University Magdeburg, Universitätsplatz 2, D-39106 Magdeburg, Germany e-mail: [email protected] C. Matern e-mail: [email protected] G. Warnecke e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_51
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It is a common practice to use continuum mechanics equations to model twophase flows. These equations are usually derived from a microscopic consideration using averaging or homogenization techniques, see Drew and Passman [3], Stewart and Wendroff [7], Saurel and Abgrall [5], as well as Baer and Nunziato [1]. We study the two-phase flow model proposed by Hantke et al. [2]. It describes the evolution of a mixture of a dispersed phase of small ball-shaped bubbles of water vapor, immersed in a carrier fluid, the corresponding liquid water phase. The model was derived using averaging techniques and it is completely in divergence form, unlike those studied in [1] as well as [7]. The type of mathematical models for two-phase flows is highly complex. Therefore, to gain an insight into the solution of these systems, numerical methods are indispensable to obtain the approximate solutions. Finite volume methods (FVM) have been widely used to construct numerical solutions for systems of conservation laws, see Toro [8] and the literature therein. Center to these methods is the calculation of numerical fluxes at the intercell boundaries. The choice of numerical fluxes leads to different numerical methods such as those of Lax–Friedrichs, Lax–Wendroff as well as Beam– Warming. In some cases, numerical fluxes are determined from approximate solutions to Riemann problems at intercell boundaries. Methods derived in this way include the Roe solver, the Osher scheme, and the HLL solver, as well as the HLLC solver. The aim of this work is to construct an efficient numerical algorithm for solving an isothermal one-dimensional and homogeneous version of the two-phase flow model introduced in [2]. The model is a weakly hyperbolic system of partial differential equations which describes the evolution of mass, momentum, and volume fraction for both the dispersed and the carrier phases. Numerical solutions are constructed using an approximate MUSCL–Hancock finite volume scheme in which intercell Riemann problems are solved using HLL and HLLC solvers. We use MINBEE slope limiters in order to control spurious oscillations at solution discontinuities.
2 The Two-Phase Flow Model We denote volume fraction, pressure, density, mean bubble radius, and velocity by c, p, ρ, R, and v, respectively. We will use the subscript C to distinguish between the carrier phase and the dispersed phase, the latter one without a subscript. The time and space variables are, respectively, represented by t and x. The Riemann problem we want to study is given by the system of partial differential equations ∂ ∂c + (c v) ∂t ∂x ∂ ∂ (c ρ) + (c ρ v) ∂t ∂x ∂ ∂ (c ρ v) + c ρ v2 ∂t ∂x ∂ ∂ (c R) + (c R v) ∂t ∂x ∂ ∂ (1 − c )ρC + (1 − c ) ρC vC ∂t ∂x
= 0, = 0, = 0, = 0, = 0,
(1)
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∂ ∂ ∂ (1 − c )ρC vC + (1 − c ) ρC vC2 + (1 − c ) pC = 0 , ∂t ∂x ∂x
together with the piecewise constant initial data c, ρ, v, R, ρC , vC (t = 0, x) =
c− , ρ− , v− , R− , ρC− , vC− for x < 0, c+ , ρ+ , v+ , R+ , ρC+ , vC+ for x > 0.
(2)
As one can see, the first four equations of the system (1) decouple from the last equations. This fact is used to find analytical solutions of this model.
2.1 Equations of State The two phases are a vapor and a liquid phase. We will consider both cases, vapor bubbles dispersed in the liquid phase and liquid droplets in the vapor carrier phase, respectively. Therefore, we assume an equation of state of the following form throughout the rest of this work: p = a 2 ρ + d0 with a 2 = const,
(3)
where a denotes the constant sound speed and d0 is some reference constant related to the constant temperature T . Considering the vapor, the ideal gas law is then a valid choice for an equation of state. In this case, we have p=
kT kTρ with a 2 = , d0 = 0 m0 m0
(4)
where k and m 0 denote the Boltzmann constant and the molecule mass, respectively. The pressure of the liquid phase p can be given by the equation of state due to Hooke’s law applied to an isotropic liquid p = p¯ + K¯
ρ −1 , ρ¯
(5)
where p¯ denotes the saturation pressure at temperature T and ρ¯ the corresponding density. The constant K¯ is the temperature-dependent modulus of compression. Written in the form of Eq. 3, we have K¯ d0 := p¯ − K¯ , a 2 := , ρ¯
(6)
where a is the speed of sound in the liquid. Note that in the isothermal case the stiffened gas equation of state and Eq. (5) coincide.
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2.2 Exact Solutions for the Riemann Problem As mentioned before, the first four equations of (1) can be decoupled from the rest of the system since they do not depend on the carrier phase quantities. Therefore, the dispersed phase equations can be solved separately. The corresponding Jacobian matrix A of the dispersed subsystem has the repeated eigenvalue λ = v of multiplicity four but only three linearly independent eigenvectors. Such a system is called weakly hyperbolic. It turns out that the solution depends mainly on the initial values of the dispersed velocity v. Depending on the sign of v+ − v− , we distinguish three cases. We observe that in the case of constant velocity, i.e., v+ − v− = 0, the equations for c, ρ and R become linear advection equations with constant speed v. Therefore, the solution for the Riemann problem is given by a contact discontinuity. The dispersed subsystem in the case v+ − v− > 0 is similar to the zero pressure gas dynamics model by Sheng and Zhang [6, p. 11]. The system possesses a unique entropy solution, see e.g., LeVeque [4, Eq. (3.28)]. In that case, the contact wave degenerates into two contacts with a vacuum state in between. Finally, in the case v+ − v− < 0, the system admits no classical solution. Instead, an entropy solution in the space of Borel measures exists which contains a δ-shock in the volume fraction c. A system of generalized jump conditions can be derived. The unique solution of this system consists of the singular values for the volume fraction, the density as well as velocity and radius. To consider the full two-phase system, the carrier fluid equations have to be taken into account. One obtains two more eigenvalues λ = vC ± aC with genuinely nonlinear fields. The corresponding waves are either shocks or rarefactions and propagate through the carrier fluid. The possible wave configurations of the full system are depicted in Fig. 1. t
t
u ∗−
u ∗−
u ∗+
u−
u−
u+ 0
u ∗0
x
(a) acoustic wave, contact -shock, acoustic wave
u ∗+ u+
0
x
(b) acoustic wave, vacuum, acoustic wave
Fig. 1 Possible wave configurations for the Riemann problem of the full system
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The full system as well as the dispersed subsystem is weakly hyperbolic, i.e., it possesses six real eigenvalues but only five eigenvectors. Due to the lack of a complete set of eigenvectors, a solver is needed that does not make use of them. It seems likely to use an HLL-type solver. However, it is impossible to determine the Riemann invariants for the carrier fluid across the middle wave/waves. Accordingly the construction of an HLLC solver, respectively, HLLCC solver requires approximations across the waves corresponding to λ = v.
3 Numerical Method Given a one-dimensional domain x ∈ [x− , x+ ], we construct a uniformly spaced grid which contains N cells. The i th cell Ii has midpoint xi and boundaries xi− 21 and xi+ 21 . The full model is solved using Strang splitting. For the given model, we solve the equation Ut + F(Ux ) = 0 to obtain the solution Un+1 after a time step Δt. We will use an approximate MUSCL–Hancock finite volume scheme with limited slopes. The scheme is given by Uin+1 = Uin −
Δt [F(U (U j , U j+1 )) − F(U (U j−1 , U j ))], Δx
(7)
where Uin are data cell averages, U (U j , U j+1 ) is an approximation to the exact solution at the cell interface position xi+ 21 and F(U (U j , U j+1 )) the corresponding numerical flux. Time steps satisfy the Courant–Friedrichs–Lewy stability condition CC F L :=
Δt max λ(uin ) ≤ 1, Δx 1≤i≤N
where for the simulation a more restrictive condition CC F L = 0.9 was used. The numerical fluxes are determined through three steps: Data reconstruction (with MINBEE flux limiter approach), Evolution (by a time step Δt/2), and the Solution of the Riemann problem, see [8, pp. 504, 506]. When the left- and rightdispersed phase velocities are equal, there are three waves leading to an HLLC scheme. In case of the right velocity exceeding the left one in the dispersed phase, the middle wave splits into two waves, creating a vacuum region. This consideration leads to a new scheme, which we will call the HLLCC solver. The numerical flux is given by
F∗ . F H L LCC = FC,H L L Due to the decoupling of the dispersed phase equations in system (1) from the rest, the flux can be split in the given manner. For the dispersed components, we get the following solution:
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(i) :
v− = v+ =: v
(ii) :
v− < v+
(iii) :
v− > v+
where vδ is given by
F(U− ) F(U+ ) ⎧ ⎪ ⎨F(U− ) F∗ = F(U+ ) ⎪ ⎩ 0
F(U− ) F∗ = F(U+ ) ∗
F =
v>0 otherwise v− > 0 v+ < 0 otherwise vδ > 0 , otherwise
√ √ v− c− ρ− + v+ c+ ρ+ vδ = √ . √ c− ρ− + c+ ρ+
To derive the flux for the liquid components from the HLLC or HLLCC solver, we impose the approximate Riemann invariants across the corresponding wave: I1 = (1 − c)ρC and I2 = (1 − c)ρC vC . The resulting scheme for the liquid components is equivalent to an ordinary HLL solver.
4 Numerical Results In the following, we discuss several numerical examples for all the three cases: Case 1—contact case, Case 2—vacuum case, and Case 3—δ-shock case. We consider bubbles in liquid as well as droplets in gas. We use parameters as noted in Table 1. These data correspond to T = 293.15 K. In order to give a condensed presentation for each test case, we only show three relevant pictures to discuss the properties of the solver and we neglect the equation describing the radius evolution. We compare the GHLL solver to the second-order HLL solver and to the exact solution, presented by a blue (dotted), red (dashed), and a black (solid) line, respectively. All simulations are performed using CCFL = 0.9. Example 1—contact case, bubbles in liquid • Uses the initial data given in Table 2. We obtain the results shown in Fig. 2. Obviously the GHLL solver gives—in contrast to the HLL—a quite good resolution for the contact wave, even on coarse grids. On the other hand, the GHLL Table 1 Equations of state parameters
a ms d0 [Pa]
Gas
Liquid
369 0
1478 −2.18 · 10−8
Numerical Solutions for a Weakly Hyperbolic Dispersed Two-Phase Flow Model Table 2 Initial data Example 1, N = 200 3 c ρ kg m Left state Right state
0.025 0.1
1.49 0.745
v
m s
−200 −200
ρC
kg 3 m
671
vC
m s
−200 200
998.29 998.24
Fig. 2 Example 1, contact case, bubbles in liquid, N = 200 Table 3 Initial data Example 2, N = 200, N = 500 3 c ρ kg v ms m Left state Right state
0.025 0.1
1.49 0.745
−50 50
ρC
kg 3 m
998.29 998.24
vC
m s
−200 200
produces small under- and overshoots near the contact in the carrier fluid which lead to small oscillations. These oscillations completely disappear on finer grids. The resolution of shock waves is of the same quality for both solvers. Example 2—vacuum case, bubbles in liquid • Uses the initial data given in Table 3. We obtain the results presented in Fig. 3 for the coarse grid and in Fig. 4 for the finer grid. Even on a coarse grid, the GHLL gives a quite good approximation of the vacuum state. Refining the mesh, the GHLL gives a nearly exact solution while the resolution of the vacuum state produced by HLL is quite poor. One can clearly see that the tiny oscillations by GHLL are totally disappeared on the finer grid. Example 3—δ-shock case, bubbles in liquid • Uses the initial data given in Table 4. Due to the formation of the singularity, the simulation for that example is performed on a finer grid. We obtain the result shown in Figure 5. While the GHLL gives a very good approximation of the exact solution, the HLL solver fails to resolve the singularity.
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Fig. 3 Example 2, vacuum case, bubbles in liquid, N = 200
Fig. 4 Example 2, vacuum case, bubbles in liquid, N = 500 Table 4 Initial data Example 3, N = 500 3 c ρ kg m Left state Right state
0.025 0.1
1.49 0.745
v
m s
10 −10
Fig. 5 Example 3, δ-shock case, bubbles in liquid, N = 500
ρC
kg 3 m
998.29 998.24
vC
m s
−200 200
Numerical Solutions for a Weakly Hyperbolic Dispersed Two-Phase Flow Model Table 5 Initial data Example 4, N = 100 3 c ρ kg m Left state Right state
0.025 0.1
998.29 998.24
v
m s
−200 200
ρC
kg 3 m
673
vC
m s
−50 −50
1.49 0.745
Fig. 6 Example 4, vacuum case, droplets in gas, N = 100 Table 6 Initial data Example 5, N = 500 3 c ρ kg m Left state Right state
0.025 0.1
998.29 998.24
v
m s
10 −10
ρC
kg 3 m
1.49 0.745
vC
m s
15 −15
Considering bubbles in a liquid, the GHLL always gives much better solutions than the HLL solver. In the following, we investigate the opposite case of droplets in gas. Because HLL and GHLL give nearly the same solution for the contact case even on a coarse grid (N = 100), we disclaim to present corresponding numerical results. Example 4—vacuum case, droplets in gas • Uses the initial data given in Table 5. The results are shown in Fig. 6. As in the opposite case of bubbles in liquids, the GHLL solver gives a much better resolution of the vacuum state. Note that in the droplet case the GHLL does not produce any over- and undershoots. Example 5—δ-shock case, droplets in gas • Uses the initial given in Table 6. Finally, we present the results for the δ-shock case for droplets in Fig. 7. As before, the HLL solver is not able to resolve the singularity while the GHLL gives a very good approximation.
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Fig. 7 Example 5, δ-shock case, droplets in gas, N = 500
5 Summary and Conclusions We constructed a new HLL-type Riemann solver which takes into account-specific properties of the system under consideration. In particular, we use a combined solver which handles the different phases in a different manner. We performed several calculations and compared the results to the ordinary HLL-Riemann solver. In all the examples presented, the GHLL gives very good results for the dispersed phase, while the HLL solver cannot fairly resolve contact waves, vacuum states, or singularities. If the carrier fluid is modeled as an ideal gas, the new GHLL always gives satisfactory results. On the other hand, if the carrier fluid is modeled as a liquid, the GHLL produces small over- and undershoots in the solution of carrier fluid on coarse grids. The reason is that for parameter |d0 | >> 0 the approximation used in the HLLCC solver is too imprecise. Accordingly, for that case, one should find a better approximation. Acknowledgements The second author gratefully acknowledges financial support by the DFGGraduiertenkolleg 1554.
References 1. M.R. Baer, J.W. Nunziato, A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. Int. J. Multiph. Flows 12, 861–889 (1986) 2. W. Dreyer, M. Hantke, G. Warnecke, Bubbles in liquids with phase transition - part 2: on balance laws for mixture theories of disperse vapor bubbles in liquid with phase change. Contin. Mech. Thermodyn. 26, 521–549 (2014) 3. D.A. Drew, S.L. Passmann, Theory of Multicomponent Fluids (Springer, New York, 1999) 4. R.J. LeVeque, Numerical Methods for Conservation Laws (Birkhäuser Verlag, 1999) 5. R. Saurel, R. Abgrall, A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150, 425–467 (1999)
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6. W. Sheng, T. Zhang, The Riemann problem for the transportation equations in gas dynamics. Mem. Amer. Math, Soc., 137(654) (1999) 7. H.B. Stewart, B. Wendroff, Two phase flow: models and methods. J. Comput. Phys. 56, 363–409 (1984) 8. E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics (Springer, Heidelberg, 2009)
Optimal Controls in Flux, Source, and Initial Terms for Weakly Coupled Hyperbolic Systems Maryse Hawerkamp, Dietmar Kröner and Hanna Moenius
Abstract In this paper, we prove the existence of an optimal control in flux-, source-, and initial terms for weakly coupled nonlinear hyperbolic systems. The basic idea for proving existence of an optimal control is to show first by parabolic regularization the existence of an entropy solution of the weakly coupled hyperbolic system for given control functions. Then, one has to show the stability of the entropy solution with respect to the control functions in order to get the optimal control. Keywords Optimal control · Weakly coupled hyperbolic system · Conservation laws
1 Introduction Weakly coupled hyperbolic systems have numerous applications in fields like traffic modeling and combustion theory. In this paper, a basis is laid to the theory of optimal control problems governed by a class of weakly coupled hyperbolic systems. To simplify the notations, we restrain ourselves to two space dimensions. Higher dimensions can be obtained similarly. First, we introduce the following weakly coupled hyperbolic initial value problem in the space–time slab: ΩT = R2 × (0, T ), T > 0. Let the control functions u0 : R2 → R M , u1 : Ω T → Rr , u2 : Ω T → Rm , the fluxes f 1 , f 2 : Rr × R → R M and the source terms g : Ω T × R M × Rm → R M be given for m, M, r ∈ N. Furthermore, we use the notation f i = (f i1 , f i2 ) = ( f 1i , f 2i ) , for i = 1, . . . , M. Then, we state the following problem: For i = 1, ..., M find the states y i : Ω T → R, y := (y 1 , ..., y M ) such that we have in ΩT
M. Hawerkamp · D. Kröner (B) · H. Moenius University of Freiburg, Freiburg, Germany e-mail: [email protected] URL: https://www.aam.uni-freiburg.de/agkr/index.html © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_52
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∂t y 1 (x, t)
+
div( f 1 ( u1 (x, t), y 1 (x, t)))
=
g 1 (x, t, y(x, t), u2 (x, t))
∂t y 2 (x, t) .. .
+
div( f 2 ( u1 (x, t), y 2 (x, t))) .. .
=
g 2 (x, t, y(x, t), u2 (x, t)) .. .
∂t y M (x, t)
+
div( f M ( u1 (x, t), y M (x, t)))
=
g M (x, t, y(x, t), u2 (x, t)) (1)
in R2 .
(2)
y(x, 0) = u0 (x)
Second, let f 1 , f 2 , g, and the set of admissible controls Uad in the control space U be given. For a given functional J (y(u), u), we analyze the following optimal control problem: Minimize J (y(u), u) subject to u = (u0 , u1 , u2 ) ∈ Uad ,
(3)
where u is the control and y = y(u) is the state function, i.e., the entropy solution of (1) and (2). Let us present related works. Using techniques of [9, 12], the author in [14] developed a general framework to weakly coupled systems with flux functions depending on space, time, the solution and a source function depending on space, time, and the vector-valued solution. In [3, 11], stability results on flux and source terms, each dependent of space, time, and the entropy solution, have been shown. The author in [15] studied optimal control problems governed by scalar nonlinear inhomogeneous hyperbolic conservation laws with control in the source term. In this paper, we combine the results of [14] for systems and [15] for scalar equations. In [7], there are results presented concerning traffic control. This paper is organized as follows. In Sect. 2, we collect the main assumptions. In Sect. 3, we treat the associated parabolic initial value problem to (1), show existence for this parabolic system, and derive estimates to get compactness. In Sect. 4, we first give a definition of the entropy solution and deal with the existence of an entropy solution to hyperbolic problems using the results of the sections before. Section 5 is devoted to stability results in the flux term using the Kruzkovs method of doubling variables [3, 9, 11]. The results of existence of an entropy solution, the L ∞ -bound, and stability are used in the last Sect. 6 to show existence of the optimal controls to (3) under certain additional assumptions. All the results refer to the initial value problem (1) and (2). For applications, the initial boundary value problem might be more interesting. We expect that all the arguments can also be used for the initial boundary value problem. In this overview, we will present only the main results. More details and the proofs can be found in the forthcoming paper [6].
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2 The Initial Value Problem for Weakly Coupled Hyperbolic Systems For the flux and source function, we need the following assumptions concerning regularity [14]. Assumption 1 (Regularity of flux and source functions) For i = 1, ..., M let: The flux functions satisfy f 1i , f 2i ∈ C 3 (Rr × R, R). For any K T ≥ 0 there exists a constant C f (K T ) ≥ 0 such that ∂ α ∂ p f i u1 y
|α|+ p≤3 |α|, p≤2
1 L ∞ (Rr ×[−K T ,K T ])
+ ∂uα1 ∂ yp f 2i L ∞ (Rr ×[−K
T ,K T ])
≤ C f (K T ).
(4)
For all y ∈ R uniformly in a ∈ Rr the following estimate is valid: M ∂ y f l (a, y) + ∂ y f l (a, y) ≤ L f . 1 2
(5)
l=1
The source functions satisfy g i ∈ C 2 (Ω T × R M × Rm , R) and for any K T ≥ 0 there exists a constant C g (K T ) ≥ 0 such that
j k β α i ∂ ∂ ∂ ∂ g ∞ x1 x2 y u 2 L (Ω
j+k+|α|+|β|≤2 j,k,|α|,|β|≥0 |β|,|α|≤1
T ×B K T
×Rm )
≤ C g (K T ).
(6)
For i = 1, ..., M the functions g i (·, t, 0, 0), ∂x1 g i (·, t, 0, 0), ∂x2 g i (·, t, 0, 0) have a compact support
(7)
for all t ∈ [0, T ]. For all y, yˆ ∈ R M and for all u2 , uˆ 2 ∈ Rm uniformly in (x, t) ∈ Ω T the following is valid: M m l l i y − yˆ l + L i u − uˆ l . g (x, t, y, u2 ) − g i (x, t, yˆ , uˆ 2 ) ≤ L i g u 2 2 l=1
(8)
l=1
With α, β and γ we denote the multi-index (α1 , ..., αr ) ∈ Nr0 , (β1 , ..., β M ) ∈ N0M resp. (γ1 , ..., γm ) ∈ Nm 0 with |α| = α1 + ... + α M .
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We introduce the set D to define an initial value problem to the weakly coupled hyperbolic system: D : = (u0 , u1 , u2 , f 1 , f 2 , g) ∈ L ∞ (R2 , R M ) × L ∞ (Ω T , Rr )∩C 2 (Ω T , Rr ) 2 × L ∞ (Ω T , Rm ) × C 3 ( Rr × R, R M ) × C 2 (Ω T × R M × Rm , R M )
| (f 1 , f 2 , g) satisfy(4), (6) .
3 Classical Solutions of the Weakly Coupled Parabolic Initial Value Problem In this chapter, we introduce the viscous approximation of (1), (2) to prove existence and uniqueness of entropy solutions of the weakly coupled hyperbolic initial value problem (1), (2).
3.1 The Weakly Coupled Viscous Initial Value Problem The parabolic problem, defined as follows with a viscosity parameter ε > 0, can be considered as the regularized counterpart of the hyperbolic initial value problem (1), (2). For i = 1, ..., M find yεi : Ω T → R such that ∂t y 1ε
+ div( f 1 (u1 , yε1 ))
=
g 1 (x, t, yε , u2 )
+
∂t y 2ε .. .
+ div( f 2 (u1 , yε2 )) .. .
=
g 2 (x, t, yε , u2 ) .. .
+
∂t y εM
+ div( f M (u1 , yεM ))
=
g M (x, t, yε , u2 ) +
⎫ εΔyε1 ⎪ ⎪ ⎪ ⎪ ⎬ εΔy 2 ⎪ .. .
ε
εΔyεM
⎪ ⎪ ⎪ ⎪ ⎪ ⎭
in ΩT and
(9) yε (x, 0) = u0 (x)
in R2 . (10)
Let the set Cc (Ω T , R M ) be given by Cc (Ω T , R M ) := { yε ∈ C 0 (Ω T , R M ) | yε is bounded, ∂t yε , ∂x1 yε , ∂x2 yε , ∂x21 yε , ∂x1 ∂x2 yε , ∂x22 yε exist and are continuous in Ω T } . To achieve sufficient regularity, we need assumptions for the initial function u0 , the functions u1 , u2 , the flux functions f 1 , f 2 and the source function g. For the flux functions f 1 , f 2 and the source function g, we assume that (4), (6) are valid.
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Assumption 2 Let us assume that the following conditions are satisfied: ∂ j ∂ k ui ∞ 2 ≤ C < ∞, u0 W 1,1 (R2 ,R M ) ≤ C x1 x2 0 L (R )
(11)
j+k≤3 j,k≥0
for i = 1, ..., M and 0 < C < ∞. Furthermore we assume u1 W 2,∞ (Ω T ,Rr ) ≤ C, ∞ 1 2 u2 W 1,∞ (Ω T ,Rm ) ≤ C , u 12 , ..., u m 2 are bounded in L ([0, T ] , L (R )),
and
m ∂x u l 1 + ∂x2 u l2 L 1 (Ω T ) ≤ C. 1 2 L (Ω T )
(12) (13)
(14)
l=1
3.2 The Basic Existence and Uniqueness Result For the next theorem and its proof, we use the ideas of [14]. Theorem 3. Let (u0 , u1 , u2 , f 1 , f 2 , g) ∈ D be given and assume (4), (5), (8), (11), (12), (13). Then there exists for each ε > 0 and each T > 0 a unique classical solution yε of the weakly coupled parabolic initial value problem (9) and (10) to (u0 , u1 , u2 , f 1 , f 2 , g) in Ω T . Furthermore there exists for each ε > 0 a constant K Tε ≥ 0 such that yε W 2,∞ (Ω T ,R M ) < K Tε (15) holds. If additionally (7) hold, then there exists for all t ∈ [0, T ] and i = 1, ..., M a constant C Tε ≥ 0 such that yε (·, t)W 1,1 (R2 ,R M ) < C Tε .
(16)
Remark 1. Since the solution of yε of the weakly coupled parabolic initial value problem (9) and (10) is bounded, we can omit the conditions (8), (5) in Theorem 3 by cutting off the nonlinearities f 1i , f 2i and g i . Finally, we need an estimate in time as in [14]. Lemma 1. Let (u0 , u1 ,u2 , f 1 , f 2 , g) ∈ D be given. Let Assumptions (7), (8), (11), (12), (13) are valid and let R > 0. Then there exist a nondecreasing function ω R ∈ C 0 ([0, ∞)) with ω R (0) = 0 such that the following inequality is valid for each ε ∈ (0, 1] and for each t, Δt ≥ 0 with t, t + Δt ∈ [0, T ] BR
i y (x, t + Δt) − y i (x, t) dx ≤ ω R (Δt) for i = 1, ..., M. ε ε
(17)
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4 Weak Solutions and Entropy Solutions Let T > 0 be arbitrary, but fixed. Definition 1 (Weak solution). Let (12) be satisfied by u1 ∈ C 2 (Ω T , Rr ), let u2 ∈ L ∞ (Ω T , Rm ) and (4), (6) be satisfied by (f 1 , f 2 , g). Let y = (y 1 , ..., y M ) ∈ L ∞ (Ω T , R M ). Then y is called weak solution of (1) to (u1 ,u2 , f 1 , f 2 , g), if for all i = 1, ..., M and for all Φ ∈ C0∞ (ΩT ) the following equality is true: y i ∂t Φ + f 1i (u1 , y i )∂x1 Φ + f 2i (u1 , y i )∂x2 Φ + g i (x, t, y, u2 )Φ dx dt = 0. (18) ΩT
Weak solutions are not unique, and therefore we introduce the entropy solution. Definition 2 (Entropy). Let η ∈ C 2 (R) be a convex function which is called entropy function. The components of the entropy fluxes p1 and p2 : Rr × R → R M to the entropy function η are defined for i = 1, ..., M, j = 1, 2, k ∈ R and a ∈ Rr by y pij (a,
y) :=
η (z)∂z f ji (a, z)dz.
(19)
k
The tripel (η, p1 , p2 ) is called entropy to the flux f 1 , f 2 . Let the functions h1 and h2 : Ω T → R M are given for i = 1, ..., M, j = 1, 2, a : Ω T → Rr and k ∈ R by y h ij (x, t,
y) : =
∂x j pij (a(x, t),
y) =
η (z)∂z ∂x j f ji (a(x, t), z)dz.
(20)
k
With this notation we will define the entropy solution. Definition 3 (Entropy solution). Let us assume the same assumptions as in Definition 1 and let u0 ∈ L ∞ (R2 , R M ). Then we call y entropy solution of (1), (2), if for i = 1, ..., M, for all entropies (η, p1 , p2 ) and all Φ ∈ C0∞ (ΩT ), Φ ≥ 0 the following entropy inequality is valid:
η(y i )∂t Φ + p1i (u1 , y i )∂x1 Φ + p2i (u1 , y i )∂x2 Φ + h i1 (x, t, y i ) + h i2 (x, t, y i ) Φ
ΩT
r i i l i i l i ∂ul1 f 1 (u1 , y ) ∂x1 u 1 + ∂ul1 f 2 (u1 , y ) ∂x2 u 1 − g (x, t, y, u2 ) Φ ≥ 0 − η (y )
i
l=1
(21) and the initial condition (2) is satisfied in the weak sense, i.e. there exist a set Ly ⊂ [0, T ] with Lebesgue measure 0 such that y(·, τ ) is defined a.e. in R2 for
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τ ∈ [0, T ] \ Ly and for all R > 0 the following is valid: lim
τ →0 τ ∈[0,T ]\L y B R
i y (x, τ ) − ui (x)dx = 0 (i = 1, ..., M). 0
4.1 Globally Limited Growth Now we need some further assumptions. Assumption 4 For the flux functions f 1 and f 2 there exist constants L 1d f , ..., L dMf > 0 such that for i = 1, ..., M: r ∂ul1 ∂ y f 1i l=1
L ∞ (Rr ×R)
+ ∂ul1 ∂ y f 2i
L ∞ (Rr ×R)
≤ L id f , let L d f =
M
L ld f . (22)
l=1
M Let C0i = sup g i (x, t, 0, 0) (x, t) ∈ Ω T and let C0 = C0l . l=1
We can show that the solution of the hyperbolic problem is bounded. Theorem 5 (Existence and uniqueness: globally limited growth). Let (u0 , u1 ,u2 , f 1 , f 2 , g) ∈ D be given. Assume that (7), (8), (11), (13), (14) are valid. Furthermore let Assumption 4 for f 1 , f 2 be satisfied. Then for each T > 0 (1), (2) have a unique entropy solution y ∈ L ∞ (Ω T , R M ) in Ω T which satisfies for almost all t ∈ [0, T ] and for i = 1, ..., M: i y (·, t)
L ∞ (R2 )
≤
M l C0 + L u C(Cu 2 ) u0 L ∞ (R2 ) exp (C + L g )t + C(L d f , Cu 1 ) + L g l=1 −
C0 i i =: M∞ (t); M∞ (t) := max M∞ (t). i=1,...,M Ld f + Lg
5 Total Variation and Stability In this section, we will investigate the total variation and the stability of the solution y [3, 11]. Assumption 6 Let the flux functions f ji : Rr × R → R be linear in the first argument for j = 1, 2 and i = 1, . . . , M. Furthermore, we need a further boundedness condition.
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Assumption 7 Let Cs : the following be valid for a constant 2 i ∂z g (z, t, ·, ·) ∞ ∂ dzdt < C , u (z, t) dzdt < Cs . s z 1 L (B K ×Rm ,R2 )
ΩT
ΩT
T
Theorem 8. (See also [3, 11].) Let (4), (6), (12), (14) and the Assumptions 6, 7 are valid. Let u0 ∈ (L ∞ ∩ L 1 ∩ BV )(R2 , R M ). Then the entropy solution y of (1), (2) satisfies y i (t) ∈ BV (R2 ) for i = 1, . . . , M. Let T0 > 0 and κ0 = ( 23 π + 1)Cu 1 C f (K T0 ) + M C g (K T0 ), then for all T ∈ [0, T0 ] the following estimate holds: M
T V (y i (T ))
i=1
T M π fg κ0 T i e T V (u 0 ) + C (K T ) eκ0 (T −t) ∂z g i (z, t, ·, ·) L ∞ (BK ×Rm ,R2 ) ≤ T0 2 i=1 R2
0
+
m
2 l ∂z u 2 (z, t) + ∂z u1 (z, t) dzdt .
l=1
(23) Now we need the following stability estimate. Theorem 9. Let (u0 , u1 , u2 , f 1 , f 2 , g) ∈ D and (uˆ 0 , uˆ 1 , uˆ 2 , f 1 , f 2 , g) ∈ D. Let u0 , uˆ 0 ∈ (L ∞ ∩ L 1 ∩ BV )(R2 , R M ) and let (11), (12), (14), Assumption 6 and 7 are valid. Assume y, yˆ ∈ L ∞ (Ω T , R M ) are entropy solution of (1), (2) with respect to (u0 , u1 , u2 , f 1 , f 2 , g) and (uˆ 0 , uˆ 1 , uˆ 2 , f 1 , f 2 , g), respectively. Furthermore assume that the functions y and yˆ fulfill y L ∞ (Ω T ) , ˆy L ∞ (Ω T ) ≤ K T . Then there exist for all R > 0, x0 ∈ R2 and t ∈ [0, T ] constants C˜ 1 , C˜ 2 > 0 such that the following estimate is valid M
y i (x, t) − yˆ i (x, t) L 1 (B R (x0 )) ≤
M
i=1
eκ
∗
T
u i0 (x) − uˆ i0 (x) L 1 (B R+N T (x0 ))
i=1
+ C˜ 1
r
r l l ∇x uˆ l − ∇x u l ∞ ˜ ∞ u 1 − uˆ 1 L (ΩT ) + C2 1 1 L (ΩT )
l=1
l=1
+ eκ
∗
T
C g (K T )
m
u l2 (x, t) − uˆ l2 (x, t) L 1 (B R+N (T −t) (x0 )×[0,T ]) ,
l=1
where C˜ 1 = C(Cu 1 , T V (u i0 ), Cu 2 , C f g (K T ), Cs , T, M), C˜ 2 = C(T, R + N T, C f g (K T ), Cu 1 ), and κ ∗ = C C f (K T ), C g (K T ), Cu 1 .
(24)
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Remark 2. Assume u0 , uˆ 0 ∈ (L ∞ ∩ L 1 ∩ BV )(R2 , R M ) and the assumptions in Theorem 9. Then Theorem 9 implies that for almost all t ∈ [0, T ] there exists a constant C = C(C˜ 1 , C˜ 2 ) ≥ 0 such that the following estimate holds. M
y (·, t) i
L 1 (R2 )
≤C
i=1 r
M
ui0 L 1 (R2 )
+
i=1
u i1 L ∞ (ΩT )
l=1
+
r
∇x u l1 L ∞ (ΩT )
m
∗ u l2 L 1 (Ω T ) eκ T +
l=1
(25)
=: M1 (T ).
l=1
From the stability estimate (24), we get the uniqueness of the entropy solution. Theorem 10. Let the assumptions of Theorem 9 be fulfilled. Then there exists at most one entropy solution of (1), (2) with respect to (u0 , u1 , u2 , f 1 , f 2 , g).
6 A Class of Optimal Control Problems We consider now the optimal control problem (3). The purpose of this section is to give sufficient conditions for the existence of an optimal control. Regarding notations and the idea of the proofs, we will follow [15]. We remind that we restrict ourselves to two space dimensions to simplify notations.
6.1 Assumptions Let U = (L ∞ ∩ L 1 )(R2 , R M ) × L ∞ (Ω T , Rr ) × (L ∞ ∩ L 1 )(Ω T , Rm ). The following assumptions will be essential below. Assumption 11 The admissible set Uad is bounded in L ∞ (R2 , R M ) ×L ∞ (Ω T , Rr ) 1 1 (R2 , R M ) × L ∞ (0, T ; W 1,∞ (R2 , Rr )) × L loc ×L ∞ (Ω T , Rm ) and closed in L loc (Ω T , Rm ). For α > 0 let the set Yα be given by
M l ∞ 2 M y C([0,T ],L ∞ (R2 )) ≤ α . Yα := y ∈ C [0, T ] , L (R , R )
(26)
l=1
Assumption 12 For some p ∈ [1, ∞) the functional J from (3) p p p J : Yα ⊂ C([0, T ] ; L loc (R2 , R M )) × Uad ⊂ L loc (R2 , R M ) × L loc (ΩT , Rr ) p × L loc (ΩT , Rm ) −→ R
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is sequentially lower semicontinuous for all α with Yα from (26). Remark 3. The validity of Assumption 12 for some p ∈ [1, ∞) implies under Assumption 11 the validity for all p ∈ [1, ∞). The next assumption ensures the existence of an optimal control to (3). 1 Assumption 13 The admissible set Uad is compact in L loc (R2 , R M ) × L ∞ (0, T ; 1 1,∞ 2 r m W (R , R )) × L loc (Ω T , R ).
6.2 Existence of Optimal Controls First, we show a lemma which makes it easy to prove the existence of optimal controls [15]. Lemma 2. Let u = (u0 , u1 , u2 ), uˆ = (uˆ0 , uˆ1 , uˆ2 ) ∈ Uad , u0 ∈ BV (R2 , R M ) and Assumptions (7), (11) and Assumptions 4, 6, 7, 11 be fulfilled. Let K R := K (0, T ) be the characteristic cone K := K (τ1 , τ2 ) := (x, t) ∈ Ω T |x| ≤ R + N (τ2 − t), τ1 ≤ t ≤ τ2 . and St the cross section of K R at time t. Then for all R > 0 we have: 1. Let u ∈ (Uad ⊂ (L 1 (S0 , R M ) × L ∞ (0, T ; W 1,∞ (R2 , Rr )) × L 1 (K R , Rm ))) and y(u) ∈ C([0, T ] ; L p (B R (0), R M )). The mapping u → y(u) is 1/ p-Höldercontinuous for p ∈ [1, ∞), i.e. there are nonnegative constants C, 1/ p such that y(u) − y(u) ˆ C([0,T ];L p (B R (0),R M )) 1/ p ≤ C u − uˆ L 1 (S0 ,R M )×L ∞ (0,T ;W 1,∞ (R2 ,Rr ))×L 1 (K R ,Rm )
(27)
for all u, uˆ ∈ Uad . 1 (R2 , R M ) × 2. If the functional J satisfies Assumption 12, then for u ∈ (Uad ⊂ (L loc 1 ∞ 1,∞ 2 r m L (0, T ; W (R , R )) × L loc (ΩT , R ))) the mapping u → J (y(u), u) is well defined and lower semicontinuous. Note that the ball B R (0) is equal to the cross section ST of K R at time T . Proof to Lemma 2, 1.: Because of Theorem 5 the unique entropy solution y i (·, t) with y i (·, t) L ∞ (R2 ) ≤ M∞ (t) for all u ∈ Uad exists. Using an interpolation inequality and the bounds M∞ (t) (see Theorem 5), M1 (t) see (25), which are monotone p increasing in t, it follows y ∈ C([0, T ]; L loc (R2 , R M )). Explicitly for any compact 2 set K ⊂ R
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y i C([0,T ];L p (K )) 1
1− 1
p = max y i (·, t) L p (K ) ≤ max y i (·, t) Lp 1 (K ) max y i (·, t) L ∞ (K )
t∈[0,T ]
t∈[0,T ]
1 p
≤ M1 (T ) · M∞ (T )
1− 1p
t∈[0,T ]
< ∞.
Let yˆ ∈ L ∞ (Ω T , R M ) be an entropy solution of the weakly coupled hyperbolic initial value problem to (uˆ 0 , uˆ 1 , uˆ 2 f 1 , f 2 , g). An interpolation inequality and the stability result of Theorem 9 imply for z i (·, t) := y i (·, t) − yˆ i (·, t) 1 1− 1p p i i i z C([0,T ];L p (B R (0))) ≤ max z (·, t) L 1 (B R (0)) z (·, t) L ∞ (B R (0)) t∈[0,T ]
M 1 p i u i0 − uˆ i0 L 1 (S0 ) + ≤ C max z (·, t) L 1 (B R (0)) ≤ C max t∈[0,T ]
t∈[0,T ]
i=1
r m u l1 − uˆ l1 L ∞ (ΩT ) + ∇x u l1 − ∇x uˆ l1 L ∞ (ΩT ) + u k2 − uˆ k2 L 1 (K l=1
1p (0,T ))
.
k=1
Proof to Lemma 2, 2.: It is y ∈ Yλ for all u ∈ Uad , since y i C([0, T ];L ∞ (R2 ,R M )) = max y i (·, t) L ∞ (R2 ,R M ) ≤ max M∞ (t) ≤ M∞ (T ). t∈[0,T ]
t∈[0,T ]
The mapping u → J (y(u), u) is well defined because of Lemma 2 (1) and Assumption 11. Let (u k )k∈N ⊂ Uad be a sequences with u k → u¯ in L 1loc (R2 , R M ) × L ∞ (0, T ; 1 (Ω T , Rm ). Through Assumption 11 Uad is closed in W 1,∞ (R2 , Rr )) ×L loc 1 (Ω T , Rm ) and thus u¯ ∈ Uad . Now L 1loc (R2 , R M ) × L ∞ (0, T ; W 1,∞ (R2 , Rr )) ×L loc p p p we have to show that u k → u¯ in L loc (R2 , R M ) × L loc (ΩT , Rr ) ×L loc (Ω T , Rm ). The interpolation inequality leads for any compact set K ⊂ R2 and D ⊂ ΩT and θ ∈ (0, 1) to ¯ L p (K ,R M )×L p (D,Rr )×L p (D,Rm ) u k − u ≤ u k − u ¯ θL 1 (K ,R M )×L 1 (D,Rr )×L 1 (D,Rm ) u k − u ¯ 1−θ L ∞ (K ,R M )×L ∞ (D,Rr )×L ∞ (D,Rm ) −→ 0
for k → ∞.
Assumption 11 gives for the second term a constant Mu > 0 such that ¯ L ∞ (K ,R M )×L ∞ (D,Rr )×L ∞ (D,Rm ) ≤ Mu . u k − u ¯ in Yλ ⊂ C [0, T ]; L p (B R (0), R M ) , since, Moreover, y(u k ) ∈ Yλ and y(u k ) → y(u) according to part 1, we have ¯ C ([0,T ];L p (B R (0),R M )) y(u k ) − y(u) 1
≤ Cu k − u ¯ Lp 1 (S0 ,R M )×L ∞ (0,T ;W 1,∞ (R2 ,Rr ))×L 1 (K R ,Rm )
−→ 0 for k → ∞.
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Assumption 12 leads to lim inf k→∞ J (y(u k ), u k ) ≥ J (y(u), ¯ u). ¯ Theorem 14. Let the assumptions of Lemma 2 be valid and additionally Assumptions 12 and 13. Then there exists a solution u¯ ∈ Uad of the control problem (3). Proof Similar as in [15]. To round off we give an example of functionals fulfilling Assumption 12. For more information compare to [15]. Example 1. Let D, K ⊂ R2 and K ⊂ Ω T be compact sets with Lipschitz bound and let yd ∈ L ∞ (D, R M ) be a given function. Then the functional J (y, u) = J1 (y) + R1 (u) := (y(x, T ) − yd (x))2 dx D + α0 u 0 2L 2 (K ,R M ) + α1 u 1 2L 2 (K ,Rr ) +α2 u 2 2L 2 (K ,Rm ) satisfies with α0 , α1 , α2 ≥ 0 Assumption 12 and is continuous. The second term R1 (u) is a regularization term, and thus this functional is also called cost functional. At last, we present an application of our theory. Conservation laws describing traffic flow can be found in e.g., [8], [7] and for multilane problems in [4]. Here, we consider a multilane LWR model with Greenshield flux. Example 2. Let ρ i be the density on lane i, ρmax the maximal density and vi s , the velocity at the i-th lane. For w, s, z ∈ R, we define vi (w, s) := w 1 − ρmax f˜i (w, s) := svi (w, s) and f i (z, s) := f˜i (z i , s). If we insert these fluxes into (1) in one dimension, we get the following LWR model with Greenshield flux with source term in R × [0, T ]: ∂t ρ 1
+
∂t ρ 2
+
.. . ∂t ρ M
+
ρ1 ∂x ρ 1 u 11 1 − ρmax ρ2 ∂x ρ 2 u 21 1 − ρmax .. . ρM ∂x ρ M u 1M 1 − ρmax ρ(x, 0)
=
g 1 (x, t, u2 , ρ)
=
g 2 (x, t, u2 , ρ) .. .
=
g M (x, t, u2 , ρ)
=
u0 (x) in R.
(28)
(29)
Systems (28) and (29) can be used to formulate an optimal control problem as in (3) and we can infer existence from Theorem 14. For realistic applications of (28), it is necessary to generalize the results of this paper also to bounded domains.
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References 1. H.W. Alt, Lineare Funktionalanalysis, 6, überarbeitete edn. (Springer-Verlag, Wien, 2011) 2. L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity problems (The Clarendon Press, Oxford University Press, New York, Oxford Mathematical Monographs, 2000) 3. R.M. Colombo, M. Mercier, M. Rosini, Stability and Total Variation Estimates on General Scalar Balance Law. Comptes Rendus Mathematique 347(1), 45–48 (2009) 4. R. M. Colombo and A. Corli, On Source Terms in Multilane Traffic Models, Communications to SIMAI Congress, Vol 2, 2007 5. A. Friedman, Partial differential equations of parabolic type (Prentice-Hall, inc., Englewood Cliffs, N.J., USA, 1964) 6. M. Hawerkamp, D. Kröner, H. Moenius, Existence of optimal controls in flux-, source and initial terms for weakly coupled hyperbolic systems University Freiburg, Preprint 2017 7. P. Kachroo, Pedestrian dynamics : mathematical theory and evacuation control, Boca Raton, Fla. [u.a.]: CRC Press, 2009 8. D. Kröner, Numerical Schemes for Conservation Laws, Wiley-Teubner, 1997 9. S.N. Kruzkov, First Order quasilinear equations in several independent variables. MATH USSR SB 10(2), 217–243 (1970) 10. O. A. Ladyzhenskaya, Linear and Quasilinear Elliptic Equations, Academic press, Inc. (London) LTD. London, 1968 11. M. Lecureux-Mercier, Improved stability estimates on general scalar balance laws. Journal of hyperbolic differential equations 84, 727–757 (2011) 12. A. Levy, On Majda’ Model For Dynamic Combustion. Communications in Partial Differential Equations 17, 657–698 (1992) 13. C. Rohde, Schwach gekoppelte Systeme hyperbolischer Erhaltungsgleichungen, Universität, Dissertation, September 1996 14. C. Rohde, Entropy solutions for weakly coupled hyperbolic systems in several space dimensions. Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 49(3), 470–499 (1998) 15. S. Ulbrich, Optimal Control of Nonlinear Hyperbolic Conservation Laws with Source Terms, Technische Universität München Fakultät für Mathematik, Habilitationsschrift, 11–37, Jun 2001
On Convergence of Numerical Methods for Optimization Problems Governed by Scalar Hyperbolic Conservation Laws Michael Herty, Alexander Kurganov and Dmitry Kurochkin
Abstract We consider optimization problems governed by scalar hyperbolic conservation laws in one space dimension and study numerical schemes for the solution to arising linear adjoint equations. We analyze convergence properties of adjoint and gradient approximations on an unbounded domain x ∈ R with a strictly convex flux. This paper provides the theoretical foundation of the scheme introduced in (Herty, Kurganov and Kurochkin, Commun. Math. Sci. 51, 15–48 (2015)) [14]. We also demonstrate that using a higher order temporal discretization helps to substantially improve both the efficiency and accuracy of the overall numerical method.
1 Introduction We are concerned with numerical methods for optimization problems governed by scalar hyperbolic conservation laws in one space dimension, which also could be further generalized to nonlinear hyperbolic systems. These types of problems arise in a variety of applications where inverse problems for the corresponding initial value problems (IVP) are to be solved. We focus on numerical methods related to those presented in [14]. In this paper, we discuss convergence properties of adjoint and gradient approximations in one-dimensional (1-D) scalar problems on an unbounded domain x ∈ R with a strictly convex flux. The discussion will be based on a M. Herty (B) Department of Mathematics, RWTH Aachen University, Templergraben 55, D-52056 Aachen, Germany e-mail: [email protected] A. Kurganov Department of Mathematics, Southern University of Science and Technology of China, Shenzhen 518055, China e-mail: [email protected] A. Kurganov · D. Kurochkin Mathematics Department, Tulane University, New Orleans, LA 70118, USA e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 C. Klingenberg and M. Westdickenberg (eds.), Theory, Numerics and Applications of Hyperbolic Problems I, Springer Proceedings in Mathematics & Statistics 236, https://doi.org/10.1007/978-3-319-91545-6_53
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general existence results for first-order schemes presented in [20] utilized to establish existence for a variety of other methods, see, e.g., [1, 7, 10, 21]. The optimization problem is formulated as follows: Find an optimal initial condition u 0 (x) (control) such that the objective functional J is minimized: min J (u(·, T ); u d (·)), u0
(1)
Here, the objective functional J is 1 J (u(·, T ); u d (·)) := 2
∞ (u(x, T ) − u d (x))2 d x
(2)
−∞
and u(x, t) is the unique entropy solution of the following IVP for the 1-D scalar hyperbolic conservation law: u t + f (u)x = 0, u(x, 0) = u 0 (x),
x ∈ R, t ∈ (0, T ], x ∈ R.
(3)
Here, u : R × [0, T ] → R, u 0 (x) is an arbitrary bounded measurable function on R, the corresponding nonlinear flux is denoted by f (u), and the terminal state u d (x) is prescribed at time t = T . In case of sufficiently smooth solutions, the formal adjoint equation is given by pt + f (u(x, t)) px = 0, x ∈ R, t ∈ [0, T ),
(4)
subject to the following terminal state: p(x, T ) = pT (x),
pT (x) := u(x, T ) − u d (x), x ∈ R.
(5)
The coupled systems (3) and (4), (5) together with p(x, 0) = 0 a.e. x ∈ R
(6)
represent the first-order optimality system for smooth solutions of the problem (1)– (3), in which (3) should be solved forward in time from t = 0 to t = T , while the adjoint equation (4) should be solved backward in time from t = T to t = 0. There has been extensive literature on PDE-constrained problems of type (1)–(3) both analytically and numerically. The semi-group generated by the conservation law is not differentiable in L 1 , and therefore the usual notion of derivatives has to be extended to tangent vectors consisting of an L 1 -part and real part of the variation in shock position, see [4]. The studied equations (3)–(6) only capture the L 1 -variation leaving the variations in the shock positions aside. A first-order optimality system that includes shock variations is presented in [5]. The most recent review of existing literature can be found in [6–8, 10, 11, 14, 15, 20].
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Even though only the L 1 -part of the optimality system is captured by (3)–(6), it has been shown in [14] that a suitable numerical implementation allows to solve the optimization problem (1)–(3). In this paper, we formalize the observed behavior by proving convergence of a numerical scheme based on solving (3) and (4)–(6). We only prove convergence outside the regions influenced by shocks. The key discussion will be on the numerical discretization of the nonconservative transport equation (4). Theoretical discussion on transport equations can be found, e.g., in [2, 3, 18]. In order to obtain a well-posed adjoint problem, we follow [20] and assume a one-sided Lipschitz condition (OSLC) to be satisfied. The OSLC condition for v ∈ L ∞ (R × (0, T )), where v(x, t) := f (u(x, t)), reads vx (·, t) ≤ α(t), α ∈ L 1 (0, T ).
(7)
The adjoint equation is then well-posed for Lipschitz terminal data pT in the sense that there exists a unique reversible solution of (4), (5). The OSLC condition for equation (4) holds, for example, if the flux in (3) is strictly convex, that is, if f ≥ c > 0 for some c > 0.
(8)
In this paper, we consider the scheme, which is second order in time and first order in space and use the results from [20] to establish its convergence. The convergence proof for a second order in time scheme is the novel contribution of this work, which provides a theoretical base for the numerical results presented in [14].
2 Numerical Method In this section, we introduce the iterative optimization algorithm for the problem (1)–(3) based on the formal optimality system. The algorithm is a simplified version of Algorithm 3.1 from [14] and may be seen as a block Gauss/Seidel iteration. From now on, the optimal solution u 0 of the problem (1)–(3) will be called the recovered initial data, while the corresponding solution of the system (3) will be referred to as the recovered solution.
2.1 Iterative Algorithm Assuming two tolerance parameters, ε J and εΔJ (the second parameter is needed since the optimal value of the objective functional may be strictly positive), are chosen a priori, we implement the iterative algorithm to generate a sequence {u (m) 0 (x)}, m = 0, 1, 2, . . . of recovered initial data as follows.
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Algorithm 2.1 Step 1. Step 2.
Step 3.
Choose an initial guess u (0) 0 (x) for the initial data u 0 (x). Set m := 0. Numerically, solve (3) with the initial state u 0 (x) = u (m) 0 (x) forward in time from t = 0 to t = T by the semi-discrete version of the Engquist– Osher scheme described in Sect. 2.2. We denote the obtained solution by u (m) (x, t). Compute the objective functional J (u
Step 4.
(m)
1 (·, T ); u d (·)) := 2
If either
∞
u (m) (x, T ) − u d (x)
2
d x.
−∞
J (u (m) (·, T ); u d (·)) ≤ ε J ,
or m > 0 and J (u (m) (·, T ); u d (·)) − J (u (m−1) (·, T ); u d (·)) ≤ εΔJ ,
Step 5.
Step 6. Step 7.
stop the iteration process. The obtained u (m) 0 (x) will be the approximation to the optimal control. Numerically, solve the adjoint system (4), (5) subject to the terminal condition p(x, T ) = u (m) 0 (x) − u d (x) backward in time from t = T to t = 0 using a semi-discrete upwind scheme described below. The solution is denoted by p (m) (x, t). Update the control u (m) 0 (x) using either a gradient descent or quasi-Newton method [17]. Set m := m + 1. Go to Step 2.
2.2 Numerical Schemes In Step 2 of the Algorithm described in Sect. 2.1, the conservation law (3) is being solved using the semi-discrete version of the Engquist–Osher scheme, which is described in this section. We consider the IVP (3) and solve it numerically on a uniform spatial grid with xα := αΔx. We denote by λ := Δt/Δx and introduce the computed cell averages over the cells [x j− 21 , x j+ 21 ]: x j+ 1
1 u j (t) := Δx
2
u(x, t) d x, u nj := u j (t n ), x j− 1 2
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where t n := nΔt. The cell averages are then evolved in time using the following semi-discrete scheme: F j+ 21 (t) − F j− 21 (t) d u j (t) =− , dt Δx where F j+ 21 denotes the Engquist–Osher numerical flux: F j+ 21 (t) = u j+1 (t)
(9) uj (t)
f (ξ )+ dξ +
0
f (ξ )− dξ . The semi-discretization (9) is a system of ODEs, which should be
0
integrated using a (nonlinearly) stable and sufficiently accurate ODE solver. For example, the system (9) can be solved using the second-order strong stability preserving (SSP) Runge–Kutta method [12, 13] also known as the Heun method [16]: n+ 1 n+ 21 n 2 , (10) u n+1 = u − λ H − H 1 1 j j j+ j− 2
2
where n+ 1
1 n F j+ 1 + , Fn+1 1 j+ 2 2 2 n n u j u j+1 = f (ξ )+ dξ + f (ξ )− dξ,
H j+ 12 :=
(11a)
Fnj+ 1
(11b)
2
2
0
0
u jn+1
= Fn+1 j+ 1
2
u j+1
n+1
f (ξ )+ dξ +
0
f (ξ )− dξ.
(11c)
0
Here, we denote by (·)+ := max{·, 0} and (·)− := min{·, 0}, and the intermediate value u jn+1 is defined by u n+1 := u nj − λ Fnj+ 1 − Fnj− 1 , j 2
(12)
2
and is, in fact, a solution obtained after a forward Euler step. In the following, we describe the semi-discrete upwind scheme used in Step 5 of Algorithm 2.1 for solving the adjoint equation (4). Since u(x, t) has been computed in Step 2, the adjoint problem (4) is, in fact, the following linear equation with variable coefficients: (13) pt + v(x, t) px = 0, x ∈ R, t ∈ [0, T ),
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subject to the terminal conditions (5), where v(x, t) := f (u(x, t)).
(14)
According to [20], Algorithm 2.1 will converge provided the numerical method for the adjoint problem (4), (5) is induced by the numerical method for the conservation law (3). Introducing the notation p nj := p(x j , t n ), the corresponding discretization of the adjoint problem can be written as follows: p nj = p n+1 +λ j
2
k=−1
where vnj+ 1 ,k = 2
n+1 vnj−k+ 1 ,k p n+1 j−k+1 − p j−k ,
(15)
2
∂ n+ 1 H j+ 12 u nj−1 , . . . , u nj+2 , n 2 ∂u j+k
(16)
n+ 1
and the numerical flux H j+ 12 is defined in (11), (12). The coefficients vnj+ 1 ,k can 2 2 be obtained explicitly by substituting (11b), (11c), and (12) into (11a) and then computing the partial derivatives in (16), which result in λ n+1 + n + u j ) f (u j−1 ) , (17a) f ( 2 λ 1 1 n+1 − n + = f (u nj )+ + f ( u jn+1 )+ 1 − λ| f (u nj )| + f ( u j+1 ) f (u j ) , 2 2 2 (17b) 1 n − 1 n+1 − λ = f (u j+1 ) + f ( u j+1 ) 1 − λ| f (u nj+1 )| − f ( u jn+1 )+ f (u nj+1 )− , 2 2 2 (17c) λ n+1 − n − = − f ( u j+1 ) f (u j+2 ) . (17d) 2
vnj+ 1 ,−1 = 2
vnj+ 1 ,0 2
vnj+ 1 ,1 2
vnj+ 1 ,2 2
Notice that when we solve the adjoint problem, both the second-order, {u nj }, and first-order, { u nj }, solutions of the forward problem are available for all n since they have been computed in Step 2 of Algorithm 2.1. Other numerical flux functions F j+ 21 are possible. We require that their derivative obtained by Eq. (16) yields a discretization of (13). Suitable conditions are stated in the following section. Remark 1. The scheme (15), (17) can be derived in an alternative way. Consider the following semi-discrete upwind scheme for the adjoint equation (13): dp j (t) + p j+1 (t) − p j (t) − p j (t) − p j−1 (t) = − f (u j (t)) + f (u j (t)) , (18) dt Δx Δx
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where p j (t) := p(x j , t). Using the vector notations, the ODE system (18) can be written as d p(t) = g(u(t), p(t)), (19) dt where p(t) = { p j (t)}T and g(·, ·) = {g j (·, ·)}T is the right-hand side (RHS) of (18). We now apply the second-order Heun method to the system (19) and obtain the following fully discretize scheme for (13): pn = pn+1 −
Δt g(un+1 , pn+1 ) + g(un , p˜ n ) , 2
(20)
where pn := p(t n ) and the intermediate value p˜ n is a solution obtained after one step of forward Euler method (applied backward in time) and defined as p˜ n := pn+1 − Δt g(un+1 , pn+1 ).
(21)
One can see that the adjoint scheme, introduced in (15), (17), can be written (see Appendix 4) in the form very similar to (20), (21): Δt g( un+1 , pn+1 ) + g(un , pn ) , 2 pn = pn+1 − Δt g( un+1 , pn+1 ). pn = pn+1 −
(22) (23)
Therefore, the induced scheme (22), (23) can be seen as a modification of the scheme is the first-order approximation of u(x j , t n+1 ), while u n+1 is (20), (21). Since u n+1 j j the second-order one, the approximation used in (20), (21) should be a little more accurate. However, the formal order of accuracy of both schemes is the same (they are first order in space and second order in time) and the convergence proof presented in the next section is valid for the induced scheme (22), (23) only. Remark 2. We have tested both the second-order scheme (20), (21) and its modification (22), (23) in numerical experiments with a variety of initial guesses u (0) 0 (x) and terminal states u d (x), both smooth and discontinuous ones. Because the difference in the numerical results obtained by the schemes (20), (21) and (22), (23) is negligible, for the sake of brevity we omit the results obtained by the scheme (22), (23). Remark 3. Note that the first-order version of both the scheme (20), (21) and (22), (23) is simply given by un+1 , pn+1 ). pn = pn+1 − Δt g(
(24)
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3 Convergence Analysis In this section, we discuss convergence properties of the numerical method, introduced in Sect. 2. We assume that (8) holds and without loss of generality we set f (0) = 0. We assume that the scheme (10)–(12) yields entropy solutions of (3). (In fact, the semi-discrete version of the Engquist–Osher scheme (9) is entropy stable as it has been proved in [19].) To be more precise, let u Δ (x, t) :=
u j (t)χ j (x),
(25)
j
where χ j (x) is a characteristic function of the interval [x j− 21 , x j+ 21 ], be a numerical solution computed at time t. We assume that there are positive constants M and Δ0 and an entropy solution u(x, t) such that for all Δt = λΔx ≤ Δ0 u Δ ∞ ≤ M, u Δ (·, t) → u(·, t) in L 1 (R) t > 0, Δt → 0.
(26)
We also assume that for the function α in (7), the discrete OSLC condition, u nj+1
−
u nj
1 ≤ λ
t n+1 α(t) dt, ∀ j ∈ Z, ∀n,
(27)
tn
holds. Notice that the condition (27) does not allow for jumps up in the initial data u 0 . To allow such jumps, the condition (27) should be relaxed (for a weakened version of (27), see [20, Condition (D3’)]). It was proved in [20, Sect. 6.5.1] that both the conditions (26) and (27) are satisfied for the original Engquist–Osher scheme, which replaced with u n+1 on the left-hand side of (12). Since the is (12), (11b) with u n+1 j j second-order Heun method is in fact a convex combination of two forward Euler steps, the results from [20, Sect. 6.5.1] are still valid for the scheme (10), (11). We now proceed with the convergence analysis of the scheme (15), (17). We first prove the following monotonicity result needed to establish the convergence proof below. n+ 1
Lemma 1. Assume that the L ∞ -bound (26) holds. Let H j+ 12 be given by (11), (12) 2 and assume that the time step is restricted by the following CFL condition: λ≤
1 . 2 max | f (u)|
(28)
|u|≤M
Then, the coefficients vnj+ 1 ,k defined in (16) are nondecreasing functions of u n for 2 ∈ { j − 1, j, j + 1, j + 2}.
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Proof. Note that by substituting (11a) into (12) and by differentiating (12) one can show that provided the condition (28) is satisfied, ∂ n+1 u ≥ 0 for k ∈ { j − 1, j, j + 1}. ∂u nk j Hence, u n+1 is nondecreasing with respect to all of its arguments, u nj−1 , u nj , and u nj+1 . j Since f is convex, f (·)± are nondecreasing functions. Further, (17a) clearly implies + n + u n+1 that vnj+ 1 ,−1 is nondecreasing with respect to f ( j ) and f (u j−1 ) . Similarly, 2
− from (17d) we obtain that vnj+ 1 ,2 is nondecreasing with respect to f ( u n+1 j+1 ) and 2
f (u nj+2 )− . Given the condition (28), one can show that vnj+ 1 ,0 is nondecreasing with 2
+ n+1 − respect to f (u nj )± , f ( u n+1 u j+1 ) by differentiating (17b), namely, j ) and f (
1 ∂vnj+ 1 ,0 1 + n+1 − 2 = 1 − 2λ f ( 1 + 2λ f + ≥ 0, u n+1 ) ( u ) j j+1 4 4 ∂ f (u nj )+ ∂vnj+ 1 ,0 λ + 2 = f ( u n+1 j ) ≥ 0, n − 2 ∂ f (u j ) ∂vnj+ 1 ,0
1 = 1 − λ| f (u nj )| ≥ 0, n+1 2 ∂ f ( u j )+ 2
∂
∂vnj+ 1 ,0 2
− f ( u n+1 j+1 )
=
λ n + f (u j ) ≥ 0, 2
where the identity | f (u nj )| = f (u nj )+ − f (u nj )− is taken into account. Similarly, the differentiation of (17c) shows that vnj+ 1 ,1 is nondecreasing with respect to f (u nj+1 )± , 2
+ n+1 − f ( u n+1 u j+1 ) , provided (28) is satisfied. Finally, using the chain rule j ) and f ( we conclude that vnj+ 1 ,k are nondecreasing functions of u n for ∈ { j − 1, j, 2 j + 1, j + 2}.
Next, we rewrite the discrete adjoint scheme (15) as p nj =
2
B nj,k p n+1 j−k ,
(29)
k=−2
where the corresponding coefficients are B nj,−2 = λvnj+ 3 ,−1 , B nj,−1 = λ vnj+ 1 ,0 − vnj+ 3 ,−1 , 2 2 2 n n n n B j,0 = 1 + λ v j− 1 ,1 − v j+ 1 ,0 , B j,1 = λ vnj− 3 ,2 − vnj− 1 ,1 ,
(30b)
B nj,2 = −λvnj− 3 ,2 .
(30c)
2
2
2
2
(30a)
2
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According to [20], the L ∞ -stability of the adjoint scheme will follow from the positivity of B nj,k , which will be guaranteed by the following lemma. n+ 1
Lemma 2. Let H j+ 12 be given by (11), (12) and the adjoint scheme given by (29), 2 (30). Assume that the L ∞ -bound (26) holds and the time step is restricted by λ≤
1 . max | f (u)|
(31)
|u|≤M
Then, the coefficients B nj,k are nonnegative: B nj,k ≥ 0 ∀ j ∈ Z, k ∈ {−2, −1, 0, 1, 2}. Proof. First, we obtain that B nj,±2 ≥ 0 from their definitions (30a), (30c) and from the definition of vnj+ 1 ,k given in (17). 2
n We now note that | u n+1 j | ≤ M, since |u j | ≤ M and the solution of the first-order Engquist–Osher scheme (12), (11b) satisfies the maximum principle. Then, B nj,0 can be estimated using (17) and (31) as follows:
λ n n | f (u j )| + | f ( u n+1 B nj,0 = 1 − j )|(1 − λ| f (u j )|) 2 + n − n+1 − n + +λ f ( u n+1 ) f (u ) + f ( u ) f (u ) j j j−1 j+1 λ n ≥1− | f (u j )| + | f ( u n+1 j )| ≥ 0. 2 Similarly, from (30a), (30b), (17), and (31), we obtain λ n + n+1 + f (u j ) 1 − λ| f ( u n+1 u j ) 1 − λ| f (u nj )| ≥ 0, j+1 )| + f ( 2 λ n − n+1 − n − f (u j ) 1 − λ| f ( 1 − λ| f = u n+1 )| − f ( u ) (u )| ≥ 0, j j−1 j 2
B nj,−1 = B nj,1
which completes the proof of the lemma.
We further obtain bounds on the discrete difference approximation p nj+1 − p nj , computed by the adjoint scheme (15), (17). Using the equivalent form (29), (30) of the adjoint scheme, we rewrite the difference as follows: p nj+1 − p nj =
2
2
n+1 n+1 n B nj+1,k p n+1 C nj,k p n+1 j−k+1 − B j,k p j−k = j−k+1 − p j−k ,
k=−2
k=−2
where the coefficients C nj,k := B nj,k + λ vnj−k+ 1 ,k+1 − vnj−k− 1 ,k+1 , for − 2 ≤ k ≤ 1,
(32a)
C nj,2 := B nj,2 ,
(32b)
2
2
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are obtained by simply regrouping the summands. The following lemma shows the positivity of the coefficients C nj,k . Lemma 3. Assume that the L ∞ -bound (26) holds and the CFL condition (28) is satisfied. Then, the coefficients C nj,k given by (32) are nonnegative: C nj,k ≥ 0 ∀ j ∈ Z, k ∈ {−2, −1, 0, 1, 2}.
(33)
Proof. Lemma 2 implies C nj,2 = B nj,2 ≥ 0. Then, (32a) together with (30a) and (17a) give C nj,−2 = B nj,−2 + λ vnj+ 5 ,−1 − vnj+ 3 ,−1 = λvnj+ 5 ,−1 ≥ 0. 2
2
2
The estimate on C nj,0 follows from (32a), (17) and the CFL condition (28): C nj,0 = B nj,−2 + λ vnj+ 1 ,1 − vnj− 1 ,1 = 1 + λ vnj+ 1 ,1 − vnj+ 1 ,0 2
2
2
2
λ n − − n n+1 + f (u j+1 ) − f (u nj )+ + f ( u n+1 u j ) (1 − λ| f (u nj )|) ≥1+ j+1 ) (1 − λ| f (u j+1 )|) − f ( 2 λ n − − n+1 + f (u j+1 ) − f (u nj )+ + f ( u n+1 u j ) ≥ 0. ≥1+ j+1 ) − f ( 2
Similarly, from (32), (30), (17), and (28), we obtain C nj,−1 = B nj,−1 + λ v n
j+ 23 ,0
− vn
j+ 21 ,0
= λ vn
j+ 23 ,0
− vn
j+ 23 ,−1
λ n u n+1 )− + f ( u n+1 )+ (1 − λ| f (u nj+1 )| − λ f (u nj )+ ) ≥ 0, f (u j+1 )+ 1 + λ f ( j+2 j+1 2 n C j,1 = B nj,1 + λ v n 1 − v n 3 = λ vn 1 − vn 1 =
j− 2 ,2
j− 2 ,2
j− 2 ,2
j− 2 ,1
λ n+1 − − f ( u j ) 1 + λ f (u nj+1 )− − λ| f (u nj )| − f (u nj )− 1 − λ f ( u n+1 )+ ≥ 0, = j−1 2
so that the proof of the lemma is complete.
Finally, we apply the convergence proof of [20, Theorem 6.4.4] to the introduced numerical method. Theorem 1. Assume f ∈ C 2 (R) satisfies (8). Let the terminal state pT , defined in (5), be Lipschitz continuous and u Δ satisfies (26) and (27). Assume that the discretization pTΔ of the terminal state is consistent, that is, there exist constants K > 0 and L > 0 such that Δ pT (x + Δx) − pTΔ (x) Δ ≤L pT ∞ ≤ K , sup Δx x∈R and
pTΔ → pT in [−R, R] ∀R > 0, Δx → 0.
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Assume the condition (28) holds. Then, the numerical solution (15), (17) converges locally uniformly to the unique reversible solution p ∈ Li p(R × (0, T )) of the adjoint problem (13), (14) as Δt = λΔx → 0. Proof. Lemmas 1, 2, and 3 ensure that all of the assumptions of [20, Theorem 6.4.4] are satisfied and thus the convergence result follows. At the end of this section, we state the convergence result for the discrete gradients justifying the presented algorithm for a smooth version of the objective functional. δ δ For a given nonnegative function φδ ∈ 1Li p02(R) with the support in [− 2 , 2 ] and R φδ (x) d x = 1, and for a given ψ ∈ C loc (R ), we define the functional Jδ as ∞ Jδ (u 0 ) :=
ψ (φδ ∗ u)(x, T ), (φδ ∗ u d )(x) d x,
(34)
−∞
where, as before, u(x, t) is the entropy solution of the IVP (3), u d (x) is a terminal state prescribed at time t = T , and ∗ denotes a convolution in x. For Jδ to be well-posed, we assume u d ∈ L ∞ (R). We discretize Jδ by Jδ (u Δ 0)=
ψ (φδ ∗ u Δ )(xk , T ), (φδ ∗ u Δ d )(x k ) Δx,
(35)
k Δ Δ where u Δ 0 , u d , and u denote the corresponding piecewise constant approximations defined in (25). The gradient of Jδ exists in the sense of Fréchet differentials, see [20, Theorem 5.3.1]. Using Lemmas 1, 2 and 3 and Theorem 1 the following convergence result immediately obtained from [20, Theorem 6.4.8].
Theorem 2. Assume that f ∈ C 2 (R) satisfies (8). Let Jδ be defined by (34), where function with the support in [− 2δ , 2δ ] and φ δ ∈ Li p0 (R) is a given nonnegative 1 2 ∞ R φδ (x) d x = 1, and ψ ∈ C loc (R ). Assume also that u 0 ∈ L (R × (0, T )) such that (u 0 )x ≤ K . Let p Δ (x j , T ) =
φδ (x j − xk )∂1 ψ (φδ ∗ u Δ )(xk , T ), (φδ ∗ u Δ d )(x k ) Δx,
(36)
k
where ∂1 ψ denotes a partial derivative of ψ with respect to its first component. Let u Δ be an approximate solution of (3) obtained by (10)–(12) and thus satisfies (26) and (27). Let p Δ be a piecewise constant approximation of the solution computed by (15), (17) subject to the terminal data (36), and assume that the CFL condition (28) holds. Then, p Δ (·, 0) is an approximation to the Fréchet derivative of Jδ with respect to u 0 in the following sense: p Δ (·, 0) → p(·, 0) = ∇ Jδ (u 0 ) in L r (R) as Δt = λΔx → 0,
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for all r ≥ 1. Herein, p is the reversible solution of (13) with the terminal data ∞ pT (x) =
φδ (x − z)∂1 ψ (φδ ∗ u)(z, T ), (φδ ∗ u d )(z) dz.
−∞
4 Numerical Results In this section, we compare the performance of the optimization method described in Sect. 2.1 using the first-order schemes (12), (11b) and (24) and the second-order schemes (10)–(12) and (20), (21). We refer the reader to [14] for further numerical results (including a much more complicated case of systems of hyperbolic conservation laws). In particular, in [14] we consider the examples, in which the control u 0 is recovered exactly. Here, on the contrary, we compare the convergence of first- and second-order (in time) schemes. The convergence analysis yields the convergence of the first-order scheme, whereas the numerical results indicate the same qualitative results for its second-order extension. The rate of convergence is improved in the second-order case. We consider the problem (1), (2) governed by the inviscid Burgers equation ut +
u2 2
x
=0
(37)
with the terminal state sin(6π(x − 13 )), if 13 ≤ x ≤ 23 , u d (x) = 0, otherwise
(38)
prescribed at T = 1/3. We solve the problem in the interval [0, 1] subject to the periodic boundary conditions using the uniform mesh with Δx = 1/400 and the following initial guess: (39) u (0) 0 = sin(2π x). The recovered initial data, u (20000) (x), and the corresponding recovered solution 0 (x, T ), computed using the studied first- and second-order schemes are shown u in Figs. 1 and 2, respectively. Finally, in Fig. 3, we show the behavior of the computed objective functional (2) for m = 1, . . . , 20000 iterations using a logarithmic scale. The obtained results clearly demonstrate the advantage of a second-order temporal discretization even when the same first-order semi-discrete schemes in space are used. (20000)
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Fig. 1 First-order results. Left: Recovered initial data u 0 (x); Right: Recovered solution u (20000) (x, T ) (plotted with points) and the terminal state u d (x) (dashed line)
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Fig. 2 Second-order results. Left: Recovered initial data u 0 (x); Right: Recovered solution u (20000) (x, T ) (plotted with points) and the terminal state u d (x) (dashed line) Fig. 3 Dependence of the computed objective functional (measured in a logarithmic scale) on the number of iterations for the first- (dashed line) and second-order (solid line) schemes
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−1.4 −1.6 0
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Acknowledgements The work of M. Herty was supported by BMBF KinOpt and the cluster of excellence EXC128, HE5386/13,14,15-1. The work of A. Kurganov was supported in part by the NSF Grants DMS-1115718 and DMS-1521009. The authors also acknowledge the support by the KI-Net research network, NSF RNMS grant DMS-1107444.
Appendix Here, we demonstrate equivalence of the backward scheme (15), (17) and the modified second-order Heun method (22), (23). First, we notice that p j+1 − p j p j − p j−1 + f (u j )− g j (u, p) = − f (u j )+ Δx Δx is linear in its second argument and hence g(un , pn ) = g(un , pn+1 ) − Δt g(un , g( un+1 , pn+1 )),
(40)
On Convergence of Numerical Methods for Optimization Problems Governed …
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where pn is defined in (23). Then the scheme (22), (23) can be written as follows: Δt g( un+1 , pn+1 ) + g(un , pn+1 ) − Δt g(un , g( un+1 , pn+1 )) . 2 (41) We then use (40) to rewrite (41) in a componentwise form: pn = pn+1 −
λ n+1 + n+1 f ( pnj = pn+1 + u j ) ( p j+1 − pn+1 ) + f ( u n+1 )− ( p n+1 − pn+1 ) j j j j j−1 2 λ n + n+1 f (u j ) ( p j+1 − pn+1 ) + f (u nj )− ( p n+1 − pn+1 ) + j j j−1 2 λ2 n + n+1 + n+1 f (u j ) f ( u j+1 ) ( p j+2 − pn+1 ) + f ( u n+1 )− ( p n+1 − pn+1 ) + j+1 j+1 j+1 j 2 λ2 n n+1 + n+1 n+1 − n+1 n+1 ( u j ) ( p j+1 − pn+1 ) + f u ) ( p − p ) − | f (u j )| f ( j j j j−1 2 λ2 n − n+1 + n+1 n+1 − n+1 n+1 ( f (u j ) f ( u j−1 ) ( p j − pn+1 ) + f u ) ( p − p ) − j−1 j−1 j−1 j−2 . 2
(42) Rearranging the terms in (42), we finally get the adjoint scheme in the following form: λ n + n+1 + n+1 f (u j ) f ( p nj = p n+1 +λ u j+1 ) ( p n+1 j j+2 − p j+1 ) 2 1 n+1 + 1 n + λ n + n+1 − λ n n+1 + +λ f ( u j ) + f (u j ) + f (u j ) f ( u j+1 ) − | f (u j )| f ( u j ) ( p n+1 j+1 − 2 2 2 2 1 n+1 − 1 n − λ n n+1 − λ n − n+1 + +λ f ( u j ) + f (u j ) − | f (u j )| f ( u j ) − f (u j ) f ( u j−1 ) ( p n+1 − j 2 2 2 2 λ n − n+1 − n+1 f (u j ) f ( u j−1 ) ( p n+1 −λ j−1 − p j−2 ), 2
p n+1 j ), p n+1 j−1 )
(43) which coincides with (15), (17). Note that the coefficients on the RHS of (43) are the ones given by (17).
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