Continuum mechanics, applied mathematics and scientific computing: Godunov's legacy 9783030388690, 9783030388706

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Gennadii V. Demidenko Evgeniy Romenski Eleuterio Toro Michael Dumbser   Editors

Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy A Liber Amicorum to Professor Godunov

Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy

Gennadii V. Demidenko Evgeniy Romenski Eleuterio Toro Michael Dumbser •

•

•

Editors

Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy A Liber Amicorum to Professor Godunov

123

Editors Gennadii V. Demidenko Sobolev Institute of Mathematics Novosibirsk, Russia

Evgeniy Romenski Sobolev Institute of Mathematics Novosibirsk, Russia

Eleuterio Toro Laboratory of Applied Mathematics, DICAM University of Trento Trento, Italy

Michael Dumbser Department of Civil, Environmental and Mechanical Engineering University of Trento Trento, Italy

ISBN 978-3-030-38869-0 ISBN 978-3-030-38870-6 https://doi.org/10.1007/978-3-030-38870-6

(eBook)

© Springer Nature Switzerland AG 2020 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, expressed 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. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

July 17, 2019 marked the 90th anniversary of the outstanding scientist, one of the leaders of modern applied mathematics, Sergei Konstantinovich Godunov. S. K. Godunov belongs to that generation of Soviet people who at the cost of huge efforts restored the Soviet Union after the hardest losses suffered during the Second World War. In the postwar years, the leadership of the Soviet Union paid special attention to nuclear and space projects, the success of which gave a chance to preserve the peace and independence of the country. S. K. Godunov was one of those scientists who actively participated in these projects. In 1951 after graduating from Moscow State University (MSU), S. K. Godunov was accepted into the Calculation Bureau of the Steklov Mathematical Institute of the USSR Academy of Sciences. Outstanding scientists B. N. Delaunay and I. G. Petrovskii were his scientific advisers at MSU. From the very first days, Sergei Godunov took an active part in the work of the team of scientists created to solve the most important practical problems in mathematical modeling and carry out calculations of various processes in the field of nuclear physics. The mathematical problems that needed to be solved arose as a result of discussions of renowned mathematicians and physicists. Yu. B. Khariton, Ya. B. Zeldovich, I. E. Tamm, A. D. Sakharov, E. I. Zababakhin, D. A. Frank-Kamenetskii, and other scientists came to the Institute from the scientific centers of Sarov and Snezhinsk to discuss the results of experiments and problem statements. In 1953, Sergei Godunov was given a task by M. V. Keldysh and I. M. Gelfand to create a variant of the numerical method of J. Neumann and R. Richtmyer for the gas dynamics equations using artificial viscosity. However, the young scientist proposed his own method and invented a difference scheme, which over time gained worldwide fame as the “Godunov scheme”. In 1954, he defended his candidate dissertation, the basis of which was precisely this method. This dissertation of S. K. Godunov was discussed by world-famous scientists I. M. Vinogradov, I. M. Gelfand, M. V. Keldysh, I. G. Petrovskii, S. L. Sobolev, and many more. In the following years, “Godunov scheme” had a huge impact on the development of modern computational mathematics. Nowadays, there is a variety of numerical methods for different classes of nonlinear equations of mathematical v

vi

Preface

physics and continuum mechanics. Many of them, including the “Godunov schemes” of high order of accuracy, are based on the initial ideas of S. K. Godunov. A significant role in the development of modern computational mathematics was also played by the monograph “Introduction to the theory of difference schemes”, written by S. K. Godunov and V. S. Ryabenkii (1962). During his work at the Mathematical Institute and the Institute of Applied Mathematics of the USSR Academy of Sciences (since 1966), S. K. Godunov created many methods for approximate solving various problems of continuum mechanics and laid the foundations of new areas of applied mathematics. Here are just a few examples. To solve numerically two-dimensional problems of continuum mechanics in domains with complex boundaries, special requirements for computational grids arise. The problem of automating the construction of curvilinear difference grids was formulated by S. K. Godunov, and various methods for solving it were developed under his leadership. S. K. Godunov created the method of time marching to steady state for the numerical analysis of the gas flow around bodies. S. K. Godunov created and justified the method of orthogonal successive substitution for solving boundary value problems for systems of ordinary differential equations. Significant results were obtained by S. K. Godunov in the theory of quasilinear equations. He studied the problem of uniqueness of generalized solution to gas dynamics equations and considered questions on the place of continuum mechanics equations in the theory of hyperbolic equations in conservative form, as well as the generalization of the concept of entropy, the law of its increase and thermodynamic relations. The obtained results formed the basis of his doctoral dissertation (1965). His work “An interesting class of quasilinear systems” (1961) on the connection between the laws of thermodynamics and the well-posedness of problems for models of continuum mechanics gave rise to a new direction of research of hyperbolic systems of conservation laws in mathematical physics. In 1969, S. K. Godunov moved to Novosibirsk at the invitation of M. A. Lavrentyev and was appointed the head of the laboratory at the Computer Center of the Siberian Branch of the USSR Academy of Sciences. His first work in Siberia was related to the study of the behavior of metals in explosion welding, which was carried out jointly with experimental physicists from the Institute of Hydrodynamics of the Siberian Branch of the USSR Academy of Sciences. To describe the high-speed deformation of metals, S. K. Godunov proposed a new approach to the modeling continuous media, which led to the creation of a unified model of continuous medium capable of describing its elastic, plastic, and liquid state using a single system of governing differential equations. Numerical modeling carried out under his leadership allowed to predict a new effect such as the formation of a submerged jet when metal plates collide. In 1980 at the invitation of S. L. Sobolev, he began to work in the Institute of Mathematics of the Siberian Branch of the USSR Academy of Sciences. At the Institute S. K. Godunov together with his disciples carried out active research of mixed problems for hyperbolic systems, computational problems in stability theory, and dichotomy for ordinary differential equations and in linear algebra. Despite the

Preface

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apparent differences, these areas are interrelated and are the continuation of the work carried out by S. K. Godunov during the Moscow period of his activity. Many new original and unexpected results have been obtained in all these areas. This has always been a characteristic feature in the scientific work of the Godunov school. Here are just a few examples. In the theory of partial differential equations, the problem of symmetrization of two-dimensional strictly hyperbolic equations was solved, complete description of the energy integrals was given, and new criteria for the well posedness of mixed problems for some classes of hyperbolic equations were obtained in terms of coefficients. In computational linear algebra, S. K. Godunov for the first time formulated the concept of guaranteed accuracy, thereby laying the foundations of a new direction in computational mathematics. He together with his disciples created a number of new computational algorithms with guaranteed accuracy for solving the spectral problem for non-Hermitian matrices, as well as for solving systems of linear equations on a computer. Within the framework of the concept of guaranteed accuracy, new fundamental concepts were introduced to ensure the rigor of computer calculations: e-spectrum, spectral portrait, dichotomy quality criterion, etc. In the theory of ordinary differential equations, S. K. Godunov together with his disciples developed new algorithms that allow carrying out numerical studies of asymptotic stability of stationary solutions to autonomous differential equations with guaranteed accuracy. Based on the developed mathematical apparatus, S. K. Godunov together with colleagues and disciples actively continues developing algorithms and carries out numerical analysis of various models of continuum mechanics, sometimes returning to the problems considered earlier and carrying out the research at higher level. Sergei Konstantinovich Godunov is selflessly devoted to science. He is always full of energy, ideas, and creative plans. His conviction, high culture, and broad erudition always attract colleagues, disciples, and followers to him. On August 4–10, 2019, in Akademgorodok of Novosibirsk (Russia), it was held a major international conference in honor of the 90th birthday of S. K. Godunov. It was attended by more than 300 scientists from 24 countries and 34 regions of Russia. Such great interest in the conference is undoubtedly connected with the scale of the personality of Sergei Konstantinovich. At the Opening Ceremony of the conference, S. K. Godunov gave the plenary lecture “Memories of difference models in hydrodynamics”. On this day, invited lectures were also given by eight leading Russian and foreign scientists. Each of them somehow mentioned the ideas, results, and methods of S. K. Godunov. This can be concluded by the titles of the lectures: “On S. K. Godunov’s works on the uranium problem” (Yu. N. Deryugin, R. M. Shagaliev, Russian Federal Nuclear Center, Sarov, Russia); “Godunov symmetric systems and rational extended thermodynamics” (T. Ruggeri, University of Bologna, Bologna, Italy); “Godunov methods” (E. F. Toro, University of Trento, Trento, Italy), etc.

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The range of topics that were within the scope of the conference included partial differential equations, equations of mathematical physics, differential-difference equations, mathematical modeling, difference schemes, computational methods of linear algebra, mathematical questions of gas dynamics, hydrodynamics, aerodynamics, and some other sections of continuum mechanics. Leading specialists in applied mathematics and mechanics from different countries of the world gave invited lectures at the conference. There were 42 invited lectures and more than 200 short communications and poster presentations. It is gratifying that young researchers took an active part in the conference. All of them had a unique opportunity to receive lessons from one of the classics of modern applied mathematics S. K. Godunov, his closest colleagues, disciples, and other famous scientists. Detailed information about the conference is available at: http://www.math.nsc. ru/conference/gsk/90/en. The present book contains selected articles written on the materials of the talks presented at the conference. Novosibirsk, Russia Novosibirsk, Russia Trento, Italy Trento, Italy

Gennadii V. Demidenko Evgeniy Romenski Eleuterio Toro Michael Dumbser

Contents

Determination of Discontinuity Surfaces of Transport Equation Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Yu. Balakina Nonlocal Problems of Asymptotic Methods of Perturbation Theory . . . . V. S. Belonosov

1 7

Numerical Solution of the Axisymmetric Dirichlet–Neumann Problem for Laplace’s Equation (Algorithms Without Saturation) . . . . . . . . . . . . V. N. Belykh

13

Contributions to the Mathematical Technology Transfer with Finite Volume Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Bermúdez, S. Busto, L. Cea and M. E. Vázquez-Cendón

21

An Easy Control of the Artificial Numerical Viscosity to Get Discrete Entropy Inequalities When Approximating Hyperbolic Systems of Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Christophe Berthon, Arnaud Duran and Khaled Saleh Development of the Matrix Spectrum Dichotomy Method . . . . . . . . . . . Elina A. Biberdorf

29 37

MHD Model of an Incompressible Polymeric Fluid. Stability of the Poiseuille Type Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. M. Blokhin and D. L. Tkachev

45

Mathematical Models of Plasma Acceleration and Compression in Coaxial Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. V. Brushlinskii and E. V. Stepin

53

Pseudospectrum and Different Quality of Stability Parameters . . . . . . . H. Bulgak

61

ix

x

Contents

Well-Balanced High-Order Methods for Systems of Balance Laws . . . . Manuel J. Castro and Carlos Parés

69

An All-Regime and Well-Balanced Lagrange-Projection Scheme for the Shallow Water Equations on Unstructured Meshes . . . . . . . . . . Christophe Chalons, Samuel Kokh and Maxime Stauffert

77

On the Dynamics of a Free Surface that Limits the Final Mass of a Self-Gravitating Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. P. Chuev

85

Exponential Stability of Solutions to Delay Difference Equations with Periodic Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. V. Demidenko and D. Sh. Baldanov

93

On Estimates of Solutions to One Class of Functional Difference Equations with Periodic Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 G. V. Demidenko and I. I. Matveeva Integral Operators at Settings and Investigations of Tensor Tomography Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Evgeny Derevtsov, Yuriy Volkov and Thomas Schuster Lagrangian Godunov Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Bruno Després On Numerical Methods for Hyperbolic PDE with Curl Involutions . . . . 125 M. Dumbser, S. Chiocchetti and I. Peshkov Modeling Sodium Combustion with Liquid Water . . . . . . . . . . . . . . . . . 135 Damien Furfaro, Richard Saurel, Lucas David and François Beauchamp Geodesic Mesh MHD: A New Paradigm for Computational Astrophysics and Space Physics Applied to Spherical Systems . . . . . . . . 145 Sudip K. Garain, Dinshaw S. Balsara and Vladimir Florinski The Works of the S. K. Godunov School on Hyperbolic Equations . . . . 153 Valeriĭ M. Gordienko On Two-Flow Instability of Dynamical Equilibrium States of Vlasov–Poisson Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Yuriy G. Gubarev Analytical Results for the Riemann Problem for a Weakly Hyperbolic Two-Phase Flow Model of a Dispersed Phase in a Carrier Fluid . . . . . . 169 Maren Hantke, Christoph Matern and Gerald Warnecke A Numerical Method for Two Phase Flows with Phase Transition Including Phase Creation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Maren Hantke and Ferdinand Thein

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An Optimization Method for Feedback Stabilization of Linear Delay Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Guang-Da Hu and Renghong Hu Ten Good Reasons for Using Polyharmonic Spline Reconstruction in Particle Fluid Flow Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Armin Iske Generalized Solutions of Galileo-Invariant Thermodynamically Consistent Conservation Laws, Constructed Using Ideas of Fundamental Works by S. K. Godunov . . . . . . . . . . . . . . . . . . . . . . . 201 M. J. Ivanov Godunov-Type Schemes for Diffusion and Advection-Diffusion . . . . . . . 209 Steven Jöns and Claus-Dieter Munz Direct and Inverse Problems for Conservation Laws . . . . . . . . . . . . . . . 217 S. I. Kabanikhin, D. V. Klychinskiy, I. M. Kulikov, N. S. Novikov and M. A. Shishlenin On a Heat Wave for the Nonlinear Heat Equation: An Existence Theorem and Exact Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 A. L. Kazakov, P. A. Kuznetsov and A. A. Lempert Algorithm for Solving the Inverse Problem of 2-D Diffraction of a Linearly Polarized Elastic Transverse Wave for an N-Layer Medium with Composite Hierarchical Inclusions . . . . . . . . . . . . . . . . . . 229 A. Yu. Khachay and O. A. Hachay Relaxation-Projection Schemes, the Ultimate Approximate Riemann Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Christian Klingenberg New Efficient Preconditioner for Helmholtz Equation . . . . . . . . . . . . . . 243 Dmitriy Klyuchinskiy, Victor Kostin and Evgeny Landa Edge-Based Reconstruction Schemes for Unstructured Meshes and Their Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Tatiana K. Kozubskaya The Collision of Giant Molecular Cloud with Galaxy: Hydrodynamics, Star Formation, Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Igor Kulikov, Igor Chernykh, Dmitry Karavaev, Viktor Protasov, Vladimir Prigarin, Ivan Ulyanichev, Eduard Vorobyov and Alexander Tutukov Model Dirac and Dirac–Hestenes Equations as Covariantly Equipped Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 N. G. Marchuk

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Contents

Developments of the Godunov Method: Explicit–Implicit Scheme, Aeroacoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Igor Menshov A Note on Construction of Continuum Mechanics and Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Michal Pavelka, Ilya Peshkov and Martin Sýkora A Method of Boundary Equations for Unsteady Hyperbolic Problems in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 S. Petropavlovsky, S. Tsynkov and E. Turkel Solution of Deformable Solid Mechanics Dynamical Problems with Use of Mathematical Modeling by Grid-Characteristic Method . . . . . . . 299 Igor B. Petrov Set of Solutions of Ordinary Differential Equations in Stability Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 A. N. Rogalev Two-Phase Computational Model for Small-Amplitude Wave Propagation in a Saturated Porous Medium . . . . . . . . . . . . . . . . . . . . . 313 Evgeniy Romenski, Galina Reshetova, Ilya Peshkov and Michael Dumbser Godunov Symmetric Systems and Rational Extended Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 Tommaso Ruggeri Variational Inequalities in the Dynamics of Elastic-Plastic Media: Thermodynamic Consistency, Mathematical Correctness, Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 Vladimir M. Sadovskii and Oxana V. Sadovskaya Estimates of Solutions to a System of Differential Equations with Two Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 M. A. Skvortsova Moving-Coordinate Method and Its Applications . . . . . . . . . . . . . . . . . . 345 Y. Takakura Numerical Methods for Model Kinetic Equations and Their Application to External High-Speed Flows . . . . . . . . . . . . . . 353 Vladimir A. Titarev The ADER Path to High-Order Godunov Methods . . . . . . . . . . . . . . . . 359 Eleuterio Toro Near-Wall Domain Decomposition for Modelling Turbulent Flows: Opportunities and Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 S. Utyuzhnikov

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Quasilinear Integrodifferential Riccati-Type Equations . . . . . . . . . . . . . 375 V. L. Vaskevich and A. I. Shcherbakov S.K. Godunov and Kinetic Theory in KIAM RAS . . . . . . . . . . . . . . . . . 381 V. V. Vedenyapin, S. Z. Adzhiev, Ya. G. Batischeva, Yu. A. Volkov, V. V. Kazantseva, I. V. Melikhov, Yu. N. Orlov, M. A. Negmatov, N. N. Fimin and V. M. Chechetkin Modeling the Explosive Phenomena Driven by Rapid Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 S. E. Yakush Explicit-Iteration Scheme for Time Integration of the Navier–Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 V. T. Zhukov and O. B. Feodoritova

Determination of Discontinuity Surfaces of Transport Equation Coefficients E. Yu. Balakina

Abstract The transport equation in an inhomogeneous medium is considered. The coefficients of this equation undergo a gap at the boundaries of inhomogeneities. To search for discontinuity surfaces of these coefficients, it is proposed to consider a function that will be unbounded on them.

1 Introduction Consider a nonstationary linear integro-differential equation ∂ f (t, r, ω, E) + ω · ∇r f (t, r, ω, E) + μ(t, r, E) f (t, r, ω, E) = ∂t E  2 = k(t, r, ω · ω , E, E  ) f (t, r, ω , E  )d E  dω + J (t, r, ω, E)

(1)

 E1

where t is a temporary variable, t ∈ T = [0, T ∗ ]; r is a spatial variable, r ∈ G ⊂ R 3 ; G is a convex bounded area; ω ∈  = {ω ∈ R 3 : |ω| = 1}; E ∈ I = [E 1 , E 2 ], E 1 > 0, E 2 < ∞. Function f (t, r, ω, E) is interpreted as particle flux density at time t in point r with energy E flying toward ω. Functions μ, k and J characterize the environment G. Function μ(t, r, E) means the coefficient of full interaction (this values the inverse of the free path and is the sum of the coefficients scattering and absorption), function k(t, r, ω · ω , E, E  ) means scattering indicatrix, and function J (t, r, ω, E) means density of internal sources. Environment G in which the process proceeds is heterogeneous. To characterize this heterogeneity, we introduce a subset G 0 of set G. The set G 0 assumed open in R 3 , dense in G (G 0 = G) and is the union of a finite number of areas: E. Yu. Balakina (B) Sobolev Institute of Mathematics, Acad. Koptyug avenue 4, 630090 Novosibirsk, Russia e-mail: [email protected] Novosibirsk State University, Pirogov street 1, 630090 Novosibirsk, Russia © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_1

1

2

E. Yu. Balakina

G0 =

p 

G i , p < ∞, G i ∩ G j = ∅ when i = j.

i=1

Subsets G i can be interpreted as parts of a heterogeneous medium G filled with ith matter. The paper considers the problem of finding discontinuity surfaces of the coefficients of an Eq. (1). In other words, the question arises of determining the internal structure of environment G. The transport equation has been the subject of study by many researchers. Of particular note is the fundamental work of Vladimirov [1] devoted to the theory of direct problems for the stationary transport equation, and a book by Germogenova [2], where a qualitative theory of solving direct problems is presented. An equation of this form for the nonstationary monoenergetic case was considered in [3, 4] (direct and inverse problems); for the stationary monoenergetic case and the coefficients that undergo a first-order discontinuity, the direct and inverse problems have been solved [5, 6]; in [7, 8] the direct and inverse problems were solved for the stationary polyenergy case and for the coefficients that undergo a discontinuity of the first kind. In this paper, we consider the transport equation, firstly, nonstationary (the coefficients of the equation and the unknown function depend on time), secondly, polyenergy, and thirdly, the coefficients of the transport equation can undergo a discontinuity of the first kind in the spatial variable (in other words, the medium, in which the process proceeds, is heterogeneous).

2 Direct Problem Introduce the function d(r, ω). This is the distance from the point r ∈ G to the border ∂G in the direction ω: d χ (r + νω)dν d(r, ω) = 0

where χ (r ) is characteristic function of area G, d = diam G. Then for arbitrary r ∈ G and ω ∈  points r ± d(r, ω)ω ∈ ∂G. More precisely, if we define the sets ∂− G(ω) = {r ∈ ∂G : n(r ) · ω > 0} and ∂+ G(ω) = {r ∈ ∂G : n(r ) · ω < 0} where n(r ) is the inner normal to the surface ∂G at the point r then r + d(r, ω)ω ∈ ∂+ G(ω), r − d(r, ω)ω ∈ ∂− G(ω). Add to Eq. (1) the initial condition f (0, r, ω, E) = ϕ(r, ω, E), (r, ω, E) ∈ G 0 ×  × I, and the boundary condition

Determination of Discontinuity Surfaces of Transport Equation Coefficients

3

f (t, ξ, ω, E) = h(t, ξ, ω, E), (t, E) ∈ T × I, ξ ∈ ∂− G(ω), ω ∈ . The solution of the direct problem f in the case where the functions μ, k, J , ϕ, and h are known exists, is unique, and representable by a uniformly convergent series. This is shown under the following assumptions regarding functions μ, k, J , h, and ϕ. Let Y be an arbitrary bounded set in R m . To the class D(Y ) we assign real functions φ(y), continuous and bounded on Y, additionally defined at the boundary points y ∈ Y \ Y. The set D(Y ) with the norm φ = sup |φ(y)| y∈Y

forms a Banach space. Suppose that all these functions μ, k, J , ϕ, and h are non-negative, extend by zero by r outside G and also μ(t, r, E) ∈ D(T × G 0 × I ), k(t, r, ω · ω , E, E  ) ∈ D(T × G 0 ×  ×  × I × I ), J (t, r, ω, E) ∈ D(T × G 0 ×  × I ), h(t, r − d(r, −ω) ω, ω, E) ∈ D(T × G ×  × I ), ϕ(r, ω, E) ∈ D(G 0 ×  × I ). The following conditions are also satisfied: h(0, ξ, ω, E) = ϕ(ξ, ω, E) for ξ = r − d(r, −ω)ω, (r, ω, E) ∈ G 0 ×  × I ;    ∂h(t, ξ, ω, E)  +ω · ∇r h(0, r, ω, E)  +μ(0, ξ, E)h(0, ξ, ω, E) =  ∂t t=0 r =ξ  E2 = k(0, ξ, ω · ω , E, E  ) f (0, ξ, ω , E  )d E  dω + J (0, ξ, ω, E)  E1

for ξ = r − d(r, −ω)ω, (r, ω, E) ∈ G 0 ×  × I. In order not to clutter up the text with calculations, we consider the case when the coefficient k is 0, and this corresponds to the motion of particles without collisions. Then the following theorem holds. Theorem 1 The solution to the direct problem exists, is unique, and representable in the form  f (t, r, ω, E) = ex p  × +



t −

μ(ν, r + (ν − t)ω, E)dν

×

α(t,r,ω)

U (t, r, ω, E)+  s t  ex p α(t,r,ω)

α(s,r,ω)

 μ(ν, r + (ν − t)ω, E)dν

J (s, r + (s − t)ω, ω, E)ds (2)

where

4

E. Yu. Balakina

 α(t, r, ω) =  U (t, r, ω, E) =

0 if 0 ≤ t ≤ d(r, −ω), t − d(r, −ω) if d(r, −ω) < t ≤ T ∗ ,

ϕ(r − tω, ω, E) if 0 ≤ t ≤ d(r, −ω), h(t − d(r, −ω), r − d(r, −ω)ω, ω, E) if d(r, −ω) < t ≤ T ∗ .

This theorem is a special case of the theorem proved in [9].

3 Inverse Problem We solve the following inverse problem. Problem. Find ∂G 0 —border the sets G 0 from the equation ∂ f (t, r, ω, E) + ω · ∇r f (t, r, ω, E) + μ(t, r, E) f (t, r, ω, E) = J (t, r, ω, E), ∂t initial condition f (0, r, ω, E) = ϕ(r, ω, E), (r, ω, E) ∈ G 0 ×  × I, and boundary conditions f (t, ξ, ω, E) = h(t, ξ, ω, E), (t, E) ∈ T × I, ξ ∈ ∂− G(ω), ω ∈ , E2 f (t, η, ω, E)d E = H (t, η, ω), t ∈ T, η ∈ ∂+ G(ω), ω ∈ ,

(3)

E1

where only the function H (t, η, ω) is known. It is required to find only the discontinuity surfaces of the coefficients μ and J of the equation, that is, to determine the internal structure of the medium G. For this, we consider a special function  T   H (t, r + d(r, ω)ω, ω)dωdt I nd(r ) = ∇ d

   

(4)



depending on the known data. It is shown that the function I nd takes unlimited values only near the desired surface ∂G 0 . This is achieved by a detailed consideration of the function I nd when we substitute into the integrand the representation of the function H in explicit form from (2). Introduce the following notation:

Determination of Discontinuity Surfaces of Transport Equation Coefficients

 mv(t, r, ω, E) = ex p



d(r,ω) 

−



5

μ(t + ν − d(r, ω), r + νω), E)dν

×

0

−[μ(t + |y − r | − d(r, ω), y, E)]l, j f (t − d(r, ω), r, ω, E)+ +[J (t + |y − r | − d(r, ω), y, ω, E)]l, j , ×

      y −r y −r y −r m t, r, , E = mv t, r, − , E + mv t, r, ,E , |y − r | |y − r | |y − r | 0



T E2  M(r, b) =

m d E 1 z

0

r,

ζ (bs) − r |ζ (bs) − r |

 , E dsd Edt,

(5)

where z is the intersection of the unit sphere in R 3 with center at the point z and the plane tangent to ∂G l at the same point. Suppose that the surface ∂G l,z is represented in the graph of (ζ, φ(ζ )), |ζ | ≤ δ, where the function φ has continuous quotients derivatives up to and including second order, and φ(0) = D1 φ(0) = D2 φ(0) = 0. Then the following estimates are valid: |φ(ζ )| ≤ const |ζ |2 , |Dk φ(ζ )| ≤ const |ζ |, k = 1, 2. Such points z will be called contact points and suppose that they form a set dense in ∂G 0 \∂G. Let us state the main results obtained in this work. Theorem 2 For any contact point z, z ∈ ∂G l ∩ ∂G j , and for r = z + τ n l , 0 < |τ | ≤  ( is a fairly small number), the function I nd(r ), defined in (4), can be represented as I nd(r ) = M(r )| ln(ρ(r, ∂G 0 ))| + O(1). A consequence of this theorem is the following. Theorem 3 The function I nd(r ) is continuous and bounded on every compact set in G 0 . Let z be an arbitrary contact point. Consider the points r = z + τ n(z), 0 < |τ | ≤ , where n(z) is some the unit normal vector to G 0 at the point z,  is a sufficiently small positive number. If the inequality M(r ) ≥ const > 0 holds, then I nd(r ) tends to infinity at τ → 0, i.e., at r → ∂G 0 . Thus, the function I nd(r ) can be unbounded only near the desired surface ∂G 0 . This property serves the basis that the function I nd(r ) is called an indicator heterogeneities of the medium G. Next we state the uniqueness theorem. For this we introduce the following notation. Suppose there are two systems of subdomains {G (i) j }, j = 1, . . . , p, func(i) (i) (i) (i) tions μ , J , h and ϕ , i = 1, 2. Substituting into Eq. (2) μ(i) (t, r, E) instead μ(t, r, E), J (i) (t, r, ω, E) instead J (t, r, ω, E), h (i) (t, r, ω, E) instead h(t, r, ω, E),

6

E. Yu. Balakina

get functions f (i) (t, r, ω, E), i = 1, 2. Substituting these functions in (3), we obtain H (i) (t, r + d(r, ω)ω, ω), i = 1, 2. We also get M(k) (r ), k = 1, 2, using (5). Theorem 4 Let functions H (1) (t, r + d(r, ω)ω, ω) and H (2) (t, r + d(r, ω)ω, ω) coincide for all (t, r, ω) ∈ [d, T ] × G × . Then the contact points z ∈ ∂G (i) 0 , i = 1, 2, for which M(i) (r ) ≥ const > 0 (r = z + τ n(z), 0 < |τ | ≤ ) are common (2) for surfaces ∂G (1) 0 and ∂G 0 . A consequence of this theorem is the following. Theorem 5 If under the conditions of Theorem 4 demand inequalities M(i) (r ) ≥ (2) const > 0 for all contact points z of surfaces ∂G (1) 0 and ∂G 0 , r = z + τ n(z), 0 < (1) (2) |τ | ≤ , then ∂G 0 = ∂G 0 .

References 1. Vladimirov, V.S.: Mathematical problems of the one-speed theory of particle transport (in Russian). Tr. Steklov Math. Inst. USSR (61), 3–158 (1961) 2. Germogenova, T.A.: Local properties of the solution of the transport equation (in Russian). Nauka, Moscow (1986) 3. Prilepko, A.I., Ivankov, A.L.: Inverse problems of determining the coefficient and the right-hand side of the non-stationary multispeed transport equation by redefinition at a point (in Russian). Differ. Equ. 22(5), 109–119 (1985) 4. Prilepko A.I., Ivankov A.L. Inverse problems of determining the coefficient, scattering indicatrix, and the right side of the non-stationary multispeed transport equation (in Russian). Differ. Equ. 22(1), 870–885 (1985) 5. Anikonov, D.S., Kovtanyuk, A.E., Prokhorov, I.V.: Using the Transport Equation in Tomography. Logos, Moscow (2000) 6. Anikonov, D.S., Nazarov, V.G., Prokhorov, I.V.: Integrodifferential indicator for the single-angle tomography problem (in Russian). Sib. J. Ind. Math. 17(2), 3–10 (2014) 7. Balakina, E.Y.: Nonclassical boundary value problem for the transport equation. Differ. Equ. 45(9), 1219–1228 (2009) 8. Anikonov, D.S., Balakina, E.Y.: A polychromatic inhomogeneity indicator in an unknown medium for an x-ray tomography problem. Sib. Math. J. 53(4), 573–590 (2012) 9. Balakina, E.Y.: Existence and uniqueness of a solution for the non-stationary transport equation (in Russian). Sib. J. Ind. Math. 18(4), 3–8 (2015)

Nonlocal Problems of Asymptotic Methods of Perturbation Theory V. S. Belonosov

Abstract For the Mathieu–Hill equations u  (t) + A2 u = εF(t, u), where ε is a small parameter and F(t, u) is a polynomial in the variable u with almost periodic coefficients by t, we propose a modification of the asymptotic averaging method leading to ε-independent autonomous systems.

1 Motivation of the Problem This work is related to the theory of dynamical instability of differential equations and systems depending on a small parameter. The generalized Mathieu–Hill equation u  (t) + A2 u = εF(t, u)

(1)

served as the initial model for the creation of many methods of this theory. Here u(t) is the sought-for function with values in Hilbert space; A is a linear self-adjoint and positive operator; ε is a small scalar parameter. The map F(t, u) is continuous over (t, u) and periodic or almost periodic over t. This class of equation is of great applied importance. In particular, it contains a variety of hyperbolic equations and systems. For ε = 0 the Eq. (1) is stable. The loss of stability for arbitrarily small ε > 0 when the Fourier spectrum of the perturbation F(t, u) contains frequencies close to some critical values is called a parametric resonance. Numerous problems of engineering and physics lead to the study of this phenomenon—stability of structures in construction mechanics, problems of electrical engineering and radio engineering, problems of wave stimulation of oil fields, etc. Therefore, it is important to find V. S. Belonosov (B) Sobolev Institute of Mathematics SB RAS, 4, Acad. Koptyug Avenue, 630090 Novosibirsk, Russia e-mail: [email protected] Novosibirsk State University, 1, Pirogova Str., Novosibirsk 630090, Russia © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_2

7

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V. S. Belonosov

conditions for instability and describe the dynamics of solutions of the perturbed system at resonance. In the case when the map F(t, u) is linear in u, the study of the parametric resonance was started more than 100 years ago by A. M. Lyapunov. Since then, a fairly complete mathematical theory has been developed (see monographs [1, 2]). It is found that if the operator A has a compact resolvent, then for each ε > 0 the set of resonant frequencies consists of intervals that are contracted to critical frequencies at ε → 0. In contrast to the usual resonance, in parametric resonance the amplitudes of oscillations of almost all solutions are increasing exponentially with time. The principal feature of nonlinear equations is that the critical frequencies also depend on the oscillation amplitudes. If the resonance conditions are satisfied at the initial moment, then the oscillation amplitudes begin to increase. When they become large enough, the critical frequencies change, and the necessary resonance conditions are violated. Therefore, the amplitudes decrease and can return to the initial values. Then the whole process is repeated again and again. Thus, instead of unlimited growth, “pulsations” of amplitudes are observed. In the physical literature this phenomenon typical for nonlinear resonance is called phase oscillations. The mathematical theory of phase oscillations has not yet been developed. There is only a description at the physical level of rigor for Hamiltonian systems with a finite number of degrees of freedom [3]. The classical theory of parametric resonance for linear equations is based on the asymptotic averaging method. The application of this method in the nonlinear case requires its preliminary modification due to a number of difficult problems. We explain this by recalling the content of the method for equations in standard form v (t) = εg(t, v; ε) with respect to the function v(t) with values in Hilbert space X with norm  · . The function g is defined on R × X × [0, ε0 ]. Furthermore, for 0 ≤ k ≤ n, there are continuous in variables (t, v) differentials Dvk g(t, v; ε) uniformly bounded on each set R × Sρ × [0, ε0 ], where Sρ = {v : v ≤ ρ}. Averaging makes it possible, with a high degree of accuracy, to decompose the sought solution into smooth slow drift and small fast oscillations. More precisely, it is necessary to construct a change of variables v = n (t, w; ε) ≡ w + εϕ1 (t, w; ε) + ε2 ϕ2 (t, w; ε) + · · · + εn ϕn (t, w; ε),

(2)

such that the functions ϕk (t, w; ε) are bounded as t → ∞ and equation v (t) = εg(t, v; ε) assumes the form w = ε[g0 (w; ε) + εg1 (w; ε) + · · · + εn−1 gn−1 (w; ε) + εn rn (t, w; ε)]. Rejecting the last term εn rn (t, w; ε), we obtain a shortened autonomous equation that approximately describes the slow drift. The mapping w → n (t, w; ε) adds small fast oscillations to it. In the classical works of Krylov, Bogolyubov and Mitropol’skii [4, 5] it is established that such a replacement exists if the function g(t, v; ε) is almost periodic in t, and the set of its Fourier exponents is finite.

Nonlocal Problems of Asymptotic Methods of Perturbation Theory

9

Let v(t) and w(t) be solutions of the original equation and the corresponding shortened equation under the condition v(0) = n (0, w(0); ε). Then for any ρ > 0 and T > 0 there is such δ > 0 that if 0 < ε ≤ δ and w(t) ∈ Sρ at t ∈ [0, T /ε], then the following estimation is performed: v(t) − n (t, w(t); ε) ≤ C(T, ρ)εn .

(3)

The required change of variables is searched in the following way. Substituting the formal decomposition (2) into the original equation and using the Taylor formula, we obtain w (t) = ε

n−1 

 εk [ψk (t, w; ε) − Dt ϕk+1 (t, w; ε)] + εn rn (t, w; ε) .

k=0 j

Here ψ0 ≡ g, while the ψk at k > 0 also depends on Dw g(t, w; ε), ϕ j , Dt ϕ j , Dw ϕ j , where j ≤ k. Therefore, ϕk and gk can be calculated by induction. If ϕ j and g j−1 are already defined for j ≤ k and the corresponding functions ψ j are almost periodic in t, then we assume 1 gk (w; ε) = lim t→∞ t

t ψk (τ, w; ε) dτ,

Dt ϕk+1 = ψk (t, w; ε) − gk (w; ε).

0

Due to the known problem of small denominators, this construction can be interrupted at any step if the Fourier exponents of the function g(t, v; ε) form an infinite set. In [6, 7] we propose an approach to overcome this problem and consider equations with nonperiodic perturbations. However, there is another fundamental question. The phase portrait of the averaged system, generally speaking, depends on ε. At ε → 0, the essential features of this portrait (stationary points, cycles, etc.) can go beyond the area Sρ , where the error estimate (3) of the averaging method was established. Therefore, in the general situation, this method gives only a local description of the features of the solutions.

2 The Idea of Modifying the Averaging Method The problem mentioned at the end of the previous section can be overcome if we find such a change of variables that the approximate equations of slow drift do not depend on ε, but reflect the main features of phase oscillations. In general, this task is quite difficult. But for equations in finite-dimensional spaces with polynomial nonlinearities it can be solved.

10

V. S. Belonosov

To avoid cumbersome technical details, we restrict ourselves to a scalar equation u  (t) + a 2 u = ε f (t, u)

(4)

in the space R of real numbers with perturbation f (t, u) = p(u) + q(t) · u,

p(u) =



ak u k .

2≤k≤m

As q(t), we consider a real almost periodic function with a finite set of Fourier exponents, that is, a trigonometric polynomial q(t) =



(α j sin ω j t + β j cos ω j t).

1≤ j≤l

It is assumed that all frequencies ω j are nonzero, and one of them is equal to the critical resonance value 2a corresponding to the linearized Eq. (4). For definiteness, we assume that 2a = ω1 . Using the classical method of the constants variation u(t) = eiat v1 (t) + e−iat v2 (t), u  (t) = ia[eiat v1 (t) − e−iat v2 (t)], Equation (4) is reduced to a system in the standard form v1 =

ε −iat ε iat e e f (t, eiat v1 + e−iat v2 ), f (t, eiat v1 + e−iat v2 ), v2 = − 2ia 2ia

with respect to the vector v = (v1 , v2 ) in the complex space X = C2 . The solutions we are interested in have the property v2 = v 1 . The problem of nonlocal description of phase oscillations is solved most simply when the polynomial p(u) contains at least one odd degree u 2k+1 with a nonzero coefficient. In this case, it is sufficient to construct the first approximation in the Krylov–Bogolyubov sense. As a result, for the vector w with components w1 , w2 , approximately describing the slow drift of the desired solution v(t), we obtain the following system: ε ε ε ε γ w2 + Pm (w)w1 , w2 (t) = − γ w1 − Pm (w)w2 , 4a 2ia 4a 2ia  2k+1 a2k+1 w1k w2k . γ = α1 + iβ1 , Pm (w) = k

w1 (t) = −

3≤2k+1≤m

It is easy to check that if at the initial moment t = 0 the values of w1 (t) and w2 (t) are complex conjugated, they will remain conjugate for all t > 0. Assuming

Nonlocal Problems of Asymptotic Methods of Perturbation Theory

11

w1 = x + i y, w2 = x − i y, and moving to the “slow” time τ = εt, we obtain the Hamiltonian system x  = −∂ H/∂ y, y  = ∂ H/∂ x, with Hamiltonian H (x, y) =

 2k+1 a2k+1 β1 1 2 2 k+1 α1 x y + (y 2 − x 2 ) − . + y ) (x k 4a 2 k+1 3≤2k+1≤m

The trajectories of the Hamiltonian system are the level lines of the function H . In this case, |H (x, y)| → ∞ at (x 2 + y 2 ) → ∞. Therefore, the resulting system does not have unlimited trajectories. By choosing a limited area on the plane containing the essential features of such a system—stationary points, separatrices, limit cycles, etc., we find an efficient approximate description of the phase oscillations of the source Eq. (4) for all sufficiently small ε. A more complicated situation arises when the polynomial p(u) contains only even powers u 2k , 1 ≤ k ≤ n, am = a2n = 0. In this case, we apply the transformation

 √ v = ε−1/(2m−2) w + ε ϕ1 (t, w; ε) + ε ϕ2 (t, w) to the vector v = (v1 , v2 ). Following the ideas expressed in the implementation of the classical averaging method, we choose ϕ1 , ϕ2 so that the vector function w(t) is a solution to the equation w =

√

εg0 (w) + εg1 (w) + ε1+μ rμ (t, w; ε).

Here μ > 0 and the function rμ (t, w; ε) is uniformly bounded on each set R × Sρ × [0, ε0 ], where Sρ = {(w1 , w2 ) ∈ C2 : |w1 |2 + |w2 |2 < ρ 2 }. The conditions presented to the functions p(u) and q(t) allow the   to1 perform , and after specified transformation. It turned out that g0 (w) = 0, μ = min 21 , 2n−1 removing the small term ε1+μ rμ (t, w; ε), we get the shortened system ε in γ w2 + ε Q n w12n w22n−1 , 4a a ε in w2 (t) = − γ w1 − ε Q n w12n−1 w22n . 4a a

w1 (t) = −

The coefficient γ is defined above, and

2 2n−1  2n 2n−1 1 a2n Qn = 2 . k a k=0 k 4(k − n)2 − 1

12

V. S. Belonosov

Assuming w1 = x + i y, w2 = x − i y and τ = εt, we arrive at a Hamiltonian system with respect to x, y with Hamiltonian H (x, y) =

β1 1 α1 x y + (y 2 − x 2 ) + Q n (x 2 + y 2 )2n . 4a 2

The phase portrait of this system again consists of limited trajectories.

References 1. Fomin, V.N.: Mathematical Theory of Parametric Resonance in Linear Distributed Systems. Leningrad University Publishing House, Leningrad (1972) 2. Yakubovich, V.A., Starzhinskii, V.M.: Parametric Resonance in Linear Systems. Nauka, Moscow (1987) 3. Sagdeev, R.Z., Zaslavsky, G.M., Usikov, D.A.: Non-linear Physics: From the Pendulum to Turbulence and Chaos. Harwood Academic Publishers, New York (1988) 4. Bogolyubov, N.N., Mitropol’skii, Y.A.: Asymptotic methods in the theory of non-linear oscillations. Hindustan Publishing Corp., Delhi; Gordon and Breach Science Publishers, New York (1961) 5. Mitropol’skii, Yu.A.: Method of Averaging in Nonlinear Mechanics. Naukova Dumka, Kiev (1971) 6. Belonosov, V.S.: The spectral properties of distributions and asymptotic methods in perturbation theory. Sb. Math. 203, 307–325 (2012) 7. Belonosov, V.S.: Asymptotic analysis of the parametric instability of nonlinear hyperbolic equations. Sb. Math. 208, 1088–1112 (2017)

Numerical Solution of the Axisymmetric Dirichlet–Neumann Problem for Laplace’s Equation (Algorithms Without Saturation) V. N. Belykh

Abstract We construct a fundamentally new, unsaturated numerical algorithm for solving the Dirichlet–Neumann problem for Laplace’s equation in smooth axisymmetric domains of a rather general shape. The distinctive feature of this algorithm is the absence of the leading error term, which, as a result, enables us to automatically adjust to arbitrary extra (extraordinary) supplies of smoothness of the sought solutions. In the case of C ∞ -smoothness, the solutions are constructed with exponential estimate for error.

1 Introduction In the modern computational practice, new numerical methods are typically created and developed with the goal to solve some particular problem. In the area of our interest, these are nonstationary axisymmetric problems with free boundaries (see [1]), in which the boundary elliptic Dirichlet–Neumann problem for Laplace’s equation arises as important auxiliary on the time step. The need of new computational algorithms for the elliptic problems is due to the fact that in the majority of the numerical methods applied for solving these problems the specificity of the elliptic operators is not taken into account in due measure. Thereby the following rather essential fact is ignored: the solutions belong to the relatively simply determined functional compact sets X (the bigger the smoothness supply of the elements of the compact set X [2] the faster the decay rate of the Alexandroff m-widths αm (X ) with the growth of the parameter m). Neglecting the extra information about the solutions to the elliptic problems also narrows down possibilities of the numerical methods. Consequently, qualitative discretization of such V. N. Belykh (B) Sobolev Institute of Mathematics, Acad. Koptyug avenue 4, 630090 Novosibirsk, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_3

13

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V. N. Belykh

problems is distinguished by ability to inherit Schauder smoothness estimates, their characteristic property. While earlier this property was hardly demanded in computational practice, it became important now due to the pursuit of the numerical solution of guaranteed quality. Moreover, the ideal would be the situation when to each existence theorem for a solution, (in the form of Schauder smoothness estimates) there corresponds a theorem on (estimate for) the convergence of approximate solutions in the norms corresponding to the smoothness of the sought solution. Babenko was the first to propose a conceptual base for resolving the problem of efficiency of numerical computations, noting that the methods with the leading error term strongly predetermine the quality of the obtained approximations (see [2]). This defect substantially reduces the value of numerical methods with the main error term and explains why the methods of higher orders, in which the indicated defect, though not disappearing completely, is diminished, are used more often. Therefore, the computational practice is in immediate need of numerical methods lacking the indicated defect, i.e., the presence of the leading error term, which were called unsaturated by Babenko (see [2]). The main property of unsaturated numerical methods (and algorithms) is the fact that, with the growth of smoothness supply of the solutions, these methods become self-enhancing, getting the efficiency in the extraordinary smoothness of the sought solutions, the so-called (unsaturation phenomenon [3]). The unsaturated numerical methods reach their peak efficiency, exponential convergence, on the spaces of C ∞ smooth solutions. This article deals with construction of an unsaturated numerical method for solving the boundary Dirichlet–Neumann problem for Laplace’s equation in C ∞ -smooth axisymmetric domains of a rather general shape, the results from [4–9] being the mathematical foundation for this construction. Because of the restrictions on the volume of this article, we were not able to provide detailed proofs. Therefore, some moments will remain undiscussed, although all crucial steps of the method will be deliberated, with different degree of detalization.

2 Statement of the Problem Let x, ξ ∈ R3 , x = (x, y, z), ξ = (ξ, η, ζ ), and consider a connected domain ω ⊂ R3 symmetric about thez-axis and bounded by a C ∞ -smooth closed surface of revolution ∂ω. Then r = x 2 + y 2 and z are invariant under the rotations of ω about the z-axis. A meridional section of ∂ω is a parametrized curve γ : [0, π ] → {r (s), z(s)}, with r ≥ 0, dz/ds ≥ 0, and γ (s) ∈ C ∞ [0, π ]. The points γ (0) and γ (π ) are the poles of ∂ω, while the functions r (s) and z(s) have (odd and even respectively) C ∞ -smooth 2π -periodic extensions from [0, π ] onto [0, 2π ]. Define the locations of the points x = (r, z) and ξ = (ρ, ζ ) on γ by coordinates s and σ , respectively: r = r (s), z = z(s); ρ = r (σ ), ζ = z(σ ), where  0 ≤ s, σ ≤ π . Introduce the following notations: ρ  ≡ dρ/dσ , ζ  ≡ dζ /d σ , δ = ρ  2 + ζ  2 ; r  ≡ dr/ds,

Numerical Solution of the Axisymmetric Dirichlet–Neumann Problem …

15

 z  ≡ dz/ds, d = r  2 + z  2 ; n ≡ δ −1 [−ζ  , ρ  ], e ≡ δ −1 [ρ  , ζ  ], N ≡ d −1 [−z  , r  ], and E ≡ d −1 [r  , z  ]. Let F(x) be a function continuous on ω. Denote the direct value of F(x) on ∂ω (if it exists) by F(x) = F(x)| x∈∂ω . Let ψ(s) ≡ ψ(r (s), z(s)) be a sufficiently smooth 2π -periodic even function. The Dirichlet–Neumann operator ψ → ϒ ψ is defined by the equality:  ϒ(∂ω)ψ =

∂ϕ ∂ϕ , ∂E ∂N

 | ∂ω ≡ (ϕ E , ϕ N ),

where E and N are the tangent and normal vectors to ∂ω; ϕ = ϕ(x) is a harmonic on ω function equal to ψ on the boundary ∂ω. The operator ϒ acts as follows: given ∂ω and ϕ| ∂ω ≡ ψ, by solving the Dirichlet problem for Laplace’s equation on ω, a regular harmonic function ϕ(x) is recovered and then ∇ϕ|∂ω ≡ ∇ϕ = (ϕ E , ϕ N ) is calculated, where ϕ E and ϕ N are the traces of the tangent and normal components of the gradient vector ϕ(x). The solution to the Dirichlet problem on the smooth axisymmetric domain ω with ϕ| ∂ω = ψ(x) is sought in the form of a double layer potential ϕ(x) = W [χ ](x) with a sufficiently smooth density χ (ξ ) invariant under the group of rotations of ∂ω. This enables us to construct the image of ϒ using the solution ϕ(x) and the Lyapunov theorems [10] on differentiability of the double layer potential W [χ ](x). The integral equation for the interior Dirichlet problem has the following form: χ (s) + W [χ ](s) = ψ(s)/2π, 0 ≤ s ≤ π,

(1)

where W [χ ](s) = W [χ ](s)|s∈γ is the direct value of W [χ ](s) on γ . The invariance of the density χ (ξ ) and the surface ∂ω allows us to integrate once in the double integral W [χ ](x) and reduce the problem to calculation of the expression [8] 1 W [χ ](s) = π

π  0

 H·n ζ 2ρ E(q) − D(q) δ h −1 ∗ χ (σ ) dσ σ −s δ (2)

π k(s, σ ) χ (σ ) dσ.

≡ 0

Using the solution χ (s) to (1), calculate the derivatives ϕ E (s) and ϕ N (s): ϕ E (s) ≡

dϕ dϕ = ν(s) + E W [χ ](s), ϕ N (s) ≡ = N W [χ ](s). dE dN

16

V. N. Belykh

Here  π 1 H·N E W [χ ](s) = − ν(σ ) 2ρ E(q) π σ −s 0    z ζ −z r + D(q) δ h −1 − ∗ d σ, r d d  π  ν(σ ) E − ν(s) e 1 ζ − z z 2ρ H · N W [χ ](s) = − E(q) + − ν(σ ) π σ −s r d 0   ρ ρ r D(q) + 2ν(s) K (q) δ h −1 + ν(σ ) − ν(s) ∗ d σ, d δ δ

(3)

d χ (σ ) 1 dχ (σ ) d χ (s) 1 dχ (s) = , ν(s) ≡ = , de δ dσ dE d ds   r(σ, s) ρ −r ζ −z , ; H ≡ H(σ, s) = , r(σ, s) = | r(σ, s)| 2 σ −s σ −s ν(σ ) ≡

D(q) = K (q) − E(q); K (q) and E(q) are the complete elliptic integrals of the first and second kind with modulus q ∈ [0, 1]: π/2 K (q) = (1 − q sin2 θ )−1/2 dθ, 0

π/2 E(q) = (1 − q sin2 θ )1/2 dθ, q = 4 ρ r h −2 ∗ , 0

and h 2∗ ≡ h 2∗ (σ, s) = (ρ + r )2 + (ζ − z)2 . Algorithms for computing of K (q), E(q), and D(q) are given in [9].

3 Discretization of Eq. (1) and the Statement of the Result Discretize problem (1). Denote by C ≡ C[ 0, 2π ] the space of continuous real 2π -periodic functions equipped with the Chebyshev norm · and by C+ ≡ C+ [ 0, π ] ⊂ C, the set of even functions. Denote by T m ⊂ C+ the subspace of trigonometric polynomials of degree at most m for an integer m ≥ 0, by Q m : C+ → T m the projection onto it, and by em (g) = inf m g − Tm the best (Chebyshev) approximation of g ∈ C+ . Tm ∈T

Numerical Solution of the Axisymmetric Dirichlet–Neumann Problem …

17

The operator W : C+ → C+ is compact [4] with the norm in C+ calculated by the formula π W ≡ max | k(s, σ )| dσ. 0≤s≤π 0

The discretization of (1) will be constructed according to the following scheme. Take the nodes s i = 2πi/(2m + 1), 0 ≤ i ≤ m, and the linear mapping J : C+ → R m+1 ,

J g = (g(s0 ), . . . , g(sm )) ≡ (g0 , . . . , gm ), |J g| ∞ = max | g i |. 0≤i≤m

Consider the interpolation Lagrange polynomial of g ∈ C+ : (Q m g)(s) ≡ Q m (s ; J g) =

m

g(sk ) wk (s), Q m = max

0≤s≤π

k=0

m

|wk (s)|,

(4)

k=0

where  2 Dm (s)/(2m + 1), wk (s) = 2(Dm (s − sk ) + Dm (s + sk ))/(2m + 1), while Dm (s) = 1/2 +

m

k = 0, k = 1, 2, . . . , m,

cos ks is the Dirichlet kernel, and w k (s j ) = δk j for 0 ≤

k=1

k, j ≤ m. The polynomial Q m (s ; J g) determines the projection Q m : C+ → T m , and Q m is its norm. It is clear that Q m wk ≡ wk and J Q m g ≡ J g. By Lebesgue’s inequality [2], we obtain the estimate for the error g(s) − Q m (s; J g) ≤ (1 + Q m ) em (g), Q m ≤ 3 + 2π −2 ln m. The polynomial Q m (s; J χ ) (see (4)) can also be used for approximation of the derivatives g ( j) (s), and the following theorem helps to estimate the error (see [11]). Theorem 1 If g ∈ C+k and 0 < j ≤ k, then g ( j) (s) − Q (mj) (s; J g) ≤ α m j e m (g ( j) ), where α m j =

π (1 2

+ Q m ) + (1 + π2 )[4 + π e +

4 π2

ln(k + 1)].

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By the linearity of W : C+ → C+ , we have W [Q m g](s) =

m

g(sk ) ak (s), ak (s) = W [wk ](s),

k=0

W Q m = max

0≤s≤π

m

|ak (s)|.

k=0

Let u = J χ , ρm (s) = χ (s) − Q m (s; J χ ),  = −J W [ρm ], and F = (1/2π )J ψ. Applying J to both sides of (1), we obtain u + Au = F + ,  = −J W [χ − Q m χ ], u,  ∈ R m+1 ,

(5)

where A = (ai k ) is the matrix with entries ai k = W [wk ](si ) for 0 ≤ k ≤ m and 0 ≤ i ≤ m. The matrix A determines a linear operator A : R m+1 → R m+1 , which is regarded as the discretization of W : C+ → C+ . The matrix A is completely filled, in contrast to the very sparse matrices used in the finite difference methods. Neglecting the error  ∈ Rm+1 in (5) and denoting the approximate value of J χ ∈ Rm+1 by χ ∈ Rm+1 , we obtain the discretization of (1) (I + A) χ = F

(6)

with the identity (m + 1) × (m + 1) matrix I . Denote by |B| ∞ the Chebyshev norm of an invertible matrix B : Rm+1 → Rm+1 ; the condition number of B is defined as κ(B) = |B| ∞ |B −1 | ∞ . If (1) is uniquely solvable in C+ and χ ≤ Q ψ , then the following nontrivial results [4, 5] are valid: Theorem 2 There exist constants m 0 , q0 , κ0 , and c0 such that for m ≥ m 0 the matrix (I + A)−1 exists and W Q m ≤ q0 , κ(I + A) ≤ κ0 ; em (χ ) ≤ | J χ − χ| ∞ ≤ c0 ( 1 + Q m ) em (χ ); em (χ ) ≤ χ (s) − Q m (s; J χ ) ≤ c0 Q m ( 1 + Q m ) em (χ ).

(7)

Here the constants q0 , κ0 , c0 are independent of m and can be calculated efficiently. Theorem 3 The sequence of functions χ

k+1

= (1 − β) χ

k

− β Aχ

k

−1 + β F, β = 1 + W Q m , k = 0, 1, . . . ,

Numerical Solution of the Axisymmetric Dirichlet–Neumann Problem …

19

obtained by solving (6) by an iterative method converges to the solution χ of (6) as a geometric progression with the common ratio less than 1. Theorem 4 If |(I + A)χ k − F| ∞ ε ≤ , | F| ∞ κ(I + A) then

|χ

− χ|∞ ≤ ε. | χ| ∞ k

By (7), the constructed method for numerical solution of the integral equation (1) is unsaturated [5]. This means that with the growth of the smoothness supply of χ (other things being equal) the decay rate of the error increases with the growth of the parameter m and, in contrast to the methods with the leading error term, reaches its peak efficiency (exponential convergence) on the spaces of C+∞ -smooth solutions of (1). For example, if χ ∈ C+∞ and χ ( k) ≤ A k k αk (A > 1, α ≥ 1), √ α then e (χ ) ≤ c e−β m with positive constants c and β [5]. As a result, the C ∞ m

+

smoothness of the solution χ to problem (1) is particularly important in practice, which is a fundamental difference between the unsaturated numerical methods and numerical methods with main error terms, i.e., saturated methods. The analytic expression (2) for the kernel of the integral operator W demonstrates that not all quadrature formulas for approximation of aik = W [ wk ](si ) are of any practical interest. On the one hand, the kernel has a “moving” logarithmic singularity on the diagonal σ = s due to the property of modulus q(σ, s) of the elliptic integrals E(q) and D(q). On the other hand, at points s near the axis of symmetry z, which is the zone of intensive growth of h −1 ∗ (σ, s), arise the phenomena of the boundary layer [8], while at the poles γ (0) and γ (π ) representation (2) has no singularities. Thus, the advantages of the constructed unsaturated numerical method for integral equation (1) are present only under the condition that the accuracy of the approximation of (2) has the same order as || ∞ = |J W [χ − Q m χ ]| ∞ for m ≥ m 0 . It is possible to satisfy this requirement by using unsaturated quadrature formulas. Such formulas, which take into account the specificity of the axisymmetric problem (1), namely, its boundary are constructed in [7, 8]. 

layers, The matrix A = ai j is calculated by these formulas with the predetermined accuracy (1 + 2 Q m ) em (χ ), since the numerical characteristics em (χ ) of the exact solution χ to Eq. (1) vanish with the exponential growth of the parameter m if χ ∈ C+∞ . This circumstance is a fundamental difference between the unsaturated discretization of problem (1) and its discretizations with main error terms, i.e., saturated methods. The tangent ϕ E (s) and normal ϕ N (s) components of ϒ are approximated similarly by Theorem 1 and formulas (3).

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References 1. Belykh, V.N.: On the evolution of a finite volume of ideal incompressible fluid with a free surface. Dokl. Phys. 62(4), 213–217 (2017). https://doi.org/10.7868/S0869565217120052 2. Babenko, K.I.: Basics of Numerical Analysis. RKhD, Izhevsk (2002) [in Russian] 3. Belykh, V.N.: Properties of the best approximations of C ∞ -smooth functions on an interval of the real axis (to the phenomenon of unsaturated numerical methods). Sib. Mat. J. 46(3), 483–499 (2005) 4. Belykh, V.N.: Exterior axisymmetric Neumann problem for the Laplace equation: numerical algorithms without saturation. Dokl. Math. 76(3), 882–885 (2007). https://doi.org/10.1134/ S1064562407060208 5. Belykh, V.N.: An unsaturated numerical method for the exterior axisymmetric Neumann problem for Laplace’s equation. Sib. Math. J. 52(6), 980–994 (2011). https://doi.org/10.1134/ S0037446611060036 6. Belykh, V. N.: Particular features of implementation of an unsaturated numerical method for the exterior axisymmetric Neumann problem. Sib. Math. J. 54(6), 984–993 (2013). https://doi. org/10.1134/S0037446613060037 7. Belykh, V.N.: On the problem of construction of unsaturated quadratures on intervals. Sb. Math. 210(1), 24–58 (2019). https://doi.org/10.4213/sm8984 8. Belykh, V.N.: The problem of numerical realization of integral operators of axisymmetric boundary value problems (algorithms without saturation). Ufim. Math. Zh. 4(4), 22–37 (2012) 9. Belykh, V.N.: Algorithms for computing complete elliptic integrals and some related functions. J. Appl. Ind. Math. 6(4), 410–420 (2012). https://doi.org/10.1134/S1990478912040023 10. Gunter, N.M.: Potential Theory and its Applications to Basic Problems of Mathematical Physics. Gostekhteoretizdat, Moscow (1953); Unfar, New York (1967) 11. Belykh, V.N.: On the problem of numerical solution of Dirichlet problem by harmonic potential of single layer (algorithms without saturation). Russ. Acad. Sci. Dokl. Math. 47(2), 252–256 (1993)

Contributions to the Mathematical Technology Transfer with Finite Volume Methods A. Bermúdez, S. Busto, L. Cea and M. E. Vázquez-Cendón

Abstract At the early 1980s, the research group in Mathematical Engineering, mat+i, started working on finite volume methods for the simulation of environmental issues concerning Galician rias (Spain). The focus was on the study of hyperbolic balance laws due to the presence of source terms related to the bathymetry. A correct treatment of these terms, an upwind discretization, was presented in [2, 7, 22]. The transfer of this knowledge has motivated the registration of the software Iber (http://www.iberaula.es). Latterly, under different research problems, we have been working on the development of a numerical algorithm for the resolution of Euler and Navier–Stokes equations. A hybrid projection finite volume/finite element method is employed making use of unstructured staggered grids (see [3, 9]). To attain second order of accuracy ADER methodology is employed [10]. On the other hand, with numerical simulation of gas transportation networks in view, a first-order well balanced finite volume scheme for the solution of a model, for the flow of a multicomponent gas in a pipe on non-flat topography, is introduced. The mathematical model consists of Euler equations, with source terms, coupled with the mass conservation equations of species. We propose a segregated scheme in which Euler and species equations are solved separately [6]. A. Bermúdez · M. E. Vázquez-Cendón (B) University of Santiago de Compostela (USC) and Technological Institute for Industrial Mathematics (ITMATI), Santiago de Compostela, Spain e-mail: [email protected] A. Bermúdez e-mail: [email protected] S. Busto University of Trento, Trento, Italy e-mail: [email protected] L. Cea University of A Coruña, A Coruña, Spain e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_4

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1 Environmental Shallow Water Flows At the early 1980s, the research group in Mathematical Engineering, mat+i (http:// www.usc.es/ingmat/?lang=en, USC), started working on finite volume methods for the simulation of environmental issues concerning Galician rias (Spain). The focus was on the study of hyperbolic balance laws due to the presence of source terms related to the bathymetry of Galician coast. A correct treatment of these terms, an upwind discretization, was initially presented in [7] and extended to bidimensional problems in [2, 22], see Fig. 1 (left). The relevance of this contribution was recognized by the scientific community, namely, the application of the ‘Conservation property’ to check the correct discretization of the balance law by means of steady solutions. Furthermore, a wetting–drying condition for unsteady shallow water flow in two dimensions leading to zero numerical error in mass conservation is presented in [11] and it has been also validated with real data in the Crouch estuary (Essex, England) in [13], see Fig. 1 (center). The correct treatment of the boundary conditions has led us to introduce in [2] a new type of finite volumes for a dual mesh, the centres of which are the midpoints of the edges of the triangles; this is why we call them finite volumes of edge-type. This type of finite volumes is extended in [3] and detailed below in this paper for tridimensional problems. The relation between the numerical scheme used to solve the hydrodynamic equations and the scheme used to solve a scalar transport model linked to the shallow water equations is also analysed in [15]. Turbulence modelling in shallow water flows was not treated so profusely as in other fluid dynamics areas. Some contributions are detailed in [14]. The transfer of this knowledge was also proved with the joint registration of TURBILLON in 2005, by the Environmental and Water Engineering Group (GEAMA, University of A Coruña) and mat+i research group of the USC. In a second step TURBILLON has evolved into the software Iber in the framework of the project Iberaula (http://www.iberaula.es) which also involves the FLUMEN group from the Polytechnic University of Catalonia and the CEDEX, see Fig. 1 (right). Moreover, it is also the registered the code MODUS (Sustainable urban drainage model, C-196-218) in collaboration with GEAMA research group.

Fig. 1 Velocity field and tidal wave propagation at the Pontevedra ria [2] (left), Crouch estuary [13] (center) and Iber software [8] (right)

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23

2 Low Mach Number Flows More recently, under project SIMULOX and under different research projects, we have been developing, within our research group, a numerical 3D code with a projection hybrid finite volume/element method for low Mach number flows of viscous or inviscid fluids [3]. We have added a turbulence module to this code. To attain second order of accuracy ADER methodology is employed (see, [9, 10]). The second part of this project is devoted to these methods and the connection with the previous ones. The transfer of this knowledge has been done with the recent registration of the code VolFEM3D (SC-62-2018). Finite volume methods combined with approximate Riemann solvers have been successfully developed for compressible flows in the 1980s (see, [21] and the references therein). The difference compressible/incompressible deeply changes the physics and also the numerics. While for compressible flows pressure is directly related to other flow variables as density and energy via a state equation, for incompressible flows pressure is a Lagrange multiplier that adapts itself to ensure that the velocity satisfies the incompressibility condition. In order to handle the latter situation, the typical explicit stage of finite volume methods has to be complemented with the so-called projection stage, where a pressure correction is computed in order to get a divergence-free velocity. There are many papers devoted to the analysis of projection finite volume methods for incompressible Navier–Stokes equations (see, for instance, [1]). To get stability, staggered grids have to be used to discretize velocity and pressure. While this can be done straightforwardly in the context of structured meshes, the adaptation to unstructured meshes is more tricky (see [2, 16]). In this section, we summarize without details a projection hybrid finite volume/element method for low Mach number flows of viscous or inviscid fluids. Starting with a 3D tetrahedral finite element mesh, the equation of the transport-diffusion stage is discretized by a finite volume method associated with a dual mesh where the nodes of the volumes are the barycenters of the faces of the initial tetrahedra, an extension of the dual mesh introduced in [2] for the shallow water flows. The transport-diffusion stage is explicit. Upwinding of convective terms is done by classical Riemann solvers as the Q-scheme of van Leer or the Rusanov scheme. Concerning the projection stage, the pressure correction is computed by a piecewise linear finite element method associated with the initial tetrahedral mesh. Passing the information from one stage to the other is carefully made in order to get a stable global scheme. We refer the reader to [3]. Moreover, the projection hybrid FV/FE method mentioned is extended to account for species transport equations (see Fig. 2). Furthermore, turbulent regime is also considered thanks to the k − ε model. Regarding the transport-diffusion stage new schemes of high order of accuracy are developed. The CVC Kolgan-type scheme (see [23]), defined in [22], and ADER methodology are extended to 3D. The latter is modified in order to profit from the dual mesh employed by the projection algorithm and the derivatives involved in the diffusion term are discretized using a Galerkin approach. The accuracy and stability analysis of the new method are carried out for the advection-diffusion-

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Fig. 2 Profile of the exact Gaussian sphere (blue) and the computed solutions using LADER (green) and CVC-G (red) at plane y = 0 (left), elevated surfaces using LADER (right) [10]

reaction equation. Within the projection stage, the pressure correction is computed by a piecewise linear finite element method. Numerical results are presented, aimed at verifying the formal order of accuracy of the scheme and the performance of the method on several realistic test problems, like the Gaussian sphere in Fig. 2. For further examples see [10].

3 Gas Flow in Transport Networks In the last years, several papers have been devoted to transient models for the case where the gas flowing in the network is homogeneous. However, in real networks neither temperature nor entropy remains constant because, first, there is heat exchange with the environment and second, there is viscous dissipation in the boundary layer near the wall of the pipelines. These features complicate the model since they make necessary to use the full Euler system of equations and to include two respective source terms. Moreover, in real networks it is quite common that the height of the pipeline with respect to a reference level changes according to the topography. This fact leads to another source term in the Euler equations similar, in some sense, to the one corresponding to bathymetry in variable-bottom elevation shallow water flow models, mentioned in Sect.1. For Euler equations with gravity the contributions are more recent, being the work by Cargo and LeRoux [12] considered a pioneering paper. The case of gas flow in one single pipeline on non-flat topography, also including friction and heat exchange source terms, has been considered in [4]. All the source terms are upwinded so as to ensure that the full scheme is well balanced. Moreover, passing from a pipeline to a network is not an easy task, in [5], a method for numerically handling pipe junctions has been introduced based on modelling them as 2D containers. Initially, we consider an homogeneous gas, as done in [4, 5] and references therein. However, in real-gas networks, several gases from different origins are injected and subsequently mixed at network junctions (see [18, 20]). This makes the modelling problem more difficult because the gas composition at any network point and at any time must be computed. More important, as the equation of state involve the gas composition, in principle Euler equations are coupled with the mass conservation

Contributions to the Mathematical Technology Transfer with Finite Volume Methods

25

Fig. 3 Mass flow at node 01A (left) and pressure at node I-013 (right). Blue: real measurement. Red: computed with a homogeneous gas composition model. Green: computed with a variable gas composition model

equations for species. Nevertheless, we propose a segregated scheme in which the Euler and the species equations are solved separately (see [6]). By doing so we have to face two new difficulties. Firstly, we have to solve an Euler equation system involving a flux vector that depends not only on the conservative variables but also on the time and space variables through the gas composition. Secondly, when solving the mass conservation equations of species to determine the gas composition, we need to preserve positivity of concentrations. Regarding the second one, in order to preserve positivity of species concentrations we follow [15, 17, 19]. The flux in the mass conservation equations is approximated in accordance with the upwind discretization of the flux we have made in the total mass conservation equation [6]. The ultimate goal of the methodology proposed in this section is the prediction of the physical variables involved in real-gas transportation networks. In order to check if this is made accurately, we present a test involving real data. In [5] our industrial partner Reganosa allowed us to use a large data basis with the measurements made in real time on its own gas network. With this information, we are able to compare the numerical results of our model against their respective experimental values. The network, consists of 11 nodes, joined by 10 pipes. Node 01A represents the Reganosa regasification plant, where we impose a pressure boundary condition taken from the data basis, see Fig. 3 (left). This is the only gas inlet into the whole network: the rest of the nodes are outlets. The main gas outlet is located at node I-013 which is a terminal node of the network where an outflow boundary condition is considered. In this experiment, the consumptions of the rest of the nodes are very small in comparison with this one. Let us suppose that the composition of the inlet gas changes along the time. In this case, we compare the evolution along the time of computed pressure and methane mass fraction with their respective measurements, at nodes 01A and I-013, see Fig. 3. The transfer of this knowledge is a proprietary tool GANESO developed by Reganosa in collaboration with ITMATI (http://www.reganosa.com/en/ganeso). Acknowledgements The authors are indebted to E.F. Toro, from the Laboratory of Applied Mathematics, University of Trento, for the useful discussions on the subject. This project was partially supported by Spanish MECD, grant FPU13/00279; by Xunta de Galicia, grant PRE/2013/031; by

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Spanish MICINN projects MTM2013-43745-R and MTM2017-86459-R; by Xunta de Galicia and FEDER under project GRC2013-014 and by Fundación Barrié; by the Reganosa company.

References 1. Bell, J.B., Day, M.S., Almgren, A.S., Lijewski, M.J., Rendleman, C.A.: A parallel adaptive projection method for low Mach number flows. Int. J. Numer. Methods Fluids 40, 209–216 (2002) 2. Bermúdez, A., Dervieux, A., Desideri, J.A., Vázquez-Cendón, M.E.: Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes. Comput. Methods Appl. Mech. Eng. 155, 49–72 (1998) 3. Bermúdez, A., Ferrín, J.L., Saavedra, L., Vázquez-Cendón, M.E.: A projection hybrid finite volume/element method for low-Mach number flows. J. Comput. Phys. 271, 360–378 (2014) 4. Bermúdez, A., López, X., Vázquez-Cendón, M.E.: Numerical solution of non-isothermal nonadiabatic flow of real gases in pipelines. J. Comput. Phys. 323, 126–148 (2016) 5. Bermúdez, A., López, X., Vázquez-Cendón, M.E.: Treating network junctions in finite volume solution of transient gas flow models. J. Comput. Phys. 344, 187–209 (2017) 6. Bermúdez, A., López, X., Vázquez-Cendón, M.E.: Finite volume methods for multi-component Euler equations with source terms. Comput. Fluids 156, 113–134 (2017) 7. Bermúdez, A., Vázquez-Cendón, M.E.: Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23, 1049–1071 (1994) 8. Bladé, E., Cea, L., Corestein, G., Escolano, E., Puertas, J., Vázquez-Cendón, M.E., Dolz, J., Coll, A.: Iber: herramienta de simulación numérica del flujo en ríos. Rev. Int. Metod. Numer. 20(1), 1–10 (2014) 9. Busto, S., Toro, E.F., Vázquez-Cendón, M.E.: Design and analysis of ADER-type schemes for model advection-diffusion-reaction equations. J. Comput. Phys. 327, 553–575 (2016) 10. Busto, S., Ferrín, J.L., Toro, E.F., Vázquez-Cendón, M.E.: A projection hybrid high order finite volume/finite element method for incompressible turbulent flows. J. Comput. Phys. 353, 169–192 (2018) 11. Brufau, P., Garcıa-Navarro, P., Vázquez-Cendón, M.E.: Zero mass error using unsteadywettingdrying conditions in shallow flows over dry irregulartopography. Int. J. Numer. Methods Fluids 45(10), 1047–1082 (2004) 12. Cargo, P., LeRoux, A.Y.: A well balanced scheme for a model of atmosphere with gravity. C. R. Acad. Sci., Ser. 1 Math. 318, 73–76 (1994) 13. Cea, L., French, J.R., Vázquez-Cendón, M.E.: Numerical modelling of tidal flows in complex estuaries including turbulence: an unstructured finite volume solver and experimental validation. Int. J. Numer. Methods Eng. 67(13), 1909–1932 (2006) 14. Cea, L., Puertas, J., Vázquez-Cendón, M.E.: Depth averaged modelling of turbulent shallow water flow with wet-dry fronts. Arch. Comput. Method Eng. 14(3), 303–341 (2007) 15. Cea, L., Vázquez-Cendón, M.E.: Unstructured finite volume discretisation of bed friction and convective flux in solute transport models linked to the shallow water equations. J. Comput. Phys. 231(8), 3317–3339 (2012) 16. Gau, W., Li, H., Liu, R.: An unstructured finite volume projection method for pulsatile flows through an asymmetric stenosis. J. Eng. Math. 72, 125–140 (2012) 17. García-Navarro, P., Vázquez-Cendón, M.E.: On numerical treatment of the source terms in the shallow water equations. Comput. Fluids 29(8), 951–979 (2000) 18. Kosch, T., Hiller, B., Pfetsch, M.E., Schewe, L. (eds.): Evaluating Gas Network Capacities. SIAM, Philadelphia (2015) 19. Larrouturou, B.: How to preserve the mass fractions positivity when computing compressible multi-component flows. J. Comput. Phys. 95(1), 59–84 (1991)

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20. Osiadacz, A.J.: Simulation and Analysis of Gas Networks. Gulf Publishing Company, Houston (1987) 21. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer, New York (2009) 22. Vázquez-Cendón, M.E., Cea, L.: Analysis of a new Kolgan-type scheme motivated by the shallow water equations. Appl. Numer. Math. 62(4), 489–506 (2012) 23. van Leer, B.: A historical oversight: Vladimir P. Kolgan and his high resolution scheme. J. Comput. Phys. 230, 2378–2383 (2011)

An Easy Control of the Artificial Numerical Viscosity to Get Discrete Entropy Inequalities When Approximating Hyperbolic Systems of Conservation Laws Christophe Berthon, Arnaud Duran and Khaled Saleh

Abstract When considering the numerical approximation of weak solutions of systems of conservation laws, the derivation of discrete entropy inequalities is, in general, very difficult to obtain. In the present work, we present a suitable control of the numerical artificial viscosity in order to recover the expected discrete entropy inequalities. Moreover, the artificial viscosity control turns out to be very easy and the resulting numerical implementation is very convenient.

1 Main Motivations The present work is devoted to the numerical approximation of the weak solutions of hyperbolic systems made of N ≥ 1 conservation laws in the form ∂t w + ∂x f (w) = 0,

x ∈ R, t > 0,

(1)

where w ∈ R N stands for the unknown vector and f : R N → R N denotes the flux function. According to some physical restriction, the solution may be imposed to C. Berthon (B) Laboratoire de Mathématiques Jean Leray, CNRS UMR 6629, Université de Nantes, 2 rue de la Houssinière, BP 92208, 44322 Nantes, France e-mail: [email protected] A. Duran · K. Saleh CNRS UMR 5208, Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France e-mail: [email protected] K. Saleh e-mail: [email protected]

© Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_5

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belong to an invariant admissible domain  ⊂ R N , so that we have w ∈ . The model is complemented with an initial data w(x, t = 0) = w0 (x) where w0 : R → R N is given. Since the system (1) is assumed to be hyperbolic, in a finite time the solution may contain discontinuities. Such discontinuities are governed by the well-known Rankine–Hugoniot relations (for instance, see [12]). Unfortunately, by only adopting these relations, the uniqueness of the solution is lost. In order to rule out nonphysical discontinuities, the system is endowed with entropy inequalities given by ∂t η(w) + ∂x G(w) ≤ 0,

(2)

where η :  → R is a convex function, called the entropy function, and G :  → R is the entropy flux function such that t ∇w f (w) = t ∇w η(w)∇w G(w). A solution of (1) is said entropy satisfying if the entropy inequalities (2) are verified for all entropy pairs (η, G). Now, let us consider the numerical approximation of w. To address such an issue, we first introduce a discretization of the space by adopting a uniform mesh made of cells (xi−1/2 , xi+1/2 ) of constant size x > 0 so that xi+1/2 = xi−1/2 + x for all i in Z. We denote xi = (xi−1/2 + xi+1/2 )/2 for i in Z. Concerning the time discretization, we fix t n+1 = t n + t with t > 0 the time increment restricted according to a suitable CFL condition (for instance, see [12]). At time t n , we consider an approximation of the solution given by a piecewise constant function as follows: w (x, t n ) = win

if x ∈ (xi−1/2 , xi+1/2 ).

(3)

In order to evolve with respect to time this approximation, the updated states (win+1 )i∈Z are evaluated by the following numerical scheme: win+1 = win −

 t  n n f  (win , wi+1 ) − f  (wi−1 , win ) , x

(4)

where f  (w L , w R ) is the numerical flux function. From now on, we underline that this numerical flux function must satisfy f  (w, w) = f (w) for consistency reasons. During the fifty last years, numerous works were devoted to propose relevant formulas to define f  (w L , w R ). Of course, it is not possible to give an exhaustive list of the numerical methods available to approximate the weak solutions of (1), but the reader is referred to [3, 12] for an overview of some well-known numerical techniques. For stability reasons, the scheme (4) is complemented with a CFL condition (for t  ≤ 21 , where  ≥ 0 is determined instance, see [3, 12]) to restrict the time step by x according to the numerical flux function definition. At this level, according to the celebrated Lax–Wendroff Theorem [7], we may expect a convergence of w (x, t n ) to a weak solution w(x, t) of system (1) as t and x tend to zero. In order to avoid some possible nonphysical solutions, the

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31

scheme must be enriched with discrete entropy inequalities. As a consequence, the updated states (win+1 )i∈Z , given by (4), must satisfy the following estimation: η(win+1 ) − η(win ) +

 t  n n G  (win , wi+1 ) − G  (wi−1 , win ) ≤ 0, x

(5)

where G  (w L , w R ) denotes the numerical entropy flux function. From now on, we underline that this numerical flux function must satisfy G  (w, w) = G(w) for all w in . If the discrete entropy inequalities (5) are satisfied by all entropy pairs (η, G), the scheme is said entropy stable or entropy preserving. The establishment of (5) may be very difficult to obtain as soon as N > 1. The Godunov scheme [4] or the HLL scheme [5] are known to be entropy preserving. Next, considering the isentropic gas dynamics or the Euler model, some schemes such as the HLLC scheme [13] or the Suliciu relaxation scheme [1] are also proved to be entropy preserving. But, in general, the proof of (5) is not reachable or, eventually, is violated. For instance, the Roe scheme [8] or the VF-Roe scheme [2] are known to be entropy violating and suitable entropy fixes must be adopted (for instance, see [6]). In this paper, we introduce a simple extension of the artificial viscosity technique [9, 11] in order to easily recover the required entropy stability given by (5). In the next section, we propose a reformulation of the artificial viscosity within the framework of Godunov-type schemes. In Sect. 3, we give a direct control of the artificial viscosity to get (5) for a given entropy pair. The last section is devoted to illustrate the relevance of the suggested numerical procedure.

2 Godunov-type Scheme with Artificial Viscosity In order to enforce the required entropy stability, we introduce artificial viscosity into the scheme (4). As a consequence, the improved numerical method of interest now reads win+1 = win −

   t  n ) − f (w n , w n ) + γ t w n − 2w n + w n f  (win , wi+1 ,  i−1 i i+1 i i−1 x 2 x

(6) where γ ≥ 0 is a parameter to govern the artificial viscosity. Now, we propose to reformulate (6) as a Godunov-type scheme. For this purpose, we introduce the following approximate Riemann solver:

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⎧ wL ⎪ ⎪ ⎪ ⎪ ⎪ w¯ ⎪ ⎪ L ⎨ w x  ⎪ L ; wL , w R = wR  ⎪ t w ⎪ R ⎪ ⎪ ⎪ ⎪ w¯ R ⎪ ⎪ ⎩ wR

if x < −( + γ )t, if − ( + γ )t < x < −t, if − t < x < 0, if 0 < x < t,

(7)

if t < x < ( + γ )t, if x > ( + γ )t,

where we have set 1 1 1 1 w¯ L = w L + (w R − w L ) = (w L + w R ), w¯ R = w R − (w R − w L ) = (w L + w R ), 2 2 2 2 1 1 wL = w L − ( f  (w L , w R ) − f (w L )) , wR = w R + ( f  (w L , w R ) − f (w R )) .  

(8) To simplify the notations, we set w¯ = w¯ L = w¯ R . Next, equipped with this approximate Riemann solver, we evolve the approximate solution (3) in time as follows:

w (x, t n + t) = wR

x − xi+1/2 n n ; wi , wi+1 t

where t is restricted by

 for (x, t) ∈ [xi , xi+1 ) × (0, t),

1 t ( + γ ) ≤ . x 2

(9)

Such a time evolution coincides to a juxtaposition of approximate Riemann solvers wR stated at each interface xi+1/2 . Because of the CFL condition (9), two successive approximate Riemann solvers never interact. After a straightforward computation, we notice that the updated states win+1 , given by (6), reformulate as follows: win+1

1 = x =

1 x



xi+1/2

w (x, t n + t)d x,

xi−1/2 x/2

wR 0

0  x   x  1 n n ; wi−1 ; win , wi+1 d x, , win d x + wR t x −x/2 t (10)

Because of the Godunov-type reformulation, we are now able to control the artificial viscosity, parameterized by γ , in order to obtain the expected discrete entropy inequality (5).

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33

3 Discrete Entropy Inequality To establish the required entropy stability (5), we first recall a result stated by Harten, Lax, and van Leer in [5]. In fact, this statement gives a local sufficient condition, interface per interface, that implies (5). The reader is referred to [5] for a proof. Lemma 1 (Harten, Lax and van Leer [5]) Let w L and w R be given in . Assume that the approximate Riemann solver satisfies x/2    x 1 1 t ; w L , w R d x ≤ (η(w L ) + η(w R )) − η wR (G(w R ) − G(w L )) , x −x/2 t 2 x

(11) for a given entropy pair (η, G). Then, the Godunov-type scheme (10) satisfies a discrete entropy inequality (5) where the numerical entropy flux function reads as follows: x 1 η(w R ) + G  (w L , w R ) = G(w R ) − 2t x

0

x/2

   x ; w L , w R d x. η wR t

From now on, let us underline that, as soon as the approximate Riemann solver is given by (7), the numerical entropy flux function rewrites ¯ − η(w R )). G  (w L , w R ) = G(w R ) + η(wR ) + ( + γ )(η(w) Now, arguing Lemma 1, it is sufficient to satisfy (11) to obtain (5). After a straightforward computation, the inequality (11) reads as follows: E0 + γ D ≤ 0,

(12)

where we have set   E0 =  η(wL ) + η(wR ) − η(w L ) − η(w R ) + (G(w R ) − G(w L )) ,

(13)

D = 2η(w) ¯ − η(w L ) − η(w R ).

(14)

Let us emphasize that E0 coincides with the entropy dissipation rate for the initial scheme (4) when the artificial viscosity γ vanishes. Of course E0 turns out to be negative as long as the scheme (4) is entropy preserving. But E0 may become positive for an entropy violating scheme. Next, since η is a convex function, we immediately get D ≤ 0 with equality to zero if and only if w L = w R . Moreover, we remark that neither E0 nor D depend on γ . As a consequence, as soon as D = 0, it is possible to fix γ ≥ 0 large enough such that the local entropy inequality (12) holds. Lemma 2 Let w L and w R be given in . Then there exists γ ≥ 0 such that (12) holds. Moreover, if the matrix ∇w2 η(w) is positive definite then γ is bounded.

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Proof The existence of γ ≥ 0 is obvious. Now, we establish that γ is bounded in a neighborhood, denoted V, of w L = w R . Indeed, since D vanishes only into V, γ is immediately bounded far away from V. First, we give the expansion with respect to w¯ of D, given by (14), in V. A direct evaluation gives D = O( w R − w L 2 ) with ¯ is positive definite. Because of the convexity of η, never optimal order since ∇w2 η(w) this quantity vanishes in V as long as w L = w R . Concerning the expansion of E0 , we first introduce wH LL =

1 1 (w L + w R ) − ( f (w R ) − f (w L )). 2 2

Following [5], we have 2η(w H L L ) ≤ η(w L ) + η(w R ) − G(w R ) + G(w L ), with equality if and only if w L = w R . Then we obtain E0 / ≤ η(wL ) + η(wR ) − 2η (w H L L ). Next, we notice that wL + wR = 2w H L L , then we have η(wL ) + η(wR ) − 2η(w H L L ) = O( w R − w L 2 ), with equality to zero if and only if w L = w R . As a consequence, we have E0 = O( w R − w L 2 ) once again with optimal order since ∇w2 η(w H L L ) is positive definite. It results γ = O(1) and the proof is achieved. 

4 Numerical Results To illustrate the procedure we adopt the Euler model, defined by w = t (ρ, ρu, E) and f (w) = t (ρu, ρu 2 + p, (E + p)u), where p is defined by the usual perfect gas law [12] with adiabatic coefficient fixed equal to 1.4. Here, the entropy function is given by η(w) = ρϕ( p/ρ 1.4 ), where ϕ is a smooth function which satisfies the restrictions determined in [10]. In Fig. 1, we display the numerical results obtained when simulating a Riemann problem with initial data given by w0 (x) = t (1, 0, 2.5) if x < 0 and w0 (x) = t (0.125, 0, 0.025) if x > 0. The adopted initial scheme is given by the VF-Roe method [2], known to be entropy violating. Then, we notice a nonphysical shock wave at the sonic point within the rarefaction. Next, we introduce the numerical artificial viscosity governed by γ fixed according to (12). We remark that the entropy violating shock wave no longer remains. In addition, Fig. 1 displays the values of γ versus time. As expected, γ is bounded. To conclude this numerical experiment, let us underline that the control of the artificial viscosity is performed according to a single entropy pair. Here, we have adopted the entropy defined with ϕ1 (θ ) = ln(θ ). As a consequence, a natural question arising concerns the behavior of the other entropies. In Table 1, we present the value of L 1 -norm of the positive part of the entropy budget, defined as the left-hand side of inequality (5), obtained for two other entropies defined by ϕ2 (θ ) = −θ 1/2.4 and ϕ3 (θ ) = θ −2/1.4 when considering the original VF-Roe scheme. Since we expect negative entropy dissipation rate, the value of the positive part must be zero. With the original VF-Roe scheme, we get positive values of the entropy dissipation rate (see Table 1) while we obtain values less than 10−16 with the derived single entropy preserving scheme.

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35

Fig. 1 Riemann solution for the density ρ, the velocity u, the pressure p and volution of γ Table 1 Evaluation of L 1 -norm of the positive part of the entropy budget Cells ϕ1 ϕ2 ϕ3 100 800 1600 12800 102400

1.05291E-6 5.53458E-8 3.42990E-8 8.27230E-9 1.23514E-9

4.40092E-7 2.30603E-8 1.42867E-8 3.44473E-8 5.14317E-9

1.48806E-6 7.90713E-8 4.90515E-8 1.18418E-8 1.76836E-9

References 1. Chalons, C., Coquel, F., Godlewski, E., Raviart, P.-A., Seguin, N.: Godunov-type schemes for hyperbolic systems with parameter-dependent source. The case of Euler system with friction. Math. Models Methods Appl. Sci. 20(11), 2109–2166 (2010) 2. Gallouët, T., Hérard, J.-M., Seguin, N.: Some recent finite volume schemes to compute Euler equations using real gas EOS. Int. J. Numer. Methods Fluids 39(12), 1073–1138 (2002) 3. Godlewski, E., Raviart, P.-A.: Numerical Approximation of Hyperbolic Systems of Conservation Laws. Applied Mathematical Sciences, vol. 118. Springer, New York (1996) 4. Godunov, K.: A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat. Sb. (N.S.), 47(89), 271–306 (1959) 5. Harten, A., Lax, P.D., van Leer, B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25, 35–61 (1983)

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6. Helluy, P., Hérard, J.-M., Mathis, H., Müller, S.: A simple parameter-free entropy correction for approximate Riemann solvers. Comptes rendus Mécanique 338(9), 493–498 (2010) 7. Lax, P., Wendroff, B.: Systems of conservation laws. Comm. Pure Appl. Math. 13, 217–237 (1960) 8. Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43(2), 357–372 (1981) 9. Tadmor, E.: Numerical viscosity and the entropy condition for conservative difference schemes. Math. Comput. 43(168), 369–381 (1984) 10. Tadmor, E.: A minimum entropy principle in the gas dynamics equations. Appl. Numer. Math. 2(3–5), 211–219 (1986) 11. Tadmor, E.: Entropy stable schemes. In: Abgrall, R., Shu, C.-W. (eds.) Handbook of Numerical Methods for Hyperbolic Problems: Basic and Fundamental Issues, vol. 17, pp. 467–493 (2016). North-Holland, Elsevier, Amsterdam (2017) 12. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction, 3rd edn. Springer, Berlin (2009) 13. Toro, E.F., Spruce, M., Speares, W.: Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 4(1), 25–34 (1994)

Development of the Matrix Spectrum Dichotomy Method Elina A. Biberdorf

Abstract The work provides a systematic review of algorithms and applications of the dichotomy problem of the matrix spectrum. This problem is a promising alternative to the asymmetric spectral problem in the classical formulation.

1 Contradictions Between the Classical Spectral Problem and Applied Problems Information on the spectra of linear operators is needed to solve a wide range of applied problems. Despite the fact that these tasks arise in various fields, many of them have similar properties. We illustrate them with the example of stability problems (for structures, flows, solutions of differential and difference equations, etc.). Moreover, for definiteness, we consider finite-dimensional linear operators matrices. (1) In many cases, stability is ensured by the fact that the spectrum of the corresponding matrix is strictly inside a certain region on the complex plane, which is called the stability region. For example, for asymptotic Lyapunov stability of a solution of the ODE system y  = Ay it is necessary and sufficient for the spectrum of the matrix A to lie strictly in the left half plane. That is, to solve such problems, it is not necessary to determine each eigenvalue individually. Instead, we need location information for eigenvalue groups relative to the curve, which is the boundary between the regions of stability and instability. E. A. Biberdorf (B) Novosibirsk State University, 1, Pirogova str., 630090 Novosibirsk, Russia e-mail: [email protected] Sobolev Institute of Mathematics, 4 Acad. Koptyug avenue, 630090 Novosibirsk, Russia

© Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_6

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(2) Despite the fact that the eigenvalues of an asymmetric matrix continuously depend on the matrix, the modulus of continuity behaves in an unpredictable way. This fact makes it reasonable to introduce the concept of a ε-spectrum or pseudospectrum [1, 2]. By definition, the complex numbers λ belong to the ε-spectrum of the matrix A if the matrix A − λI “is almost degenerate” : σmin (A − λI ) ≤ ε. The sizes of the ε-spectrum spots determine how much the eigenvalues of the original and perturbed matrices can differ |λ j (A) − λ j (A + A)| , A ≤ ε. This explains seemingly paradoxical examples when the solutions of the system y  = Ay are stable, and ones of the perturbed system y  = (A + A)y are unstable. The same explains the fact that solutions of stable systems can show large growth at the initial time intervals [1, 3], which in practice can lead to destruction of engineering structures, the development of turbulence of flows, etc. Thus, to solve applied problems, knowing the location of the points of the spectrum is not enough, information on the magnitude and location of spectral spots is needed. It is easy to notice that the spectral problem in the classical formulation is oriented exactly at calculating individual eigenvalues. That is, from the point of view of applications, it is redundant on the one hand (see clause 1), and on the other hand, it is insufficient (clause 2). Obviously, for a more adequate response to the needs of practice, the formulation of the spectral problem must be changed.

2 New Statement of the Problem In order to formulate the new spectral problem, it took many years of work of academician S. K. Godunov and his group. Although the movement toward new problem statements can be found in earlier works (for example, in the monograph [4]), it was Godunov’s ability to think independently of the paradigms that have developed over decades and even centuries that allowed us to approach a constructive solution to this issue. To date, the formulation of the dichotomy of the matrix spectrum, which is an alternative to the classical spectral problem, consists of three points: 1. Determine if there are points of the matrix spectrum on a given curve. 2. If there are no spectrum points on the curve, evaluate distance from the spectrum to the curve. 3. Determine the bases of the invariant subspaces of the matrix corresponding to the eigenvalues located on opposite sides of the curve, and bring the matrix to a cell-diagonal form. This statement of the problem is obviously more adequate to the needs of applied areas, but it has not received proper distribution over 30 years.

Development of the Matrix Spectrum Dichotomy Method

39

3 Base Algorithms For the dichotomy of the matrix spectrum with respect to some encountered in applications curves the Godunov’s group has created efficient algorithms. They could be called basic, since the algorithms for more complex curves, developed recently, are somehow reduced to these cases. The dichotomy algorithms of the matrix spectrum with respect to the circle are presented in [2, 3, 5–10]. Although they differ from each other, their main idea can be described as follows. It is known that separating the spectrum of the matrix A by the unit circumference is equivalent to the solvability of the boundary value problem for an infinite system of difference equations U j+1 = AU j + f j ,  f j  ≤ C < ∞, U j  ≤ C < ∞, −∞ < j < ∞. This means the existence of a summable matrix Green’s function G j of this problem. G j+1 = AG j (−∞ < j < ∞), G +0 − G −0 = I. The norm of this function in the space of discrete summable functions can be considered a criterion for the absence of the spectrum of the matrix A on the unit circle. Moreover, the projector on the invariant subspace of the matrix corresponding to the eigenvalues lying inside the circle is the matrix P = G +0 . Thus, the task is reduced to calculating the Green’s function and its norm ω = G. It is very important to note that the algorithms created to solve this problem have a number of significant advantages. Firstly, the basis of the algorithms is orthogonal transformations, namely, QR decomposition. Secondly, there is a variant of the algorithm that does not use matrix inversion during iterations (A. N. Malyshev algorithm [8]). Third, fast and predictable convergence. Fourth, the structure of the algorithms allows us to evaluate the accuracy of the result. We also note that such an approach is completely natural for solving stability problems. It turns out that one of the equivalent norms of the function G j coincides with the solution of a system of equations generalizing the well-known Lyapunov equations. Separating the spectrum of the matrix A relative to the imaginary axis is equivalent to separating the spectrum of the matrix exponent eτ A with respect to the unit circle. Thus, this problem reduces to the previous case. The parameter τ is chosen from the considerations of convergence of the series by which the exponent is calculated. One-dimensional spectral portraits are methods that allow visualize information about the location of the matrix spectrum in the plane relative to the family of curves [3, 11, 12]. So let γx be a family of curves, and ωγx be a numerical criterion for the dichotomy of the spectrum of the matrix A with respect to the curve γx . Then the graph of the function ω(x) = ωγx is a one-dimensional spectral portrait.

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In contrast to the image of the ε-spectrum on the whole plane, this method is more economical and at the same time very informative. By analyzing one-dimensional spectral portraits, it is possible to select areas (rings, bands, etc.) on the plane that contain groups of eigenvalues, construct the bases of the corresponding invariant subspaces, and bring the matrix to block-diagonal form by a similar transformation.

4 Dichotomy Relative to the Second-Order Curves Let us consider the ellipse (Reλ/a)2 + (Imλ/b)2 = 1, where  a = (ρ  + 1/ρ) /2, b = 1 1 (ρ − 1/ρ) /2. Then transforming the complex plane λ = 2 ξ + ξ turns it into two circles centered at the origin with radii ρ and 1/ρ. In this case, the initial matrix pencil A − λB, linear in the argument λ turns into quadratic, with respect to the argument  ξ : A − λB = A − 2−1 (ξ + 1/ξ ) B = −(2ξ )−1 B − 2ξ A + ξ 2 B . It follows that the separation of the spectrum of A − λB with respect to the ellipse is equivalent to the separation of the spectrum of the quadratic pencil B − 2ξ A + ξ 2 B with respect to the circles Cρ or C1/ρ . The latter is possible if and only if there is a solution to the boundary value problem ρ 2 BU j − 2ρ AU j+1 + BU j+2 = f j ,  f j  ≤ C < ∞, U j  ≤ C < ∞. If we group each two equations of this problem together, then it can be written as follows: AU j − BU j+1 = F j , F j  ≤ C < ∞, U j  ≤ C < ∞, 

where A=

ρ 2 B −2ρ A ρ2 B



 ,

B=

−B 2ρ A −B

 .

This means that the problem of separating the spectrum of the pencil A − λB with respect to the ellipse reduces to the problem of the dichotomy of the spectrum of the pencil A − ξ B with respect to the unit circle [9, 13]. Dichotomy relative to hyperbola ReλImλ = c bases on the fact that the transformation λ → ξ λ = icξ + ξ turns the hyperbola into the imaginary axis [14]. The dichotomy problem with respect to the hyperbola for the initial pencil reduces to a dichotomy with respect to the imaginary axis of the spectrum of the quadratic pencil with respect to the variable ξ : A − λB = A − (ξ + ic/ξ )B = ξ −1 (ξ A − (ic + ξ 2 )B) = 0. It, in turn, is equivalent to a dichotomy of the spectrum of the matrix pencil A − ξ B     −icB A 0 B A= , B= 0 I I 0 relative to the imaginary axis since det(A − ξ B) = det(ξ A − (ic + ξ 2 )B).

Development of the Matrix Spectrum Dichotomy Method

41

A similar idea is used to the dichotomy of the matrix pencil spectrum relative to the parabola Reλ = −(Imλ)2 . The transformation λ = ξ 2 + ξ reduces the problem to dichotomy of the spectrum of the quadratic pencil A − (ξ 2 + ξ )B relative to the imaginary axis, which is equivalent to the same problem for a linear matrix pencil of higher dimension     A0 B B −ξ . 0 I I 0 This algorithm was developed and tested by Popova M. A. and Biberdorf E. A. Earlier, a similar approach was proposed for the spectrum of the matrix in [13].

5 Applications The method of decomposing the polynomial into factors [15] f 0 x n + f 1 x n−1 + · · · + f n−1 x + f n = h(x)g(x), whose roots lie, respectively, inside and outside the unit circle is based on the next fact. The absence of the roots of the polynomial on the unit circle is equivalent to the solvability of the discrete boundary value problem AU j − BU j+1 = F j , F j  ≤ C < ∞, U j  ≤ C < ∞ and therefore to the dichotomy with respect to the unit circle of the spectrum of a special matrix pencil ⎛

f0 f1 . . . ⎜ f0 . . . ⎜ A=⎜ .. ⎝ .

⎞ ⎛ fn f n−1 ⎜ f n−1 f n−2 ⎟ ⎟ ⎜ ⎟ , −B = ⎜ .. .. ⎠ ⎝. . f0 f1

⎞ ⎟ fn ⎟ ⎟. .. .. ⎠ . . . . . f n−1 f n

An application of the dichotomy relative to the imaginary axis to the study of engineering constructions stability is demonstrated in [12]. Namely, the phenomenon of flutter of the wing and the oscillations of the turbine blades were investigated. In [3], a modification of this algorithm for matrices with a large norm was applied to the study of the stability of a plane-parallel Poiseuille flow of a viscous incompressible fluid. The problem  ∞ of computing the control u, d x/dt = Ax + Bu, minimizing the functional 0 (x(t)2 + Bu(t)2 )dt is solved by computing the projector P on the invariant subspace of the composite matrix

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 M=

A 2AB(B ∗ B)−1 B ∗ 2AI −A∗



 ,

P=

P11 P12 P21 P22



corresponding to the eigenvalues of the matrix M lying in the right half plane. Then −1 . (see [16]) u(t) = −(B ∗ B)−1 B ∗ x(t), where  = −P21 P11

6 Conclusion This review shows that the dichotomy of the matrix spectrum is maximally adapted for many applications and at the same time has great potential for development and modifications. It can be expected that over time it will occupy a more significant place in the computational arsenal of mathematicians. This work was supported by the RFBR, grant 17-01-00791.

References 1. Trefethen, L.N., Trefethen, A.E., Reddy, S.C., Driscoll, T.A.: Hydrodynamic stability without eigenvalues. Sci., New Ser. 261(5121), 578–584 (1993) 2. Godunov, S.K.: Modern Aspects Linear Algebra. Scientific Book, Novosibirsk (1997) (in Russian) 3. Biberdorf, E.A., Blinova, M.A., Popova, N.I.: Modifications of the dichotomy method of a matrix spectrum and their application to stability tasks. Num. Anal. Appl. 11(2), 108–120 (2018) 4. Daletsky, Y.G., Crane, M.G.: Stability of Differential Solutions Equations in a Banach Space. Nauka, Moscow (1970) (in Russian) 5. Godunov, S.K.: Circular dichotomy of the matrix spectrum. Sib. Mat. J. 27(5), 24–37 (1986) (in Russian) 6. Godunov, S.K.: The dichotomy problem of the matrix spectrum. Sib. Mat. J. 27(5), 57–59 (1986) (in Russian) 7. Bulgakov, A.Y., Godunov, S.K.: Circular dichotomy of the matrix spectrum. Sib. Mat. J. 29(5), 59–70 (1988) (in Russian) 8. Malyshev, A.N.: Introduction to Computational Linear Algebra. Nauka, Siberian Branch, Novosibirsk (1991) (in Russian) 9. Godunov, S.K., Sadkane, M.: Elliptic dichotomy of a matrix spectrum. Linear Algebr. Its Appl. 248, 205–232 (1995) 10. Godunov, S.K., Sadkane, M.: Some new algorithms for the spectral dichotomy methods. Linear Algebr. Its Appl. 358, 173–194 (2003) 11. Kostin, V.I., Godunov, S.K.: Spectral Portraits of Matrices. Preprint IM SB RAS, Novosibirsk (1990) (in Russian) 12. Bunkov, V.G., Godunov, S.K., Kurzin, V.B., Sadkane, M.: Application of the new mathematical tool (One-dimensional spectral portraits of matrices) to the problem of aeroelastic vibrations of turbine blad cascades. Sci. Notes TsAGI 40(6), 3–13 (2009) (in Russian) 13. Malyshev, A.N., Sadkane, M.: On parabolic and elliptic spectral dichotomy. SIAM J. Matrix Anal. Its Appl. 18, 265–278 (1997)

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14. Blinova, M.A., Popova, N.I., Biberdorf, E.A.: The application of the dichotomy of the matrix spectrum to the study of the stability of currents. In: Proceedings of the International Scientific Conference Marchukov’s Scientific Readings, pp. 106–112 (2017). http://conf.nsc.ru/ files/conferences/cam17/427967/ProceedingsCAM2017.pdf (in Russian) 15. Biberdorf, E.A.: A criterion for the dichotomy of the roots of a polynomial by a unit circle. Sib. J. Ind. Math. 3(1), 16–32 (2000) (in Russian) 16. Godunov, S.K.: Ordinary Differential Equations with Constant Coefficient. AMS, Providence (1997)

MHD Model of an Incompressible Polymeric Fluid. Stability of the Poiseuille Type Flow A. M. Blokhin and D. L. Tkachev

Abstract We study a generalization of the Pokrovski–Vinogradov model for flows of solutions and melts of an incompressible viscoelastic polymeric medium to nonisothermal flows in an infinite plane channel under the influence of magnetic field. For the linearized problem (when the basic solution is an analogue of the classical Poiseuille flow for a viscous fluid described by the Navier–Stokes equations) we find a formal asymptotic representation for the eigenvalues under the growth of their modulus. We obtain a necessary condition for the asymptotic stability of the Poiseuille-type shear flow.

1 Introduction In this work, we study a generalization of the structurally phenomenological Pokrovski–Vinogradov model describing flows of melts and solutions of incompressible viscoelastic polymeric media to the nonisothermal case under the influence of magnetic field. In the Pokrovski–Vinogradov model, the polymeric medium is considered as a suspension of polymer macromolecules which move in an anisotropic fluid produced, for example, by a solvent and other macromolecules. The influence of environment on a real macromolecule is modeled by the action on a linear chain of Brownian particles each of which represents a large enough part of the macromolecule. Brownian particles often called “beads” are connected to each other by elastic forces called “springs”. In the case of slow motions, the macromolecule is modeled as a chain of two particles called “dumbbell”. The physical representation of linear polymer flows described above results in the formulation of the Pokrovski–Vinogradov rheological model: A. M. Blokhin · D. L. Tkachev (B) Sobolev Institute of Mathematics, 4 Acad. Koptuyg Avenue, Novosibirsk 630090, Russia e-mail: [email protected] Novosibirsk State University, 1 Pirogova Str., Novosibirsk 630090, Russia A. M. Blokhin e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_7

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 ρ

∂ ∂ vi + vk vi ∂t ∂ xk

 =

∂ σik , ∂ xk

σik = − pδik + 3

∂vi = 0, ∂ xi

η0 aik , τ0

d 1 + (k − β)I 2 3β aik − vi j a jk − vk j a ji + aik = γik − ai j a jk , dt τ0 3 τ0

(1) (2)

(3)

where ρ is the polymer density, vi is the i-th velocity component, σik is the stress tensor, p is the hydrostatic pressure, η0 , τ0 are the initial values of the shear viscosity and the relaxation time respectively for the viscoelastic component, vi j is the tensor of the velocity gradient, aik is the symmetrical tensor of additional stresses of second rank, I = a11 + a22 + a33 is the first invariant of the tensor of additional stresses, ki is the symmetrized tensor of the velocity gradient, k and β are the γik = vik +v 2 phenomenological parameters taking into account the shape and the size of the coiled molecule in the dynamics equations of the polymer macromolecule. Structurally, the model consists of the incompressibility and motion Eqs. (1) as well as the rheological relations (2), (3) connecting kinematic characteristics of the flow and its inner thermodynamic parameters. In this work, we consider one of such generalizations that takes into account the influence of the heat and the magnetic field on the polymeric fluid motion (see Sect. 2 for more details). Our main interest is an analogue of the Poiseuille flow which is the well-known shear flow in an infinite channel. It turns out that in our case this flow has a number of features. For example, computations show that for some values of parameters the velocity profile is stretched in the direction opposite to the forces of pressure.

2 Nonlinear Model of the Polymeric Fluid Flow in a Plane Channel Under the Presence of an External Magnetic Field In a dimensionless form, the model describing magnetohydrodynamic flows of incompressible polymeric fluid reads (we keep the notations from [1]) divu = u x + v y = 0,

(4)

divH = L x + M y = 0,

(5)

  du 0 + ∇ P = div(Z ) + σm (H, ∇)H + Gr (Z − 1) , 1 dt

(6)

MHD Model of an Incompressible Polymeric Fluid

47

da11 − 2 A1 u x − 2a12 u y + L 11 = 0, dt

(7)

da22 − 2 A2 v y − 2a12 vx + L 22 = 0, dt

(8)

I a12 da12 K − A1 vx − A2 u y + = 0, dt τ¯0 (Z )

(9)

1 dZ Ar Am = x,y Z + Z D + σm Dm , dt Pr Pr Pr

(10)

dH − (H, ∇)u − bm x,y H = 0, dt

(11)

where t is the time, u, v and L, 1 + M are the components of the velocity vector u and the magnetic field H, respectively, in the Cartesian coordinate system x, y; P = ρ + σm

L 2 + (1 + M)2 , 2

ρ is the pressure; a11 , a22 , and a12 are the components of the symmetrical anisotropy tensor of second rank;

=

1 (ai j ), i, j = 1, 2; Re

L ii =

K I = W −1 +

2 ) K I aii + β(aii2 + a12 , i = 1, 2; τ¯0 (Z )

k¯ I, k¯ = k − β; 3

I = a11 + a22 is the first invariant of the anisotropy tensor; k, β (0 < β < 1) are the phenomenological parameters of the rheological model; Ai = W −1 + aii , i = 1, 2; Z = TT0 ; T is the temperature, T0 is an average temperature (room temperature; we will further assume that T0 = 300 K); I = K I + β I ; τ¯0 (Z ) = 1 , J (Z ) = exp{ E¯ A Z −1 }, K Z J (Z ) Z E¯ A = ET0A , E A is the activation energy; D = a11 u x + (vx + u y )a12 + a22 v y ; Dm = L 2 u x + L(1 + M)(vx + u y ) + (1 + M)2 v y ; other physical constants are described in [1]. Remark 1 Our main problem is the problem of finding solutions to the mathematical model (4)–(11) describing magnetohydrodynamic flows of an incompressible polymeric fluid in a plane channel with the depth 1 and bounded by the horizontal walls which are the electrodes C + and C − along which we have electric currents with the current strength J + and J − , respectively.

48

A. M. Blokhin and D. L. Tkachev

External with respect to the chanel S areas S1+ , S1− are under the influence of + + the magnetic fields with components L + 1 = 0, M1 , and M1 | y= 21 +0 = −1 + − − L− 1 = 0, M1 and M1 | y=− 21 −0 = −1 +

1+M(− 21 ) χ−

1+M( 21 ) , 1+χ0+

. Values of temperature Z on the

1+ 1+0 θ¯

sides of the chanel will be defined below with boundary conditions (12). Acquired correlations between boundary values of M1+ + 1, 1 + M( 21 ), and M1− , 1 + M(− 21 ) correspondingly arise from continuity of the normal component for the magnetic induction vector on the sides of the chanel. The domains S1+ and S1− external to the channel are magnets with the magnetic susceptibilities χ1+ and χ1− . On the walls of the channel the following boundary conditions hold: ⎧ y = ± 21 : u = 0 (no-slip condition), ⎪ ⎨ Z = 1 (T = T0 ), y = 21 : (12) ⎪ ⎩ 1 θ ¯ ¯ y = − : Z = 1 + θ (θ = , θ = T − T0 ). 2

T0

We have the temperature T = T0 in the domain S1+ , whereas on the electrode C + we have 1 θ y = − : Z = 1 + θ¯ , θ¯ = , θ = T − T0 , 2 T0 i.e., for θ¯ > 0 there is heating from below (T is the temperature in the domain S1− and on the electrode C − ), and for θ¯ < 0 there is heating from above. Remark 2 We will consider the electrodes C + and C − as the boundaries between two uniform isotropic magnetics. Therefore, on the boundaries C + and C − the following known conditions hold: y = 21 (C + ) : L = −J + , M y = 0, (13) y = − 21 (C − ) : L = −J − , M y = 0. Stationary solutions of the mathematical model (4)–(11) were studied in [1]. Particular solutions (analogous to Poiseuille and Couette solutions for the Navier–Stokes equations system) were constructed there in the following form:

ˆ U(t, x, y) = U(y), ˆ ˆ p(t, x, y) = P(y) + pˆ 0 − Ax,

(14)

where U = (u, v, a11 , a12 , a22 , Z , L , M)T , T ˆ ˆ ˆ M(y)) , U(y) = (u(y), ˆ vˆ (y), aˆ 11 (y), aˆ 12 (y), aˆ 22 (y), Zˆ (y), L(y), ˆ pˆ 0 is the pressure on the channel axis for y = 0, x = 0, and A(> 0) is the constant drop of pressure along the x axis.

MHD Model of an Incompressible Polymeric Fluid

49

We will further study Lyapunov’s stability with respect to small perturbations of the stationary flow described above. Denoting small perturbations of all values by the same symbols, after linearization we obtain the following linear problem: ⎧ ⎪ ut ⎪ ⎪ ⎪ ⎪ ⎪ vt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (α11 )t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (α12 )t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (α22 )t ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

+ uu ˆ x − Zˆ (α11 )x + Zˆ (α22 )x − Zˆ (α12 ) y + uˆ v − Zˆ α12 + 1 = 0, 



+ uv ˆ x − Zˆ (α12 )x + 2 = 0, 

+ u(α ˆ 11 )x − 2αˆ 1 u x − 2αˆ 12 u y + αˆ 11 v + R33 α11 + + R34 α12 + R35 α22 + r11 Z = 0, 

+ u(α ˆ 12 )x − αˆ 1 vx − αˆ 2 u y + αˆ 12 v + R43 α11 + + R54 α12 + R45 α22 + r12 Z = 0, 

+ u(α ˆ 22 )x − 2αˆ 12 vx − 2αˆ 2 v y + αˆ 22 v + R53 α11 +

+ R54 α12 + R55 α22 = 0, 1 Ar  Ar ˆ x,y Z + uˆ aˆ 12 Z + Z t + uˆ Z x + Zˆ  v = Z (aˆ 11 u x + Pr Pr Pr Am + aˆ 12 vx + aˆ 12 u y + aˆ 22 v y + uˆ  a12 ) + σm ( Lˆ 2 u x + Pr ˆ + λˆ )vx + L(1 ˆ + λˆ )u y + uˆ  ((1 + λˆ )L + Lˆ M) + (1 + λˆ )2 v y ), + L(1

ˆ x − (1 + λˆ )u y − uˆ  M − bm x,y L = 0, L t + uˆ L x + v Lˆ  − Lu ˆ x − (1 + λˆ )v y − bm x,y M = 0, Mt + uˆ Mx − Lv 1 u x + v y = 0, t > 0, |y| < , x ∈ R 1 ; 2 u = v = Z = L = M y = 0,

1 y = ± , t > 0, x ∈ R 1 , 2

(15) (16)

where all new values and parameters are connected to the mixed problem (4)–(14) data and base Poiseuille-type solution.

3 Periodic Perturbations. Formulation of Main Results We will be looking for solutions of system (15) in the special form ˜ U(t, x, y) = U(y) exp{λt + iωx},

(17)

where λ = η + iξ , ξ, ω ∈ R 1 , U = (u, v, α11 , α12 , α22 , , Z , L , M)T . The following statements are true. Theorem 1 If problem (15), (16) has a solution in form (17) (the parameter ω is constant!), then we have the following asymptotic representation for λ:

50

A. M. Blokhin and D. L. Tkachev ⎡

λk = ⎣

1 2

− 21

⎤−1 

1 Zˆ αˆ 2

dξ ⎦



1 2

− 21

−

 ˆ + λˆ )) × (Ar Zˆ aˆ 12 + Am σm L(1

1 2



αˆ 2 Zˆ



iωuˆ + R44 2αˆ 12 R43 + αˆ 2 αˆ 22

 + iωuˆ 

1 Zˆ αˆ 2

+

αˆ 12 × Zˆ

    Zˆ 1 σm (1 + λˆ )2  + dξ + kπi + O , k → ∞, (18) αˆ 2 k ˆ bm Z αˆ 2

where we use O as a “big O” notation. From representation (18) we obtain a necessary condition for the asymptotic stability of the Poiseuille-type flow described in Sect. 2 Theorem 2 For the asymptotic stability of the Poiseuille-type flow it is necessary that the following inequality holds true:

1 2

− 21



   1 2αˆ 12 k + 2β k + 2β W −1 + (aˆ 11 + aˆ 22 ) + χˆ¯ 0 2 aˆ 12 + αˆ 2 3 3 αˆ 2    Zˆ αˆ 12  ˆ σm (1 + λˆ )2 ˆ + λˆ )  + dξ > 0. (19) Ar Z aˆ 12 + Am σm L(1 + αˆ 2 Zˆ bm Zˆ αˆ 2 αˆ 2 ˆ χ¯ 0 Zˆ



Formulated results were acquired in the course of the study of linear stability for the Poiseuille-type flows both for the base Pokrovski–Vinogradov model and for its generalization [2–7]. Detailed information about values that are used to formulate model (4)–(14) can be found in [8]. Paper [9] is dedicated to the full proof of theorems 1 and 2. Mixed problem when in (13) instead of components of normal derivative we have its values is studied in [10]. Acknowledgements This work is supported by the RFBR grant 19-01-00261a. Authors would like to thank A. V. Yegitov for the help in preparing this paper.

References 1. Blokhin A.M., Semenko R.Y.: Stationary magnetohydrodynamic flows of a non-isothermal incompressible polymeric liquid in the flat channel. Bull. South Ural. State Univ., Ser.: Math. Model., Program. Comput. Softw. 11(4), 41–54 (2018) 2. Blokhin, A.M., Yegitov, A.V., Tkachev, D.L.: Asymptotics of the spectrum of a linearized problem of the stability of a stationary flow of an incompressible polymer fluid with a space charge. Comput. Math. Math. Phys. 56(1), 102–117 (2018) 3. Blokhin, A., Tkachev, D., Yegitov, A.: Spectral asymptotics of a linearized problem for an incompressible weakly conducting polymeric fluid. ZAMM (Z. Angrew. Math. Mech.) 98(4), 589–601 (2018) 4. Blokhin, A., Tkachev, D.: Spectral asymptotics of a linearized problem about flow of an incompressible polymeric fluid. Base flow is analogue of a Poiseuille flow. In: AIP Conference Proceedings, vol. 2017, pp. 030028, 030028-1–030028-7 (2018)

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5. Blokhin, A.M., Yegitov, A.V., Tkachev, D.L.: Linear instability of solutions in a mathematical model describing polymer flows in an infinite channel. Comput. Math. Math. Phys. 55(5), 848– 873 (2015) 6. Blokhin, A.M., Tkachev, D.L., Yegitov, A.V.: Asymptotic formula for the spectrum of the linear problem describing periodic polymer flows in an infinite channel. Comput. Math. Math. Phys. 59(9), 992–1003 (2018) 7. Blokhin, A.M., Tkachev, D.L.: Linear asymptotic instability of a stationary flow of a polymeric medium in a plane channel in the case of periodic perturbations. J. Appl. Ind. Math. 8(4), 467–478 (2014) 8. Blokhin, A.M., Tkachev, D.L.: Stability of Poiseuille-type flows for an MHD model of an incompressible polymeric fluid. Fluid Dyn. 54(8), 1051–1058 (2019) 9. Blokhin, A.M., Tkachev, D.L.: Stability of the Poiseuille-type flow for and MHD model of an incompressible polymeric fluid. Europian J. Mech./B Fluids 80, 112–121 (2020) 10. Blokhin, A.M., Tkachev, D.L.: Stability of Poiseuille-type flows for an MHD model of an incompressible polymeric fluid. J. Hyperbolic Differ. Eqn. 16(4), 1–25 (2019)

Mathematical Models of Plasma Acceleration and Compression in Coaxial Channels K. V. Brushlinskii and E. V. Stepin

Abstract Numerical simulations and computations of plasma flows in coaxial channels play an essential role in elaboration and creation of plasma accelerators and plasma compressors. The paper presents a short review of the theoretical, experimental, and numerical investigations in this area. A special attention is focused on the additional presence of a longitudinal magnetic field in the channel. Mathematical models of plasma acceleration and compression are set out in terms of numerical solving of corresponding axisymmetric MHD problems. Some Godunov type difference methods realized in computations are mentioned. The computation results obtained over the last years are related to the accelerating and compressing flow property dependence on physical conditions and parameters of the problems and on the longitudinal magnetic field magnitude.

1 Introduction Plasma flows in channels between two coaxial electrodes are a permanent object of investigations which is of interest for the theory and different scientific and technological applications. The discussion below concerns the two series of works connected with the actual applications. First, plasma as an electric current conductor moves K. V. Brushlinskii (B) · E. V. Stepin Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow 125047, Russia e-mail: [email protected] National Research Nuclear University MEPhI, Moscow 115409, Russia E. V. Stepin e-mail: [email protected] K. V. Brushlinskii Lomonosov Moscow State University, Moscow 119991, Russia © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_8

53

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in a magnetic field. The idea of plasma accelerators is based on this effect: if an accelerator channel has a form of a nozzle, plasma acceleration in the axial direction reminds gas flow in a de Laval nozzle but with higher velocity values. Second, if the central electrode is shorter than the external one, the conic shock wave is joined to it, behind which the region of compressed and heated plasma—the compression zone—on the channel axis is generated. This effect can be of interest as a dynamical trap or as a heated plasma injector into the traps constructed for plasma confinement in the controlled thermonuclear fusion projects. The wide range of plasma process investigations in both directions was performed at the initiative of Morozov [1–4]. A series of plasma accelerators is developed and created under his leadership. In the quasi-steady-state high-current plasma accelerator (QSPA), record values of output energy and velocity have been experimentally achieved [5–7]. Plasma compression in channels of plasma accelerators is investigated in [8–12]. In these investigations, mathematical modeling and computation in addition to theoretical and experimental researches played a significant role. The models are based on the two-dimensional problems with magnetohydrodynamics (MHD) equations and their approximate solving. The well-known methods created for gas dynamics by S.K. Godunov [13] and other authors [14–16] are realized in the MHD problems. The review of numerical models and obtained results is available in [17, 18]. The first calculations of compression flows are presented in [19–24]. In the above-mentioned investigations, two-dimensional plasma flows in a transverse (azimuthal) magnetic field are considered. To continue and develop these works, we consider plasma flows in the presence of a longitudinal field additionally induced by external current-carrying conductors. Mathematical nature of such flows is more difficult: there are nine types of steady-state flows significantly differing from each other by the ratio between the flow velocity and three characteristic longitudinal magnetic sound speeds—fast, slow, and Alfven [18, 25]. In the present work, the mathematical problem statement for the plasma flow in a channel is outlined, numerical methods used in the calculations are mentioned, the results of plasma acceleration and compression computations obtained recently and their analysis with corresponding commentaries are presented. These results partially are published in [26, 27], partially are obtained for the first time.

2 Mathematical Model The mathematical model of plasma flow in coaxial channels is realized in terms of two-dimensional (axisymmetric) problems with the MHD equations [18, 25, 28]: The problems are formulated in the region of the variables (z, r ): 0 < z < 1, r1 (z) < r < r2 (z),

(1)

Mathematical Models of Plasma Acceleration and Compression in Coaxial Channels

55

where r = r1 (z), r = r2 (z) are the equations of electrode cross sections by the plane ϕ = const . If the central electrode of the length z 1 < 1 is shorter than the external one, the bottom border of the region past its tip is the symmetry axis, i.e., r1 (z) = 0 when z 1 < z < 1. The equations are considered in the dimensionless form. The measurement units of all the values are composed of the dimensional constants determined by the problem statement: the channel length Z , the values of density ρ0 and temperature T0 (and consequently pressure p0 ) at the channel input, and the electric current J in the electric circuit. The boundary conditions at the channel input are as follows:

z = 0 : ρ = 1, T = 1, Hϕ =

r0 υr Hr , Hz = Hz0 , = = r z , υϕ = 0, r υz Hz

where r z = dr/dz is a natural slope of plasma trajectories to z-axis. The conditions on the impermeable equipotential electrodes are as follows: r = r1 (z), r = r2 (z) : υn = 0, Hn = 0 There are no conditions at z = 1 due to the assumption about supersonic velocity at the channel output. The main attention is focused on steady-state time relaxed solutions; therefore, initial conditions can be quite arbitrary. The problem contains three dimensionless parameters: β = 8π p0 /H02 —the ratio of the characteristic gas dynamic pressure to the magnetic one—and characteristic values of the radius r0 and the longitudinal magnetic field intensity Hz0 at the channel input related to the chosen measurement units. The new spacial coordinates (z, y) are introduced: z = z, r = r1 (z)(1 − y) + r2 (z)y transforming the region with curvilinear boundaries (1) into the square 0 < z < 1, 0 < y < 1. The equation system in the new coordinates and in the conservative form is presented in [26]. The set out problems are solved numerically by different methods. Flux corrected transport (FCT) method with the Zalesak’s correction realized in MHD recently is used most often [14, 16]. In one of the early works [24], the experience of calculation by using TVD method [15] was gained. In [29], Godunov’s method [13] well known in computational gas dynamics was realized in MHD problems solving: for this purpose the breakup of discontinuity problem in MHD with a transverse magnetic field had been resolved under the adiabatic index γ = 5/3 and γ = 2. It should be noted that many “economic” and “high resolving” explicit difference schemes for hyperbolic equation systems

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K. V. Brushlinskii and E. V. Stepin

are based on the S. K. Godunov’s idea about conservation of Riemann invariant monotonicity, and for this reason, they should be called Godunov type schemes.

3 Plasma Acceleration in Presence of Longitudinal Magnetic Field The first results relating to a longitudinal magnetic field influence on plasma acceleration in a channel are mentioned in [18, 30]. In connection with the development of plasma accelerators, the main attention of these papers and the following ones is focused on super-Alfven flows with the transition through the fast magnetosonic speed. At the same time, the examples of sub-Alfven and combined flows—superAlfven near the central electrode and sub-Alfven near the external one—are presented. In [31, 32], the flows with Alfven and close to its velocity values are specially considered. In [33, 34], the specific case of the flow transition through the Alfven speed and its properties are investigated. Sub-Alfven flows may be realized in a relatively strong longitudinal magnetic field. They are characterized by specific energy transformations: for example, in transonic (relatively to the slow magnetosonic speed) flows thermal energy transforms into kinetic and electromagnetic ones. Computation results of accelerating flows are continued in [26, 35]. The dependence of the plasma acceleration on electrode geometry is researched therein: it is more efficient in a channel with a convex inward central electrode in comparison with the opposite configuration. This result theoretically justifies the choice of QSPA channel geometry. Plasma rotation around the channel axis due to the interaction between the longitudinal magnetic field and the radial electric current is quantitatively registered in the calculations. The calculations have demonstrated that the longitudinal field deviates the electric current near the external electrode to the channel entry and presses the plasma here to the electrode. This result doesn’t depend on the electrode polarity since the MHD equations don’t change under the simultaneous change of the azimuthal magnetic field Hϕ and the electric current j signs. It means that the longitudinal magnetic field can positively influence on the flow accelerating properties. It is known that, in flows with the rather strong electric current J , experiments show current sliding and plasma rarefaction connected with it near the anode. This effect is obliged to the Hall effect, which can be taken into account in a special modification of the MHD model. It can distinguish the electrode polarity [18]. Comparing the earlier computation results of the flows with the Hall effect [18] and the ones with the longitudinal field pointed above, we can see that the current deviations in these two situations are opposite to each other in the case when the external electrode is anode. This result justifies the electrode polarity chosen in QSPA.

Mathematical Models of Plasma Acceleration and Compression in Coaxial Channels

57

4 Compression Plasma Flows in the Channels The attention to the compression effect was attracted by A. I. Morozov in the papers [2, 3] in connection with a theoretical possibility to use it in the nuclear fusion problem. The compression was detected experimentally in [8–10] and is being investigated nowadays [11, 12]. Some examples of its numerical simulation are presented in [18–24]. The recent time ones can be seen in [27]. There is theoretically justified and verified in computation that in the channels with the transverse magnetic field, the plasma compression and heating take place at the conic shock wave joint to the central electrode tip. It is known that the plasma density can increase here no more than in 4 times (γ = 5/3): (γ + 1) p2 + (γ − 1) p1 γ +1 ρ2 = ≤ , ρ1 (γ − 1) p2 + (γ + 1) p1 γ −1

(2)

where the indices (1, 2) are related with density and pressure state before and behind the shock wave [25]. The plasma is accelerated in the nozzle-type preshaped channel part and becomes more rarefied with respect to the value ρ = 1 at the channel input. Then the density increases at the shock wave under the restriction (2) and continues to increase in the adiabatic regime up to the maximum value ρmax . This value is reached at a conic surface inside the wave and then decreases again near the axis. The temperature increases to the more high values, and its maximum is located at the axis. The compression and heating values are strongly dependent on the parameter β = 8π p0 /H02 . If β = 0.05, i.e., under the sufficiently high value of the electric current J , the values ρmax ∼ 2, Tmax ∼ 30 are obtained in computations. Finally, they are determined by the channel geometry details. The velocity field is distorted at the shock wave: before it, the velocity is inclined to the axis following the central electrode geometry, and behind the wave, the flow becomes near parallel to the horizontal external one. The electric current lines are also distorted here. When the parameter β increases, i.e., under the higher pressure p0 or the weaker current J , the compression becomes weaker. It is interesting to follow its asymptotic behavior at β → ∞, i.e., the limiting transfer to gas dynamics. The recent computation shows that the compression effect takes place also in a gas flow, but in the very weak form: its maximum parameters are ρmax ∼ 0.8, Tmax ∼ 2. Compression flows with the longitudinal magnetic field are investigated for the cr at the input that provide the super-Alfven transonic flow restricted values Hz0 < Hz0 regime. The compression effect takes place here, but is weaker: the angle between the axis and the shock wave is larger, the maximum values ρmax and Tmax decrease when Hz0 grows. The reduction of the compression effect in the presence of the longitudinal magnetic field can be explained in the following way. The compression is obliged mainly to the azimuthal magnetic field that vanishes at the axis and is small in its vicinity. Hence Hϕ /Hz → 0 at r → 0, and the longitudinal field plays here the main role.

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For the same reason the compression effect relaxes when the parameter β increases. The magnetic pressure of the azimuthal field decreases in comparison with the gas one. Acknowledgements The work is supported by the Russian Science Foundation (project no. 1611-10278).

References 1. Morozov, A.I.: The acceleration of a plasma by a magnetic field. Sov. Phys. JETP. 5(2), 215–220 (1957) 2. Morozov, A.I.: Steady-state plasma accelerators and their possible applications in thermonuclear research. Nuclear Fusion. Special Suppl., 111–119 (1969) 3. Morozov, A.I.: Thermonuclear systems with dense plasma. Vestnik AN SSSR. 6, 28–36 (1969) 4. Brushlinskii, K.V.: Mathematical models of plasma in Morozov’s projects. Plasma Phys. Rep. 45(1), 33–45 (2019) 5. Morozov, A.I.: Fundamentals of coaxial stationary plasma accelerators. Sov. J. Plasma Phys. 16, 69–78 (1990) 6. Volkov, Y.F., Kulik, N.V., Marinin, V.V., Morozov, A.I. et al.: Analysis of plasma flow parameters generated by full-scale QSPA Kh-50. Sov. J. Plasma Phys. 18, 718–728 (1992) 7. Morozov, A.I.: Introduction to Plasma Dynamics. CRC, Boca Raton (2012) 8. Vinogradova, A.K., Morozov, A.I.: Plasma compressor operation regimes. Sov. J. Tech. Phys. 21, 1475 (1976) 9. Ananin, S.I., Astashinskii, V.M., Bakanovich, G.I., Kostyukevich, E.A., Kuzmitskii, A.M., Mankovskii, A.A., Min’ko, L.Y., Morozov, A.I.: Investigation of plasma flow shaping in the quasi-steady-state high current plasma accelerator. Sov. J. Plasma Phys. 16, 108–113 (1990) 10. Astashinskii, V.M., Bakanovich, G.I., Kuzmitskii, A.M., Min’ko, L.Y.: Selection of operating modes and parameters of a magneto-plasma compressor. Inzh. Fiz. Jh. 62(3), 386–390 (1992) 11. Astashinskii, V.M., Kuzmitskii, A.M., Mishuk, A.A.: Dynamics of the compression plasma flow shaping in a miniature magneto-plasma compressor. Inzh. Fiz. Jh. 84(5), 1022–1027 (2011) 12. Garkusha, I.E., Solyakov, D.G., Chebotarev, V.V., Makhlay, V.A., Kulik, N.V.: Experimental investigation of high-energy quasi-steady-state plasma flows, generated by a magneto-plasma analog of the de-Laval nozzle in compression and acceleration regimes. Plasma Phys. Rep. 45(2), 166–178 (2019) 13. Godunov, S.K.: A difference method for the numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat. Sb. 47(89)(3), 271–306 (1959) 14. Oran, E.S., Boris, J.P.: Numerical Simulation of Reactive Flow. Elsevier, Amsterdam (1987) 15. Harten, A.: Conservation schemes for hyperbolic conservation lows. J. Comp. Phys. 49, 357– 393 (1983) 16. Zalesak, S.T.: Fully multidimensional flux-corrected transport algorithms. J. Comput. Phys. 31, 335–362 (1979) 17. Brushlinskii, K.V., Morozov, A.I.: Calculation of two-dimensional plasma flows in channels. In: Leontovich, M.A. (ed.) Reviews of Plasma Physics, vol. 8, pp. 105–197. Consultants Bureau, NY (1980) 18. Brushlinskii, K.V.: Mathematical and Computational Problems in Magnetic Gas Dynamics. BINOM, Moscow (2009) 19. Brushlinskii, K.V., Gerlakh, N.I., Morozov, A.I.: Finite conductivity effect on the steady-state self-compressing plasma flows. Doklady AN SSSR. 180(6), 1327–1330 (1968) 20. Brushlinskii, K.V., Morozov, A.I., Paleichik, V.V., Savel’ev, V.V.: Two-dimensional compression plasma flow in coaxial channel. Sov. J. Plasma Phys. 2(4), 291–300 (1976)

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21. Ananin, S.I., Min’ko, L.Y.: Calculation of radiated compression plasma flow structure and its parameters taking into account plasma generating channels configurations. Sov. J. Plasma Phys. 15(8) (1989) 22. Ananin, S.I.: Calculation of the dynamics of compressive flows in a quasi-steady-state plasma accelerator in the ring inflow approximation. Sov. J. Plasma Phys. 18(3) (1992) 23. Kalugin, G.A.: Numerical calculation of compression flows in plasma accelerator channel. Matem. modelirovanie. 3(1), 25–36 (1991) 24. Kalugin, G.A.: Effect of dissipative processes on compression flows in a plasma accelerator. Fluid Dyn. 26(3), 402–408 (1991) 25. Brushlinskii, K.V.: Mathematical Foundation of Liquid, Gas and Plasma Computational Mechanics. Intellekt, Dolgoprudniy (2017) 26. Brushlinskii, K.V., Zhdanova, N.S., Stepin, E.V.: Acceleration of plasma in coaxial channels with preshaped electrodes and longitudinal magnetic field. Comput. Math. Math. Phys. 58(4), 593–603 (2018) 27. Brushlinskii, K.V., Stepin, E.V.: Numerical model of compression plasma flows in channels under a longitudinal magnetic field. Differ. Equ. 55(7), 894–904 (2019) 28. Kulikovskiy, A.G., Lyubimov, G.A.: Magnetohydrodynamics. Addison-Wesley, Boston (1965) 29. Ratnikova, T.A.: Godunov’s scheme in MHD problems with a transverse magnetic field. Matem. modelirovanie 9(8), 3–15 (1997) 30. Brushlinskii, K.V., Zhdanova, N.S.: MHD flows in the channels of plasma accelerators with a longitudinal magnetic field. Plasma Phys. Rep. 34(12), 1037–1045 (2008) 31. Styopin, E.V.: Numerical simulation of near-Alfven MHD flows relaxation with a longitudinal magnetic field. J. Plasma Phys. 81, 905810309 (2015) 32. Stepin, E.V.: Stationary magnetohydrodynamics flows in curved coaxial canals. Vestnik Natsional’nogo Yadernogo Universiteta MEPhI. 4(5), 407–420 (2015) 33. Stepin, E.V.: Steady-state transalfvenic MHD flows in coaxial channels with longitudinal magnetic field. J. Phys.: Conf. Ser. 1205, 012055 (2019) 34. Stepin, E.V.: Researches of MHD flow transition through longitudinal alfvenic velocity in coaxial channels. Vestnik Natsional’nogo Yadernogo Universiteta MEPhI. 8(1), 69–78 (2019) 35. Brushlinskii, K.V., Styopin, E.V.: Numerical simulation of plasma flows in curved coaxial ducts with longitudinal magnetic field. J. Phys.: Conf. Ser. 937, 012007 (2017)

Pseudospectrum and Different Quality of Stability Parameters H. Bulgak

Abstract Uncertainties in the data lead to the need to investigate the spots formed by the eigenvalues of matrices. These spots are known as pseudospectrum and it allows to solve Hurwitz and Schur stability problems, but do not give information about their correctness. The paper gives a view of the quality of stability parameters for Hurwitz and Schur problems. These parameters one can use as the condition numbers for corresponds problems.

1 Floating-Point Arithmetic It all started with the advent of computers. So, for example, Von Neumann and Goldstine (1947) and Turing (1948) offered various condition numbers for systems of linear algebraic equations. Now one of them is well-known condition number μ(A) = ||A||||A−1 || = σ N (A)/σ1 (A) for regular N -dimensional matrix A = (ai j ), i, j = 1, 2, . . . N . Here ||A||, σ N (A), and σ1 (A) are spectral norm, maximal, and minimal singular values of A, respectively. Godunov S. K. suggested the first version of theory of computing with guaranteed accuracy of the linear algebraic systems in the Euclidean vector spaces [1]. In the monograph, a mathematical model of floating-point numbers which are used by computers was proposed. A real number z together with its computer representation f l(z) is connected such that there exist real |α| < ε1 ; |β| < ε0 ; αβ = 0 which guarantee f l(z) = z + αz + β, |α| < ε1 ; |β| < ε0 . Here ε0 and ε1 are the constants of use floating-point standard (ε0 is the smallest positive number and 1 + ε1 is the least computer number which is greater than 1). Hence, during computing one is working with the vector f l(x) and matrix f l(A) which are close to its originals x and A: H. Bulgak (B) Department of Mathematics, Selçuk University, Cambridge, UK e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_9

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|| f l(x) − x|| ≤ ||x||ε1 +

√

N ε0 ;

|| f l(A) − A|| ≤ ||A||ε1 + N ε0 . This required a revision of some classical formulations of linear algebra problems from the point of view of they are well conditionality (the solution should not change catastrophically with a small change in the input data). The symmetric eigenvalue problem is well-posed. The computing of the spectral norm of a matrix is also a well-posed problem. The eigenvalue problem is ill-posed in general case.

2 Pseudospectrum and Spectral Portrait of a Matrix Uncertainties in the data lead to the need to investigate the spots formed by the eigenvalues of matrices sufficiently close to the given matrix. Instead of the eigenvalues for non-self-adjoint matrices, one must work with pseudospectrum (L.N. Trefethen terminology [2]) or -spectrum (Godunov’s group terminology, see preface [3]). It leads researchers to try to find a graphic representation of -spectrum of matrix A. These representations are known as spectral portrait of a matrix. The first spectral portrait of a matrix was given by Kostin V.I. There exists a number of equivalent definitions of the pseudospectrum. These definitions depend on chosen norms. In Selçuk University was elaborated application MVC [4]. This application uses the following definition: if for complex number z either A − z I is singular or ||(A − z I )−1 || >

1 ||A||

then z is an element of -spectrum. It is clear that if one chooses, for example,  > δ > 0 then -spectrum includes in δ-spectrum. This remark allows drawing the spots of the pseudospectrum one in another. In this case, MVC application colored pixels in the complex domain |z| < ||A||. This application is interesting from an educational point of view. The used algorithm is expensive. One must compute spectral norm and inverse matrix for all pixels. One can decrease cost of the algorithm if instead spectral norm ||A|| to use equivalent value max|ai j |. It is clear that max|ai j | ≤ ||A|| ≤ N max|ai j |. Also one can decrease the cost of the algorithm if applying a variant of Sherman– Morrison formula for computing inverse matrix using the known inverse matrix for a close one. A similar algorithm is suggested in [5].

Pseudospectrum and Different Quality of Stability Parameters

63

3 Hurwitz Stability Problem There is a well-known Hurwitz problem formulated as follows. Do all eigenvalues of matrix A lie in the interior of the left half-plane? Using spectral portrait of a matrix one can find the answer in question. One application of this problem is stability problem for linear differential equations with constant coefficients. Godunov S. K. decided to find condition number which will show the quality of stability. There exist two Lyapunov theorems which are concerned with Hurwitz stability of (1) x  (t) = Ax(t), t > 0. First theorem: Hurwitz stability of A depends on its eigenvalues. Second theorem: Hurwitz stability of A depends on the existence of positive definite solution of A∗ H + H A = −C for any positive definite C. For several years, Sarybekov R.A. worked on this problem under the guidance of Godunov S.K. Many numerical experiments with the calculation of the matrix Lyapunov equation showed him that the worse its condition, the worse the quality of stability A. Sarybekov R. A. died an untimely death. In [6], he proposed to use of “extremal Lyapunov functions" as a measure of stability quality A. min

A∗ H +H A=−C,C>0

λ N (H (C)) . λ1 (H (C))

Here λ N (H (C)), λ1 (H (C)) are maximal and minimal eigenvalues of symmetric H (C), respectively. The first Lyapunov theorem guarantees that for any Hurwitz matrix A there exists δ > 0 such that its eigenvalues satisfied to the inequalities Realλ j (A) ≤ −δ||A|| < 0, j = 1, 2, . . . N . It is clear that δ  ||A|| is less than |max Realλ j (A)|. The quantity δ  is known as the factor of stability of (1). But we could not use the factor of stability as a practical measure of stability for (1) Ostrowski-type example is given in [7]. For every solution of the Hurwitz stable (1) the following inequality 

||x(t)|| ≤ θ  e−δ ||A||t ||x(0)||, t > 0 is true. Here δ  is any positive number less than δ, θ  ≥ 1 and θ  does not depend on the initial vector x(0) but θ  depends on δ  . The author started work with stable problem with this estimation. The aim was to use as condition number of Hurwitz problem a couple (δ  , θ  ). The author proved the theorem of continuity of the parameters δ  , θ  with respect to matrix A. Godunov S. K. suggested instead the couple (δ  , θ  ) number θmin as a minimal θ which guarantees the inequality

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||x(t)|| ≤ θ e

−t||A|| θ

||x(0)||, t > 0

for any x(0). Along with this, the author’s Theorems took the form [8]. Theorem 1 Let A be Hurwitz stable matrix (viz., θmin < ∞) then A + B also is < 10θ12 and Hurwitz stable matrix for an arbitrary perturb matrix B such that ||B|| ||A|| min

3 |θmin (A + B) − θmin (A)| ≤ 4θmin (A) ||B|| is true. ||A||

Theorem 2 Let A be Hurwitz stable N-dimensional matrix (viz., θmin < ∞) then for condition number of Lyapunov matrix transformation L H = A∗ H + H A the 3 . following estimation holds: μ(L) ≤ N θmin Unfortunately, θmin parameter was not algebraic computable one. The author suggested algebraic parameter κ(A) such that if there exists H positive definite solution of Lyapunov matrix equation A∗ H + H A + I = 0 with identical matrix I then κ(A) = 2||A||||H ||. In the other case, κ(A) = ∞ [7]. This parameter guarantees the inequalities (t > 0) ||x(t)|| ≤



κ(A)e

||et A || ≤

−t||A|| κ(A)

||x(0)||,

 −t||A|| κ(A)e κ(A) .

Godunov S.K. found a physical interpretation of κ(A) such that [9]  κ(A) = 2||A||

sup

||x(0)||=1,T >0 0

T

||x(t)||2 dt.

If S is symmetric Hurwitz matrix then κ(S) = μ(S) and the following inequality holds: −t||S|| ||et S || ≤ e κ(S) , t > 0. The following theorem is true [10]. Theorem 3 Let A be Hurwitz stable matrix (viz., κ(A) < ∞) then A + B also is 1 < 15κ(A) and Hurwitz stable matrix for an arbitrary perturb matrix B such that ||B|| ||A||

|κ(A + B) − κ(A)| ≤ 2.15κ 2 (A) ||B|| is true. ||A||

In [9], a definition of practically asymptotically stability (practically Hurwitz stability) is given such that if κ(A) < κ ∗ then A is known as κ ∗ -practically asymptotically stable matrix. In [10, 11], an algorithm with guaranteed accuracy for computing κ(A) is given. Theorem 3 guarantees that the distance from Hurwitz stable A to the set of unstable ||A|| . It is interesting to find this distance. This question matrices is greater than 15κ(A) was discussed in [12]. Van Loan C. defined the parameter θ (A) = ||A||max−∞ 0, t 2 x ⎩ u(x, 0) = u 0 (x).

(7)

The system can be written in the form (1) with U = u,

F(U ) =

u2 , S(U ) = u 2 , 2

H (x) = x.

(8)

It can be easily checked that the stationary solutions are given by u(x) = Ce x , C ∈ R.

(9)

Observe that, given a cell value u i , the solution of the problem (5) to be solved at the first step of the reconstruction procedure can be easily solved: u i∗ (x) =

xu i ex . e xi+1/2 − e xi−1/2

Therefore, any standard reconstruction operator can be easily modified to become well-balanced. Let us solve (7) in the space interval [−1, 1] and the time interval [8] taking the stationary solution (10) u 0 (x) = e x as initial condition. The boundary condition u(−1, t) = e−1

(11)

is imposed at x = −1 and free boundary condition is considered at x = 1. First-, second-, and third-order methods of the form (3) are applied. The numerical flux and the reconstruction operators considered are, respectively, the Rusanov flux, the second-order MUSCL operator (see [13]), and the third-order CWENO operator (see [6]). Both the standard and the well-balanced modifications of the reconstruction operators are considered. TVDRK methods are used for the time discretization (see [8]).

Well-Balanced High-Order Methods for Systems of Balance Laws

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Fig. 1 Solution at t = 8s: first-, second-, and third-order well-balanced (left) and non-well-balanced (right) schemes. Number of cells: 100 Table 1 Well-balanced schemes: L 1 errors and convergence rates for the stationary solution Cells Error (1st) Order (1st) Error (2nd) Order (2nd) Error (3rd) Order (3rd) 100 200 400 800

4.21E-15 2.90E-15 1.84E-14 4.45E-16

– – – –

8.87E-16 4.42E-16 1.82E-15 1.83E-16

– – – –

3.20E-16 2.54E-16 7.40E-14 2.61E-15

– – – –

Table 2 Non-well-balanced schemes: L 1 errors and convergence rates for the stationary solution Cells Error (1st) Order (1st) Error (2nd) Order (2nd) Error (3rd) Order (3rd) 100 200 400 800

1.58E-01 7.51E-02 3.66E-02 1.81E-02

– 1.08 1.04 1.04

5.42E-02 1.48E-02 3.91E-03 1.01E-03

– 1.87 1.92 1.98

3.12E-02 5.41E-03 7.33E-04 9.49E-05

– 2.54 2.87 2.95

Figure 1 shows the numerical solutions obtained at time t = 8 s using the wellbalanced and the non-well-balanced methods. Tables 1 and 2 show the L 1 -errors at time t = 8 using meshes with increasing number of cells N . It can be observed that the well-balanced numerical methods preserve the stationary solution at machine precision while the errors given by the non-well-balanced methods behave as expected according to the order of the method.

5 Further Comments The methodology summarized here has been successfully applied to derive highorder well-balanced methods for shallow water systems [2, 4], blood flows in vessels [10], Euler system with gravity [5, 7], Ripa system ([11]), among others. Figure 2

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Fig. 2 Propagation of a tsunami in the Mediterranean Sea. Third-order well-balanced scheme. T = 30 min

shows a simulation of a tsunami wave in the Mediterranean Sea obtained by solving the shallow water equations on spherical coordinates using a third-order wellbalanced numerical method. The interested reader is addressed to [5] for aspects such as the possible difficulties related to the solution of (5), the effect of quadrature formulas on the well-balancedness of the method, or the derivation of well-balanced methods in the presence of discontinuities of the function H .

References 1. Castro, M.J., Gallardo, J.M., López, J.A., Parés, C.: Well-balanced high order extensions of Godunov method for linear balance laws. SIAM J. Numer. Ann. 46, 1012–1039 (2008) 2. Castro, M.J., López, J.A., Parés, C.: High order exactly well-balanced numerical methods for shallow water systems. J. Comput. Phys. 246, 242–264 (2013) 3. Castro, M.J., de Luna Morales, T., Parés, C.: Well-balanced schemes and path-conservative numerical methods. In: Abgrall, R., Shu, C.W. (eds.) Hand book of Numerical Methods for Hyperbolic Problems. Handbook of Numerical Analysis, vol. 18, pp. 131–175. Elsevier, Amsterdamn (2017) 4. Castro, M.J., Ortega, S., Parés, C.: Well-balanced methods for the shallow water equations in spherical coordinates. Comput. Fluids 157, 196–207 (2017) 5. Castro M.J., Parés C.: Well-balanced high-order methods for systems of balance laws, to appear in J. Sci. Comp. (2020) 6. Cravero, I., Semplice, M.: On the accuracy of WENO and CWENO reconstructions of third order on nonuniform meshes. J. Sci. Comput. 477, 1219–1246 (2016)

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7. Gaburro, E., Castro, M.J., Dumbser, M.: Well-balanced Arbitrary-Lagrangian-Eulerian finite volume schemes on moving nonconforming meshes for the Euler equations of gas dynamics with gravity. Mon. Not. Roy. Astron. Soc. 477, 2251–2275 (2018) 8. Gottlieb, S., Shu, C.-W.: Total variation diminishing Runge-Kutta schemes. Math. Comput. 67, 73–85 (1998) 9. Harteh, A., Engquist, B., Osher, S., Chakravarthy, S.R.: Uniformly high order accurate essentially non-oscillatory schemes III. J. Comput. Phys. 131, 3–47 (1997) 10. Müller, L.O., Parés, C., Toro, E.F.: Well-balanced high-order numerical schemes for onedimensional blood flow in vessels with varying mechanical properties. J. Comput. Phys. 242, 53–85 (2013) 11. Sánchez-Linares C., Morales de Luna T., Castro M.J.: A HLLC scheme for Ripa model. Appl. Math. Comput. 272, 369–384 (2016) 12. Shu, C.-W.: Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws, Institute for Computer Applications in Science and Engineering (ICASE) (1997) 13. Van Leer, B.: Towards the ultimate conservative difference scheme, V. A second order sequel to Godunov’s method. J. Comput. Phys. 32, 101–136 (1979)

An All-Regime and Well-Balanced Lagrange-Projection Scheme for the Shallow Water Equations on Unstructured Meshes Christophe Chalons, Samuel Kokh and Maxime Stauffert

Abstract We are interested in the numerical approximation of the shallow water equations in two space dimensions. We propose a well-balanced, all-regime, and positive scheme. Our approach is based on a Lagrange-projection decomposition which allows to naturally decouple the acoustic and transport terms.

1 Governing Equations and Asymptotic Limit We consider the numerical approximation of the shallow water equations 

∂t h + ∇ · (hu) = 0, 2 ∂t (hu) + ∇ · (hu ⊗ u) + ∇ gh2 = −gh∇z,

(1)

where x ∈ R2 → z(x) denotes a given smooth topography and g > 0 is the gravity constant. Both the water depth h and the velocity u = (u 1 , u 2 ) ∈ R2 depend on the space and time variables, namely, x ∈ R2 and t ∈ [0, ∞). We assume that h(x, t = 0) = h 0 (x) and u(x, t = 0) = u0 (x) are given. We are interested in the design of a numerical scheme that satisfies the well-balanced property, i.e., able to strictly preserve the well-known lake at rest solutions such that h + z = constant and u = 0. For a review on well-balanced numerical schemes, we refer, for instance, the reader to [2, 8] and the references therein. We also refer to [5] where the authors focus on the 1D case and propose a well-balanced Lagrange-projection strategy. In [3],

C. Chalons (B) · M. Stauffert Laboratoire de Mathématiques de Versailles, UMR 8100, Université de Versailles Saint-Quentin-en-Yvelines, 78035 Versailles cedex, France e-mail: [email protected] S. Kokh CEA Saclay, 91191 Gif-sur-Yvette, France e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_11

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the proposed Lagrange-projection scheme is fully well-balanced, which means that it preserves a full set of equilibrium solutions (and not only the lake at rest). The Lagrange-projection methodology is especially well suited for subsonic or near lowFroude number flows. Indeed, it allows to design a natural implicit–explicit strategy which is uniformly stable with respect to the Froude number since the CFL time step restriction is driven by (slow) material waves and not by (fast) acoustic waves, see [6]. Regarding the uniform consistency, we will follow the anti-diffusive technique on the pressure numerical flux introduced in [7] and also used in [4]. The low-Froude limit. Before going further, we give the asymptotic behavior of the solutions of (1) in the low-Froude limit. With this in mind and as it is customary, we consider the dimensionless quantities t = t/T , x = x/L, h = h/h 0 , u = u/u 0 , and z = z/z 0 where T , L, h 0 , u 0 , and z 0 are, respectively, a reference time, length, water height, velocity, and topography of the flow and such that √ u 0 = L/T and z 0 = h 0 . Defining the Froude number by Fr = u 0 /c0 where c0 = gh 0 is the reference sound speed, easy calculations give the dimensionless equations 

∂t h + ∇ · (hu) = 0, ∂t (hu) + ∇ · (hu ⊗ u) +

1 ∇p Fr 2

= − Fr12 h∇z,

(2)

where p(h) = h 2 /2. Assuming that h and z can be written as h = h (0) + h (1) Fr + h (2) Fr 2 + O(Fr 3 )

and

u = u(0) + u(1) Fr + u(2) Fr 2 + O(Fr 3 ),

we get in particular p = p (0) + p (1) Fr + p (2) Fr 2 + O(Fr 3 ) = p(h (0) ) + h (1) h (0) Fr + O(Fr 2 ). Therefore, at order −2 and −1 with respect to the Froude number, (2) gives ∇ p (0) + h (0) ∇z = 0 and ∇ p (1) + h (1) ∇z = 0, which is equivalent to h (0) + z = H (t) and h (1) = h (1) (t), so that the asymptotic behavior is given by 

∂t h (0) + ∇ · (h (0) u(0) ) = 0, ∂t (h (0) u(0) ) + ∇ · (h (0) u(0) ⊗ u(0) ) + ∇ p (2) = −h (2) ∇z.

(3)

This system is not closed since p (2) and h (2) are not defined. We then consider a bounded domain  for the space variable and periodic boundary conditions, so that integrating (3) gives ∂t h (0) = 0 and therefore h (0) + z = H is constant both in space and time. This leads to ∇ · (h (0) u(0) ) = 0 and to the low-Froude limit 

z (0) (0) ∇ · u z = ∇(0)· ( H u ) (0) 1 − H ∂t u + ∇ · (u ⊗ u(0) ) +

1 ∇ p (2) H

= ∇ · ( Hz u(0) ⊗ u(0) ) − h (2) ∇ Hz . (4) Notice that when the topography is flat, i.e., z = 0, the classical incompressible Euler equations are recovered.

An All-Regime and Well-Balanced Lagrange-Projection Scheme …

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2 Acoustic-Transport Decomposition and Algorithm Let us now turn to the acoustic-transport decomposition of (1). If we develop the spatial derivatives and isolate the transport terms (u · ∇)ϕ, ϕ = h, hu, we can use an operator splitting with respect to time to obtain the acoustic and transport systems ∂t h + h∇ · (u) = 0, ∂t (hu) + hu(∇ · u) + ∇ p = −gh∇z,

(5)

∂t h + (u · ∇)h = 0, ∂t (hu) + (u · ∇)(hu) = 0.

(6)

With these notations, the proposed Lagrange-projection algorithm is defined as follows: for a given discrete state (h, hu)nj , j ∈ Z, at time t n , we define (h, hu)n+1 j at time t n+1 by a two-step process, namely, first update (h, hu)nj to (h, hu)n+1− by j n+1 to (h, hu) by solving (6). By solving solving (5), and then update (h, hu)n+1− j j we mean here approximating the solution for a period of time t. Let us note that if we set τ = 1/ h, (5) can be recast into g ∂t τ − τ (x, t)∇ · u = 0, ∂t u + τ (x, t)∇ p = −τ (x, t) ∇z. τ Following [4], we will choose to discretize (5) thanks to a pressure relaxation ⎧ ∂t τ − τ (x, t)∇ · u = 0, ⎪ ⎪ ⎨ ∂t u + τ (x, t)∇ = −τ (x, t) τg ∇z, ⎪ ⎪ ⎩ ∂t  + τ (x, t)a 2 ∇ · u = λ( p EOS (τ ) − ),

(7)

with p EOS (τ ) = g/(2τ 2 ), in the limit λ → +∞. This limit is accounted for by setting (x, t n ) = p EOS (τ (x, t n )), and then solving (7) with λ = 0. We add another approximation by replacing τ (x, t)∂xr with τ (x, t n )∂xr , r = 1, 2. To sum up, we will define our approximation of (5) by solving (7) over the time interval [t n , t n + t), with (x, t n ) = p EOS (τ (x, t n )) and λ = 0. Note that since (7) and (6) are rotational invariant, it is sufficient to focus on the corresponding quasi-1D systems ⎧ ∂t τ − τ (x, t n )∂x u 1 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂t u 1 + τ (x, t n )∂x  = −τ (x, t n ) g ∂x z, τ ⎪ ∂t u 2 = 0, ⎪ ⎪ ⎪ ⎪ ⎩ ∂t  + τ (x, t n )a 2 ∂x u 1 = 0,

(8)

and ∂t ϕ + u 1 ∂x ϕ = 0, ϕ ∈ {h, u 1 , u 2 }, see [4] for more details. Considering a local space step x j , the following discretization strategy of (8) was presented in [5]:

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⎧ n+1− n n t ⎪ τ u = τ − τ − u ⎪ j j j+1/2 j−1/2 j x ⎪ j

⎪ ⎪ ⎨ (u )n+1− = (u )n − τ n t  L , −  R, 1 j

1 j

j x j

⎪ = (u 2 )nj (u 2 )n+1− ⎪ j ⎪ ⎪ ⎪ ⎩ n+1− = n − τ n j

j

t j x j



j+1/2



j−1/2



(9)



a 2 u j+1/2 − u j−1/2 ,

, where nj = gh nj 2 /2 and the numerical fluxes u j+1/2 ,  Lj+1/2 and  R, j−1/2 are such that R, L , − n n n u j+1/2 = u  (U j , U j , U j+1 , U j+1 ),  j+1/2 =  (U j , U j , U j+1 , Unj+1 ),  j+1/2 = +  (U j , Unj , U j+1 , Unj+1 ), where U = (h, hu, , z)t and



 (U L , U R ) = u  (U L , UnL , U R , UnR ) =

(u 1 ) R − (u 1 ) L  L +  R −a , 2 2









 − L 1 (u 1 ) L + (u 1 ) R − R − {ghz} (UnL , UnR ), 2 2a 2a

1 n n n n ±  (U L , U L , U R , U R ) =  (U L , U R ) ± {ghz} (U L , U R ) 2 and {ghz} (UnL , UnR ) = g(h nL + h nR )(z R − z L )/2. If one chooses = n (resp. = n + 1− ), we get an explicit (resp. implicit) scheme. Note that {ghz} that accounts for the gravity source term is always evaluated at time t n . As far as the transport step is concerned, we consider a classic explicit scheme for ϕ ∈ {h, hu 1 , hu 2 }, ϕ n+1 = ϕ nj − j



t t n+1− n+1− ϕ u j+1/2 ϕ n+1− u j+1/2 − u j−1/2 , (10) j+1/2 − u j−1/2 ϕ j−1/2 − x j x j j

n+1− (resp. ϕ n+1− with ϕ n+1− j+1/2 = ϕ j j+1 ) if u j+1/2 ≥ 0 (resp. < 0). In the above formulas, a is taken as a local approximation  of the Lagrangian sound speed at each interface, a j+1/2 = 1.01 max(h j gh j , h j+1 gh j+1 ). For the detailed properties of (9)–(10), we refer the reader to [5]. Let us just recall that it is conservative in the usual sense of finite volume methods. Moreover, the scheme is well-balanced for lake at rest solutions, which means that if unj = 0 and h nj + z nj = h nj+1 + z nj+1 for all j ∈ Z, then = h nj and un+1 = unj , j ∈ Z. At last, the time-implicit scheme is stable under h n+1 j j a condition which does not depend either on the acoustic system or the sound speed c, but which only depends on the transport step and its material velocity u. This is of particular interest in the low-Froude regime, since the definition of t is uniform with respect to the Froude number. Truncation error in the low-Froude regime. In this paragraph, we consider the dimensionless shallow water equations and we motivate a correction to get the uniform consistency. The correction is similar to the one in [7] for low-Mach regimes and we focus on the explicit case = n. In the following, we will say that the flow is in the low-Froude regime if Fr 1 and ∂x p + h∂x z = O(Fr 2 ). Using the dimensionless

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quantities, the pressure numerical fluxes write ,n  Lj+1/2

nj

1 = 2+ Fr 2Fr 2

 R,n j+1/2

=

nj+1 Fr 2



h nj + h nj+1 a n n (z j+1 − z j ) − (u n − u nj ),  j+1 −  j + 2 2Fr j+1 

1 − 2Fr 2

nj+1

− nj

+

h nj + h nj+1 2

 (z j+1 − z j ) −

a (u n − u nj ). 2Fr j+1

If we compute the truncation errors and use the fact that nj+1 − nj + (h nj + h nj+1 )(z j+1 − z j )/2 = O(Fr 2 x), we obtain ,n =  Lj+1/2

 R,n j+1/2

=

nj Fr 2

+

nj+1 Fr 2

1 2Fr 2



h nj + h nj+1 x nj+1 − nj + (z j+1 − z j ) + O , 2 Fr

1 − 2Fr 2



h nj + h nj+1 x n n  j+1 −  j + (z j+1 − z j ) + O . 2 Fr

It is thus clear that the consistency errors are not uniform with respect to the Froude number. In order to avoid large errors in the numerical diffusion terms when Fr → 0, we suggest to replace the definition of  (U L , U R ) by



θ (U L , U R ) =









L +  R (u 1 ) R − (u 1 ) L − θ (U L , U R )a , 2 2

which, as long as we take θ j+1/2 = consistency. In O(Fr), now gives the uniform

n practice, we will set θ j+1/2 = min |u j+1/2 |/max(c j , c j+1 ), 1 . At last, recall that the scheme easily extends to 2D unstructured meshes by rotational invariance. Moreover, it is easily proved that both fully explicit (EXEX) and implicit–explicit (IMEX) schemes are well-balanced also in 2D.

3 Numerical Experiments We first consider a traveling vortex as in [1]. For this test case, we consider a flat bottom and we use a regular Cartesian mesh of 160 × 160 cells that discretizes the physical domain [0, 1] × [0, 1]. The boundary conditions imposed are periodic along the x-direction and absorbing boundaries along the y-direction. The mapping of the velocity magnitude is displayed in Fig. 1 and we can observe that the accuracy of the solution is really improved by the low-Froude correction. Furthermore, the accuracy of the solution between the EXEX and the IMEX scheme with low-Froude correction is comparable, whereas it took about 100 times less time steps and 10 times less CPU time computation with the IMEX than with the EXEX scheme.

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Fig. 1 Flat bottom. Velocity magnitude at final time T f = 0.1 with the EXEX scheme (top) and the IMEX scheme (bottom). θ = 1 (left), θ = O(Fr) (center), and exact solution (right)

Fig. 2 Non-flat bottom. Velocity magnitude at final time T f = 0.1 with the EXEX scheme (top) and IMEX scheme (bot) with θ = 1 (left) and θ = O(Fr) (right)

We now consider a non-flat bottom. We extend the physical domain of the traveling vortex test above to the rectangle [0, 2] × [0, 1]. The boundary conditions and initial conditions for h and u are the same as before. However, we consider here a topography   defined by z(x, y) = 10 exp −5(x − 1)2 − 50(y − 0.5)2 following [1]. We compare in Fig. 2 the results between EXEX and IMEX schemes, with or without low-Froude correction. Here again, the vortex structure of the flow is completely destroyed by numerical diffusion without low-Froude corrections θ = O(Fr), with both schemes. Finally, we can remark that the EXEX scheme took about 20 times more iterations and 15 times more CPU times than the IMEX scheme, both with low-Froude correction.

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References 1. Bispen, G., Arun, K.R, Lukacova, M., Noelle, S.: IMEX large time step finite volume methods for low Froude number shallow water flows. Commun. Comput. Phys. 16, 307–347 (2014) 2. Bouchut, F.: Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws: And Well-Balanced Schemes for Sources. Springer Science & Business Media, Berlin (2004) 3. Castro Díaz, M.J, Chalons, C., Morales de Luna, T.: A Fully Well-Balanced Lagrange-ProjectionType Scheme for the Shallow-Water Equations, SINUM (2018) 4. Chalons, C., Girardin, M., Kokh, S.: An all-regime Lagrange-Projection like scheme for the gas dynamics equations on unstructured meshes. Commun. Comput. Phys. 20(1), 188–233 (2016) 5. Chalons, C., Kestener, P., Kokh, S., Stauffert, M.: A large time-step and well-balanced LagrangeProjection type scheme for the shallow-water equations. Commun. Math. Sci. (2017) 6. Coquel, F., Nguyen, Q., Postel, M., Tran, Q.: Entropy-satisfying relaxation method with large time-steps for Euler IBVPs. Math. Comput. 79(271), 1493–1533 (2010) 7. Dellacherie, S.: Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number. J. Comput. Phys. 229(4), 978–1016 (2010) 8. Gosse, L.: Computing Qualitatively Correct Approximations of Balance Laws. Springer, Berlin (2013)

On the Dynamics of a Free Surface that Limits the Final Mass of a Self-Gravitating Gas N. P. Chuev

Abstract The Cauchy problem considered for a system of nonlinear integral equations of Volterra type that describes, with the use of Lagrangian coordinates, the motion of a finite mass of an ideal self-gravitating gas limited by a free boundary. A theorem of existence and uniqueness of the solution of the problem of motion of a rarefied gas in the space of infinitely differentiable functions is formulated and proved. The solution is constructed as a series with recursively calculated coefficients. The solutions obtained are used to study the dynamics of the free boundary.

Introduction The scientific direction on the movement of liquid and gas masses with free boundaries is one of the most important sections of modern hydroaerodynamics and is the subject of rigorous mathematical research. The main results in this direction were obtained by Lavrentiev, Ovsyannikov, Nalimov, Pukhnachev, Andreev, and others [1–4].

1 Formulation of the Problem Suppose that at an instant t = 0 in R 3 a area 0 is set, filled with an ideal, polytropic and isentropic gas, the particles of which are attracted to each other according to Newton’s law. The problem of gas motion in a force field reduces to determining t ∈ R 4 , occupied by the gas at time t, as well as the law of motion of the free surface, the velocity vector u(x, t), pressures p(x, t), and densities ρ(x, t), satisfying in x ∈ t , t ∈ (0, T) system of equations of gas dynamics in the form of Euler [5]:

N. P. Chuev (B) Ural State University of Railway Transport (USURT), Ekaterinburg, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_12

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1 du(x, t) + ∇ p(x, t) = ∇(x, t), dt ρ(x, t) dρ(x, t) + ρ(x, t)divu(x, t) = 0, dt

(1)

where ∇—gradient operator, div—divergence operator. At the instant t = 0 for all x = {x, y, z} in 0 , we know the distributions: u 0 (x)—is the gas velocity, p0 (x)— is the pressure, and ρ0 (x)—is the gas density. The functions u0 (x), ρ0 (x) and the closed boundary of the region 0 are defined in C ∞ (0 )—infinitely differentiable functions in 0 . At the boundary t of t , the condition is satisfied ρ(x, t) = 0, for x ∈ t , t ≥ 0. Remark We denote the functions u, v, F, x, and the variable ξ, η, z by vectors. The equation of state for polytropic gas is given in the form p = γ1 ρ γ , where γ > 1—adiabatic index. However, under this condition and with an arbitrary γ , the term ρ1 ∇ p in the system (1) may have a singularity at the boundary points. Therefore, it is further assumed that γ = 1 + mn , for n ∈ N , m ∈ N , and n ≥ 2. 1 We introduce the functions σ = ρ m that we substitute in the system (1). We then obtain the system of equations: du(x, t) + mσ n−1 ∇σ = ∇, dt 1 σt + σ divu + ∇σ u = 0, m

(2)

where σ (x, t)|t = 0 for t ≥ 0. The force of Newtonian attraction in the right-hand sides of the vector equations of systems (1)–(2) is F(x, t) = ∇(x, t), where ∇—Newtonian potential gradient, which is given by the triple integral [6–8]:  (x, t) = G t

ρ(x  , t)  dx , |x − x  |

(3)

where G—gravitational constant, |x − x  |—distance between points in a region. Consider a gas dynamics system without pressure or a rarefied self-gravitating gas, which has the following form: du(x, t) = F(x, t), dt dρ(x, t) = −ρ(x, t)divu(x, t). dt

(4)

For x ∈ t , t ≥ 0, ρ|t = 0, σ |t = 0 systems (2) and (4) are equivalent. Therefore, to solve the problem of determining the law of motion of the free surface t for t ≥ 0, we will consider the system (4).

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We will consider the gas motion under the condition that the free boundary at all instants of time consists of the same particles, that is, the possibility of mass transfer through the free surface is excluded. This circumstance, as well as the gas rarefaction condition, makes it convenient to switch from the Euler coordinates (x, t) to the Lagrangian coordinates (ξ , t) for a one-to-one mapping ξ → x(ξ , t) is sufficiently smooth, and the existence of a differentiable inverse transform is satisfiable provided that the Jacobian of the mapping   ∂x ∂(x, y, z) = det J (ξ , t) = ∂(ξ, η, ζ ) ∂ξ

(5)

is nonzero. Jacobian of the mapping J (ξ , t) satisfies the conditions J (ξ , 0) = 1 and J (ξ , t) > 0 when t ≥ 0. Transform the gas dynamics system (4) and find its form in the Lagrangian coordinates (ξ, t) in a known manner [2–5]. If we consider x = {x, y, z} as a function of independent variables ξ, η, ζ, t, then at the time t the particle velocity will be ∂ x(ξ , t) = u (x(ξ , t), t) = v(ξ , t). ∂t

(6)

We use the differential equality for functions in the transition from Euler to Lagrangian variables f (x, t) = f (x(ξ , t), t) = g(ξ , t) df ∂g = , dt ∂t then acceleration is

du ∂ = dt ∂t



∂x ∂t

 =

∂2x . ∂t 2

(7)

The continuity equation in the Lagrange variables has the form [5] ρ (x(ξ , t), t) J (ξ , t) = ρ0 (ξ ).

(8)

Since ρ (x(ξ , t), t) = ρ (ξ , t), we deduce ρ (ξ , t)J (ξ , t) = ρ0 (ξ ),

(9)

where ρ0 (ξ ) is the initial density for each point 0 . In the vector equation of the system (4), we pass to the Lagrangian coordinates, after calculating the gradient of potential (ξ , t). Then the force function F(x, t) based on the theorem on changing a variable in a multiple integral and for x  = x(η, t), we obtain

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 F (x(ξ , t), t) = G

x(η, t) − x(ξ , t) J (x(η, t)) dη, |x(ξ , t) − x(η, t)|3

ρ (x(η, t), t) 0





(10)



∂(x ,y ,z )    where J (η, t) = ∂(ξ  ,η ,ζ  ) —Jacobian conversion (5) for η = {ξ , η , ζ } ∈ 0 . In view of the (7)–(10), we have the system (4) in the Lagrange variables:

∂2x =G ∂t 2

 ρ0 (η) 0

x(η, t) − x(ξ , t) dη = F (x(ξ , t)) , |x(ξ , t) − x(η, t)|3 ρ (ξ , t)J (ξ , t) = ρ0 (ξ ).

(11)

For the system [11], the Cauchy problem is posed with the following initialboundary conditions: x(ξ , 0) = ξ ,

∂ x(ξ , t) |t=0 = u0 (ξ ), J (ξ , 0) = 1, ρ (ξ , 0) = ρ0 (ξ ), ∂t ρ (ξ , t) = 0 when ξ ∈ 0 and t ≥ 0 for ∀ξ ∈ 0 .

(12)

The functions u0 (ξ ), ρ0 (ξ ) and closed border region 0 belong to space C ∞ (0 )— infinitely differentiable functions. The Cauchy problem for an integrodifferential system of Eq. (11) is equivalent to a system of integral equations of Volterra type:  x(ξ , t) = ξ + u0 (ξ )t +

t

(t − τ )F(x(ξ , τ ), τ )dτ

(13)

0

with initial (12) and additional conditions for ∀ξ ∈ 0 :  max

sup|ξ |, sup|u 0 (ξ ) |, sup|ρ0 (ξ ) |,  ρ0 (η) |ξη−ξ dη| sup|F 0 | = G −η|3

 = A < ∞.

(14)

0

The equivalence of Eqs. (11) and (13) is easily verified by differentiation and secondary integration. The solution of the system of integral equations (13) x = x(ξ , t) from a given initial density value allows us to determine the gas density for t ≥ 0 ρ(x(ξ , t), t) = ρ0 (ξ )J (ξ , t)−1 .

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2 The Existence and Uniqueness Theorem for the Solution of the Cauchy Problem for an Integral Equation of Volterra Type Theorem on the existence of a solution (13) proved by the method of successive approximations [9, 10] for (12), (14). Continuity of functions x = x(ξ , t) and space C ∞ (0 ) affiliation proved by the method of mathematical induction. The proof of the uniqueness of the solution using the Hadamard lemma and the Gronwall inequality [11]. ¯ 0 × [0, t1 ] for 0 ≤ t ≤ t1 has Theorem Integral equation (13) in the region Q t =  a unique solution satisfying the initial-boundary conditions (12), (14) in C ∞ (0 ) in the form of an absolutely and uniformly converging series. The general term of the series can be calculated using the recurrence formula  x n+1 (ξ , t) = ξ 0 + u0 (ξ )t +

t

(t − τ ) · F(x n (ξ , τ ), τ )dτ.

(15)

0

The solution x(ξ , t) satisfies the inequality 2

|x(ξ , t)| ≤ max|ξ | +

max|u0 (ξ )|t + max|F 0 (ξ )| t2 , 1 − Dt 2

(16)

where D—fixed number. Differentiable inverse transformation ξ = x −1 (x, t) exists for the obtained solution x = x(ξ , t) for J (ξ , t) > 0. Consequently, ρ(x, t) : ρ (ξ , t) = ρ (x −1 (x, t), t) = ρ(x, t). Similarly, from equality (16), we calculate u(x, t) :

∂ x(ξ , t) = v(ξ , t) = v x −1 (x, t), t = u(x, t). ∂t Note that ρ(x, t) = 0 is the equation of the free surface t for t ≥ 0. The functions ρ(x, t), ρ (ξ , t), v(ξ , t), and u(x, t) belong to the space C ∞ for all variables. The solution of the integral equation (13) x = x(ξ , t) is defined for ξ ∈ 0 , t ≥ 0. Then the free boundary is the image of the points 0 under the map ξ ∈ 0 to the points x ∈ t . Thus, the problem of determining the law of motion of the free boundary is also the problem of finding a map x = x(ξ , t).

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3 An Example of the Evolution of the Free Surface of a Gaseous Sphere We apply the obtained results to the study of the free surface of a rotating gaseous sphere of radius R provided that the forces of Newtonian mutual attraction act on the gas particles. At the instant of time t = 0, there is rotation of the gaseous sphere around the fixed axis O Z of an orthogonal coordinate system with angular velocity ω as a solid. The law of motion of the free surface is determined by the solution of the system of integral equations (13) for u(ξ , 0) = ω × r = {−ωη, ωξ, 0}, when ξ ∈ 0 , r = {ξ, η, ζ }, ω = {0, 0, ω}. In this example, we obtain the approximate law of motion of the free surface t for t ≥ 0. Using the first approximation (15) of the solution of the integral equation (13): x(ξ , t) ≈ x 1 = ξ + u0 (ξ )t + F0

t2 t2 = ξ + (ω × r)t + F 0 (ξ ) , 2 2

(17)

provided that with ξ ∈ 0 , ξ = {ξ, η, ζ }, F 0 (ξ ) = {−F0 ξ, − F0 η, − F0 ζ }, where F0 —positive constant. Solution of the system (17) is  2 1 − F0 t2 x + ωt y ξ=  ; η= 2 2 1 − F0 t2 + ω2 t 2

 2 1 − F0 t2 y − ωt x z ; ζ =  2 . 2 2 1 − F0 t2 1 − F0 t2 + ω2 t 2

Given that the sphere 0 for t = 0 is given by the equation ξ 2 + η2 + ζ 2 = R 2 , we obtain the approximate equation of the free surface t in the form of an ellipsoid of rotation x 2 + y2 z2

 +  2 = 1. 2 2 2 R 2 1 − F0 t2 R 2 1 − F0 t2 + ω2 t 2 We emphasize that the configurations of the free surface of the rotating gaseous  2 sphere are considered for t < F0 . Ellipsoid of rotation for t → F20 approaches to a circular disk in the plane X OY. In this case, the semi-axis c on O Z approaches to √ a zero, semi-axis a = b on O X and OY approaches  to a value of more 2R. For ω = 0, the gaseous sphere begins to collapse at t → sphere approaches to a infinity.

2 , F0

and for t >

2 F0

radius

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Note that, based on the initial conditions of the above example, Eq. (15) has an exact solution that describes the oscillatory motion of a rotating self-gravitating gaseous sphere.

References 1. Lavrentiev, M.A., Shabat, B.V.: Problems of hydrodynamics and their mathematical models, p. 416. Nauka, Moscow (1973) 2. Ovsyannikov, L.V.: General equations and examples, p. 75. Nauka. Sib. otdelenie, Novosibirsk (1967) 3. Nalimov, V.I., Pukhnachev, V.V.: Unsteady motion of an ideal fluid with a free boundary, p. 174. NGU, Novosibirsk (1975) 4. Andreev, V.K.: Stability of transient fluid movements with a free boundary, p. 136. Nauka, Novosibirsk (1992). ISBN 5-02-029967 5. Ovsyannikov, L.V.: Lectures on the basics of gas dynamics, p. 336. Izhevsk, Institut kompyuternih issledovanii (2003) 6. Gunter, N.M.: Potential theory and its application to the basic problems of mathematical physics, p. 415. Gostehizdat (1953) 7. Sretensky, L.N.: Theory of Newtonian potential, p. 318. OGIZ, Moscow-Leningrad (1946) 8. Godunov, S.K.: Equations of mathematical physics, p. 392. Nauka, Moscow (1979) 9. Vasilieva, A. B., Tikhonov, N.A.: Integral equations, p. 160. FIZMATLIT, Moscow (2002). ISBN 5-9221-0275-3 10. Smirnov, N.S.: Introduction to the theory of nonlinear integral equations, p. 125. Leningrad, Moscow (1936) 11. Hartman, F.: Ordinary differential equations, p. 720. Mir, Moscow (1970)

Exponential Stability of Solutions to Delay Difference Equations with Periodic Coefficients G. V. Demidenko and D. Sh. Baldanov

Abstract In the paper, we consider systems of linear delay difference equations with periodic coefficients. We establish sufficient conditions for the asymptotic stability of the zero solution to the systems and obtain estimates of the stabilization rate of solutions at infinity. When obtaining the results we use a discrete analog of Lyapunov– Krasovskii functional.

Introduction In the paper, we consider systems of linear delay difference equations with periodic coefficients (1) xn+1 = A(n)xn + B(n)xn−τ (n) , n = 0, 1, 2, . . . , where {A(n)}, {B(n)} are sequences of N -periodics (m × m)-matrices and τ (n) ∈ N is a delay parameter. Our aim is to study the asymptotic stability of the zero solution to systems of the form (1) and to obtain estimates for solutions {xn } characterizing the decay as n → +∞. The first results on the stability of solutions to ordinary difference equations were obtained more than a hundred years ago in the works of A. Poincare and O. Perron. However, systematical research on the stability theory for difference equations began in the 1950s (see, for example, [1–3]). During this period, the study of the stability theory of difference equations was caused by the development of numerical methods and mathematical modeling. Nowadays, there are a number of monographs describG. V. Demidenko (B) Sobolev Institute of Mathematics, Acad. Koptyug avenue 4, 630090 Novosibirsk, Russia e-mail: [email protected] Novosibirsk State University, Pirogov street 1, 630090 Novosibirsk, Russia D. Sh. Baldanov Novosibirsk State University, Pirogov street 1, 630090 Novosibirsk, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_13

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ing various results on the stability theory for difference equations and methods by which they are obtained (see, for example, [4–6]). In the last quarter of a century, active research on the stability theory for delay difference equations began to be carried out (see, for example, [7–14]). In particular, some studies on stability have been conducted for equations of the form (1) (see, for example, [9, 13, 14]). In stability studies, there were used analogs of methods from the theory of functional differential equations (spectral methods, the method of Halanay-type inequalities, the method of Lyapunov functions, construction of a solution in the operator form and establishment of a connection between differential equations and delay difference equations, and so on). In our paper, when studying the asymptotic stability of the zero solution to (1) we will use a discrete analog of the Lyapunov–Krasovskii-type functional t K (t − s)y(s), y(s)ds,

V (t, y) = H (t)y(t), y(t) +

(2)

t−τ

H (t) = H ∗ (t) > 0,

K (s) = K ∗ (s) > 0, s ∈ [0, τ ],

introduced in [15] for studying the asymptotic stability of solutions to delay differential equations d y(t) = A(t)y(t) + B(t)y(t − τ ), dt where A(t + T ) = A(t),

B(t + T ) = B(t), t > τ > 0.

The discrete analog of the functional (2) considered on solutions {xn } to system (1) has the following form: v(n, x) = H (n)xn , xn  +

n−1 

K n− j−1 x j , x j ,

(3)

j=n−τ

where H (n) = H (n + N ), K 0 , K 1 , . . . , K τ are some Hermitian positive definite matrices. Using this functional, we establish sufficient conditions for the asymptotic stability of the zero solution to system (1) and obtain estimates of the decay rate of the norm of solution xn  as n → ∞. In the case B = 0, the obtained results coincide with those well known for difference equations with constant coefficients [16].

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1 Delay Difference Equations Consider a system of linear delay difference equations with periodic coefficients (1). Assume that the delay parameter is bounded: 1 ≤ τ (n) ≤ τ < ∞. Then system (1) can be written as the following system of linear difference equations with variable coefficients: xn+1 = A(n)xn +

τ 

B j (n)xn− j , n = 0, 1, . . . ,

(4)

j=1



where B j (n) =

B(n) for τ (n) = j, 0 for τ (n) = j.

When studying the asymptotic stability of the zero solution to system (4), we will use Lyapunov–Krasovskii-type functionals of the form (3). In the following theorem, sufficient conditions of the asymptotic stability of the zero solution to system (1) are formulated. Theorem 1 Assume that there exist Hermitian positive definite matrices H (n), K j ,

j = 0, 1, . . . , τ,

(5)

such that H (0) = H (N ),  j = K j−1 − K j > 0,

j = 1, . . . , τ,

and composite matrices ⎛

A∗ (n)H (n + 1)B1 (n) C11 (n) ⎜ C(n) = − ⎜ . .. ⎝ .. . ∗ ∗ Bτ (n)H (n + 1)A(n) Bτ (n)H (n + 1)B1 (n)

where

C00 (n) B1∗ (n)H (n + 1)A(n)

⎞

. . . A∗ (n)H (n + 1)Bτ (n) . . . B1∗ (n)H (n + 1)Bτ (n) ⎟ ⎟ , (6) .. .. ⎠ . . ... Cτ τ (n)

C00 (n) = A∗ (n)H (n + 1)A(n) − H (n) + K 0 , 1 C j j (n) = B ∗j (n)H (n + 1)B j (n) −  j , 2

j = 1, . . . , τ − 1,

Cτ τ (n) = Bτ∗ (n)H (n + 1)Bτ (n) − K τ

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are also positive definite for n = 0, . . . , N − 1. Then the zero solution to system (1) is asymptotically stable. Proof Let the initial data be given for system (1): x0 , x−1 , . . . , x−τ .

(7)

Then there exists a unique solution {xn } to the initial value problem (1), (7). Using matrices (5), we consider the functional of the form (3) v(n, x) = H (n)xn , xn  + K 0 xn−1 , xn−1  + · · · + K τ −1 xn−τ , xn−τ , n = 0, 1, 2, . . . . By the conditions on H (n) and {K j }, the functional v(n, x) is strictly positive. We write the difference v(n + 1, x) − v(n, x) using the notation for matrices  j . We have v(n + 1, x) − v(n, x) = H (n + 1)xn+1 , xn+1  − H (n)xn , xn  +

n 

K n− j x j , x j  −

j=n+1−τ

n−1 

K n− j−1 x j , x j 

j=n−τ

= H (n + 1)xn+1 , xn+1  − H (n)xn , xn  + K 0 xn , xn  n−1 

−

n− j x j , x j  − K τ xn−τ , xn−τ .

j=n−τ

Taking into account that {xn } is a solution to the equivalent problem (4), (7), we obtain v(n + 1, x) − v(n, x) = A∗ (n)H (n + 1)A(n)xn , xn  −H (n)xn , xn  + K 0 xn , xn  +

τ  A∗ (n)H (n + 1)B j (n)xn− j , xn  j=1

+

τ  B ∗j (n)H (n + 1)A(n)xn , xn− j  j=1

+

τ  j,i=1

B ∗j (n)H (n + 1)Bi (n)xn−i , xn− j 

Exponential Stability of Solutions to Delay Difference Equations …

−

n−1 

97

n− j x j , x j  − K τ xn−τ , xn−τ .

j=n−τ

Using matrix (6) this expression can be rewritten in the form ⎛

⎞ ⎛ ⎞ xn xn ⎜ ⎟ ⎜ ⎟ v(n + 1, x) − v(n, x) = − C(n) ⎝ ... ⎠ , ⎝ ... ⎠ xn−τ xn−τ 

−

1 2

n−1 

n− j x j , x j  − τ xn−τ , xn−τ .

(8)

j=n−τ +1

By the conditions of the theorem all matrices C(n) are positive definite. Recall that τ (n) ≤ τ , so from the definitions of matrices B j (n) it follows that from the entire set of matrices {C(n)}, n ∈ N, the number of distinct ones is at most τ . Hence, there exists a constant c1 > 0 such that for all n ∈ N the estimate holds ⎛

⎞ ⎛ ⎞ xn xn τ  ⎜ ⎟ ⎜ ⎟ C(n) ⎝ ... ⎠ , ⎝ ... ⎠ ≥ c1 xn−i 2 . i=0 xn−τ xn−τ



(9)

Since all matrices { j } are positive definite, then from (8) we obtain the inequality 0 ≤ v(n + 1, x) ≤ v(n, x) − c1

τ 

xn−i 2 ≤ v(n, x) − c1 xn 2 .

i=0

It follows from this estimate that the sequence {v(n, x)} is monotonically decreasing and bounded from below. Therefore, according to the Weierstrass theorem, it has a limit. Hence, by virtue of the uniqueness of the limit 0 ≤ c1 xn 2 ≤ v(n, x) − v(n + 1, x) → 0 as n → ∞. This implies the asymptotic stability of the zero solution to system (1). Theorem is proved.

2 Estimates for Solutions to Delay Difference Equations In this section, we obtain estimates for solutions to the initial value problem (1), (7), from which it follows that the norm of the solution decreases at infinity with exponential rate.

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Theorem 2 Assume that the conditions of Theorem 1 hold. Let κ j ∈ (0, 1), j = 1, 2, . . . , τ , be such that 1 − i + κi K i−1 ≤ 0, i = 1, . . . , τ − 1, −τ + κτ K τ −1 ≤ 0. 2

(10)

Then for functional (3) the estimate holds v(n, x) ≤

n−1

(1 − ε j )v(0, x),

(11)

j=0

where v(0, x) = H (0)x0 , x0  +

−1 

K − j−1 x j , x j ,

j=−τ

 ε j = min κ1 , . . . , κτ ,

 c1 , H ( j)

(12)

c1 > 0 is a constant mentioned in (9). Proof For any solution {xn } to the initial value problem (1), (7) it was established identity (8). By the conditions of the theorem, matrices C(n) are positively definite, and by virtue of estimate (9), we have ⎞ ⎛ ⎞ xn xn c1 ⎟ ⎜ ⎟ ⎜ H (n)xn , xn . C(n) ⎝ ... ⎠ , ⎝ ... ⎠ ≥ H (n) xn−τ xn−τ



⎛

(13)

The inequality follows from the conditions of the theorem: 1 2

≥

n−1 

n− j x j , x j  + τ xn−τ , xn−τ 

j=n−τ +1

n−1 

κn− j K n− j−1 x j , x j  + κτ K τ −1 xn−τ , xn−τ .

(14)

j=n−τ +1

Taking into account estimates (13) and (14) from identity (8), we obtain the inequality v(n + 1, x) ≤ v(n, x) −

n−1  c1 κn− j K n− j−1 x j , x j . H (n)xn , xn  − H (n) j=n−τ

Hence, by virtue of definition (12), we have

Exponential Stability of Solutions to Delay Difference Equations …

⎡

99

⎤

n−1 

v(n + 1, x) ≤ v(n, x) − εn ⎣H (n)xn , xn  +

K n− j−1 x j , x j ⎦

j=n−τ

= (1 − εn )v(n, x). Repeating similar reasoning for {v(k, x)}, k = n − 1, . . . , 1, we obtain inequality (11). Theorem is proved. Corollary Assume that the conditions of Theorem 2 hold. Then for the solution to the initial value problem (1), (7) the inequality holds xn 2 ≤ (h 1 (n))−1

n−1

 (1 − ε j ) H (0)x0 , x0  +

j=0

−1 

 K −l−1 xl , xl  ,

(15)

l=−τ

where h 1 (n) > 0 is minimal eigenvalue of matrix H (n), ε j ∈ (0, 1) is defined in (12). Estimate (15) follows directly from inequality (11). Let n ≥ k N , then inequality (15) can be written in the form xn 2 ≤ (h 1 (n))−1

n−1

j=k N

 (1 − ε j ) exp k

N −1 

 ln(1 − εi ) v(0, x).

i=0

Since εi ∈ (0, 1), then by virtue of this inequality the norm of the solution xn  to system (1) decreases as n → ∞ with exponential rate for any initial data (7). Remark 1 In the case of constant matrices A and B, Theorem 2 was proved in [17]. Remark 2 In the case of linear difference equations with periodic coefficients, similar estimates are contained in [18, 19]. Acknowledgements The work is supported by the Russian Foundation for Basic Research (project no. 19-01-00754).

References 1. Koval’, P.I.: Reducible systems of difference equations and stability of their solutions (in Russian). Uspekhi Mat. Nauk. 12(6), 143–146 (1957) 2. Hahn, W.: Uber die Adwendung der Methode von Ljapunov auf Differenzengleichungen. Math. Ann. 136(1), 430–441 (1958)

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3. Jury, E.I.: A simplified stability criterion for linear discrete systems. ERL Report Series. No. 60, 373 (1961) 4. Hahn, W.: Theorie und Anwendung der direkten Methode von Ljapunov. Springer, Berlin (1959) 5. Halanay, A., Wexler, D.: Qualitative Theory of Impulse Systems. Mir, Moscow (1971). (in Russian) 6. Elaydi, S.N.: An Introduction to Difference Equations. Springer, New York (1999) 7. Gyori, I., Pituk, M.: Asymptotic formulae for the solutions of a linear delay difference equation. J. Math. Anal. Appl. 195(2), 376–392 (1995) 8. Erbe, L.H., Xia, H., Yu, J.S.: Global stability of a linear nonautonomous delay difference equation. J. Differ. Equ. Appl. 1(2), 151–161 (1995) 9. Yu, J.S.: Asymptotic stability of a linear difference equation with variable delay. Comput. Math. Appl. 36(10–12), 203–210 (1998) 10. Agarwal, R.P., Kim, Y.H., Sen, S.K.: Advanced discrete Halanay-type inequalities: stability of difference equations. J. Inequal. Appl. 2009, Article ID 535849 (2009) 11. Berezansky, L., Braverman, E.: Exponential stability of difference equations with several delays: recursive approach. Adv. Differ. Equ. Article ID 104310 (2009) 12. Khusainov, D.Ya., Shatyrko, A.V.: Research of absolute stability of the difference system with delay via the second method of Lyapunov (in Russian). J. Vych. Prikl. Mat. No. 4, 118–126 (2010) 13. Kulikov, A.Yu.: Stability of a linear nonautonomous difference equation with bounded delays. Russ. Math. 54(11), 18–26 (2010) 14. Kulikov, A.Yu., Malygina, V.V.: Stability of a linear difference equation and estimation of its fundamental solution. Russ. Math. 55(12), 23–33 (2011) 15. Demidenko, G.V., Matveeva, I.I.: Stability of solutions to delay differential equations with periodic coefficients of linear terms. Sib. Math. J. 48(5), 824–836 (2007) 16. Daleckii, Ju.L., Krein, M.G.: Stability of Solutions of Differential Equations in Banach Space. American Mathematical Society, Providence (1974) 17. Demidenko, G.V., Baldanov, D.Sh.: Asymptotic stability of solutions to delay difference equations. J. Math. Sci. 221(6), 815–825 (2017) 18. Aidyn, K., Bulgakov, H.Ya., Demidenko, G.V.: Numeric characteristics for asymptotic stability of solutions to linear difference equations with periodic coefficients. Sib. Math. J. 41(6), 1005– 1014 (2000) 19. Demidenko, G.V.: Matrix Equations. Novosibirsk State University, Novosibirsk (2009)

On Estimates of Solutions to One Class of Functional Difference Equations with Periodic Coefficients G. V. Demidenko and I. I. Matveeva

Abstract In the present paper, we consider a class of functional difference equations with periodic coefficients. We establish criteria for asymptotic stability of the zero solution to the equations and obtain estimates characterizing the decay rates of solutions to these equations at infinity.

1 Introduction In the present paper, we consider a class of systems of functional difference equations with periodic coefficients x(t) = A(t)x(t − τ ),

t ≥ 0,

(1)

where τ > 0 is the delay parameter, A(t) is a matrix of dimension n × n with continuous T -periodic entries, i.e., A(t + T ) ≡ A(t), and T = mτ , m > 0 is integer. Systems of the form (1) are often called continuous-time delay difference systems in the literature. Our aim is to study asymptotic stability of the zero solution to (1) and to obtain some estimates of solutions that characterize their exponential decay as t → ∞. The problem of stability of solutions is one of the basic problems in the theory of functional difference equations. At present, there is a great number of papers devoted to the study of this problem for various classes of equations. For example, for the system of linear difference equations yn+1 = B(n)yn , n = 0, 1, . . . ,

B(n) = B(n + N ),

(2)

G. V. Demidenko · I. I. Matveeva (B) Sobolev Institute of Mathematics, Acad. Koptyug avenue 4, 630090 Novosibirsk, Russia e-mail: [email protected] G. V. Demidenko e-mail: [email protected] Novosibirsk State University, Pirogov street 1, 630090 Novosibirsk, Russia © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_14

101

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it is well known that the zero solution is asymptotically stable if and only if the spectrum of the monodromy matrix X (N ) = B(N − 1) · · · B(1)B(0)

(3)

belongs to the unit disk ω = {λ ∈ C : |λ| < 1}. In particular, in the case of constant coefficients B(n) = B, n ≥ 0, the zero solution is asymptotically stable if and only if the spectrum of B belongs to ω (for instance, see [1–4]). From the theoretical viewpoint, the spectral criterion gives the comprehensive answer to the question on asymptotic stability of the zero solution to (2). However, it is well known that the problem of finding eigenvalues of non-Hermitian matrices is ill-conditioned from the viewpoint of perturbation theory (for example, see [5]). Therefore, in solving particular problems on stability of solutions to difference equations, other methods are used along with the spectral criterion, for example, the method based on the Lyapunov function. In the case of constant coefficients in (2), constructing the Lyapunov function is reduced to finding a solution to the Lyapunov discrete equation H − B ∗ H B = C, C = C ∗ > 0.

(4)

Moreover, according to the Lyapunov criterion, asymptotic stability of the zero solution to (2) is equivalent to the existence of a positive definite solution H = H ∗ > 0 to the matrix equation (4) (for instance, see [1]). Using the solution H = H ∗ > 0 to (4) with C = I , one can obtain the following estimate for solutions to (2) with constant coefficients:  yn  ≤ 1 −

1 H 

n/2



H y0 , n = 1, 2, . . .

(5)

(for example, see [6, 7]). In particular, (5) characterizes the decay rate of solutions yn  → 0 as n → ∞. Using this solution H , in the case of constant coefficients, S.K. Godunov together with his disciples elaborated algorithms for numerical study of asymptotic stability with guaranteed accuracy. Using the matrix difference Lyapunov equation H (l) − B ∗ (l)H (l + 1)B(l) = C(l), C(l) = C ∗ (l) > 0, l = 0, 1, . . . , N − 1,

(6)

asymptotic stability of the zero solution to (2) was studied in [8–11]. A criterion for asymptotic stability of the zero solution to (2) in terms of solvability of (6) was established in [12]. This criterion is an analog of the Lyapunov criterion for difference equations with constant coefficients. In the present paper, we prove criteria for asymptotic stability of the zero solution to (1) and obtain estimates for solutions to the initial value problem for (1)

On Estimates of Solutions to One Class of Functional Difference Equations …



103

x(t) = A(t)x(t − τ ), t ≥ 0, x(ξ) = ϕ(ξ), ξ ∈ [−τ , 0],

(7)

where ϕ(ξ) ∈ C([−τ , 0]) is a given vector function. It should be noted that these results have important applications to delay differential equations of neutral type d [x(t) − A(t)x(t − τ )] = B(t)x(t) + C(t)x(t − τ ). dt The present paper is a continuation of our investigations of stability of solutions to nonautonomous delay equations (for instance, see [13–19]).

2 Criteria for Asymptotic Stability Further, we suppose that the matrix A(t) is nonsingular for t ≥ 0. Let the matrix X (t) be a solution to the initial value problem for the functional difference equation  X (t) = A(t)X (t − τ ), t ≥ 0, X (ξ) = I, ξ ∈ [−τ , 0]. Using X (t), a solution to the initial value problem (7) can be written in the form x(t + (k − 1)τ ) = X (t + (k − 1)τ )ϕ(t − τ ), t ∈ [0, τ ), k = 1, 2, . . . . Obviously, X (t + (k − 1)τ ) = A(t + (k − 1)τ ) . . . A(t), t ∈ [0, τ ), k = 1, 2, . . . . Since A(t) is T -periodic and T = mτ , then X (t + pT + lτ ) = X (t + lτ )(X (t + T − τ )) p , p ≥ 0, 0 ≤ l ≤ m − 1, t ∈ [0, τ ). We call the matrix X (t + T − τ ) the monodromy matrix of (1). Obviously, the zero solution to (1) is asymptotically stable if and only if the spectrum of the monodromy matrix belongs to the unit disk ω = {λ ∈ C : |λ| < 1} for t ∈ [0, τ ]. Introduce the following matrices: Y (t, k, l) =

k−1  j=l

 A(t + jτ ) =

A(t + (k − 1)τ ) . . . A(t + lτ ) for k > l, I for k = l.

(8)

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By definition, Y (t, k, 0) = X (t + (k − 1)τ ). Then Y (t, m, 0) = X (t + T − τ ) is the monodromy matrix of (1). Consider Y (t, m + 1, 1). By (8) and T -periodicity of the matrix A(t), we obviously have Y (t, m + 1, 1) = A(t)A(t + (m − 1)τ ) · · · A(t + τ ). Since the matrix A(t) is nonsingular, taking into account (8), we obtain Y (t, m + 1, 1) = A(t)X (t + T − τ )A−1 (t). Hence, the spectrum of the matrix Y (t, m + 1, 1) coincides with the spectrum of the monodromy matrix X (t + T − τ ). Similarly, we can prove that the spectrum of the matrix Y (t, m + l, l) coincides also with the spectrum of the matrix X (t + T − τ ). We will use the matrices Y (t, k, l) in order to obtain our results. Consider the following boundary value problem for the Lyapunov functional difference equations with the unit matrix in the right-hand side H (t + lτ ) − A∗ (t + lτ )H (t + (l + 1)τ )A(t + lτ ) = I,

(9)

H (t) = H (t + T ), l = 0, 1, . . . , m − 1, t ∈ [0, τ ). The next theorem holds. Theorem 1 I. Let the zero solution to (1) be asymptotically stable. Then (9) has a unique solution {H (t + lτ )}; moreover, H (t + lτ ) = H ∗ (t + lτ ) > 0. II. Let (9) have a solution {H (t + lτ )} such that H (t + lτ ) = H ∗ (t + lτ ) > 0. Then the zero solution to (1) is asymptotically stable. Proof I. Prove the first part of the theorem. Let the zero solution to (1) be asymptotically stable, i.e., the spectrum of the monodromy matrix X (t + T − τ ) belongs to the unit disk ω for t ∈ [0, τ ]. We now prove the unique solvability of (9). If a solution {H (t + lτ )} exists then H (t) = A∗ (t)H (t + τ )A(t) + I, H (t + τ ) = A∗ (t + τ )H (t + 2τ )A(t + τ ) + I, ··················································· H (t + T − τ ) = A∗ (t + T − τ )H (t + T )A(t + T − τ ) + I, H (t) = H (t + T ). Substituting H (t + τ ) defined by the second equality to the first one, we have H (t) = I + A∗ (t)A(t) + (A(t + τ )A(t))∗ H (t + 2τ )A(t + τ )A(t). By (8), the expression can be represented as follows: H (t) = I + Y ∗ (t, 1, 0)Y (t, 1, 0) + Y ∗ (t, 2, 0)H (t + 2τ )Y (t, 2, 0).

On Estimates of Solutions to One Class of Functional Difference Equations …

105

Arguing in a similar way, we obtain H (t) = I + Y ∗ (t, 1, 0)Y (t, 1, 0) + Y ∗ (t, 2, 0)Y (t, 2, 0) + . . . +Y ∗ (t, m − 1, 0)Y (t, m − 1, 0) + Y ∗ (t, m, 0)H (t + T )Y (t, m, 0). Since H (t) = H (t + T ), then the matrix H (t) has to satisfy the Lyapunov discrete equation H (t) − Y ∗ (t, m, 0)H (t)Y (t, m, 0) = I + Y ∗ (t, 1, 0)Y (t, 1, 0) + · · · + Y ∗ (t, m − 1, 0)Y (t, m − 1, 0).

(10)

Exactly as above, we obtain that the matrices H (t + lτ ), l = 1, . . . , m − 1, have to satisfy the Lyapunov discrete equations H (t + lτ ) − Y ∗ (t, l + m, l)H (t + lτ )Y (t, l + m, l) = I + Y ∗ (t, l + 1, l)Y (t, l + 1, l) + · · · + Y ∗ (t, l + m − 1, l)Y (t, l + m − 1, l). (11) But the spectra of the matrices Y (t, m + l, l) belong to the unit disk ω. Hence, each of the equations is uniquely solvable and its solution H (t + lτ ) is positive definite. Thus, if a solution to (9) exists, then it is unique; moreover, H (t + lτ ) = H ∗ (t + lτ ) > 0, l ≥ 0. We now show that (9) has a solution. By analogy with [12], introduce the matrix series H (t + lτ ), l = 0, 1, . . . , m − 1, as follows: H (t + lτ ) =

∞ 

Y ∗ (t, k, l)Y (t, k, l).

(12)

k=l

In view of asymptotic stability of the zero solution to (1), this series converges. By (8), we have H (t + lτ ) = I + A∗ (t + lτ )A(t + lτ ) + A∗ (t + lτ )A∗ (t + (l + 1)τ )A(t + (l + 1)τ ) ×A(t + lτ ) + A∗ (t + lτ )(A(t + (l + 2)τ )A(t + (l + 1)τ ))∗ (A(t + (l + 2)τ )A(t + (l + 1)τ )) ×A(t + lτ ) + · · · = I + A∗ (t + lτ )A(t + lτ ) + A∗ (t + lτ )Y ∗ (t, l + 2, l + 1)Y (t, l + 2, l + 1) ×A(t + lτ ) + A∗ (t + lτ )Y ∗ (t, l + 3, l + 1)Y (t, l + 3, l + 1)A(t + lτ ) + . . . .

Since H (t + (l + 1)τ ) =

∞  k=l+1

Y ∗ (t, k, l + 1)Y (t, k, l + 1), Y (t, l + 1, l + 1) = I,

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then we obviously obtain the equality H (t + lτ ) = I + A∗ (t + lτ )H (t + (l + 1)τ )A(t + lτ ) for every l. Consequently, the series H (t + lτ ) defined by (12) satisfies (9). The equality H (t) = H (t + T ) follows immediately from (12) because A(t) = A(t + T ). The formula (12) yields also H (t + lτ ) = H ∗ (t + lτ ) > 0, l ≥ 0. The first part of the theorem is proved. II. We now prove the second part of the theorem. As mentioned above, the matrices H (t), H (t + τ ), . . . , H (t + T − τ ) have to satisfy the Lyapunov discrete equations (10), (11). Since H (t) = H ∗ (t) > 0 and the right-hand side of (10) is positive definite, then, as known, the spectrum of the matrix Y (t, m, 0) = X (t + T − τ ) belongs to the unit disk ω. Hence, according to the spectral criterion, the zero solution to (1) is asymptotically stable. The theorem is proved. The next assertion follows immediately from the proof of Theorem 1. Theorem 2 The zero solution to (1) is asymptotically stable if and only if there exists a Hermitian T -periodic solution H (t) > 0 to the Lyapunov functional difference equations (13) H (t + lτ ) − A∗ (t + lτ )H (t + (l + 1)τ )A(t + lτ ) = I, l = 0, 1, . . . , t ∈ [0, τ ). Remark 1 Theorems 1, 2 give criteria for asymptotic stability of the zero solution to (1) in terms of solvability of the Lyapunov functional difference equations.

3 Estimates for Solutions to (1) In this section, we establish estimates for solutions to (1). Theorem 3 Let the zero solution to (1) be asymptotically stable, let H (t) be a T periodic solution to (13). Then, for the solution x(t) to the initial value problem (7), the following estimate holds:  x(t + (k − 1)τ )2 ≤ 1 −

1 Hmax

k Hmax max ϕ(ξ)2 , ξ∈[−τ ,0]

k = 1, 2, . . . , t ∈ [0, τ ), Hmax = max H (t). t∈[0,T ]

Proof Introduce the matrix series

(14)

On Estimates of Solutions to One Class of Functional Difference Equations …

107

⎞∗ ⎛ ⎞ ⎛ ∞ i−1 i−1    ⎝ h(t, l) = A(t + jτ )⎠ ⎝ A(t + jτ )⎠ .

(15)

i=l

j=0

j=0

Comparing (12) and (15), we obviously have H (t) = h(t, 0). We now establish some properties of the series (15). Lemma 1 The following estimates hold: 2  l−1 , l ≥ 0,

h(t, l)v, v ≤ H (t + lτ ) A(t + jτ )v j=0

(16)

for any v ∈ C n . Proof By the definitions of h(t, l) and Y (t, k, l), we obtain

h(t, l)v, v =

∞ 

X ∗ (t + (i − 1)τ )X (t + (i − 1)τ )v, v i=l

=

∞ 

(Y (t, i, l)X (t + (l − 1)τ ))∗ Y (t, i, l)X (t + (l − 1)τ )v, v i=l

=

∞ 

Y ∗ (t, i, l)Y (t, i, l)X (t + (l − 1)τ )v, X (t + (l − 1)τ )v

i=l

= H (t + lτ )X (t + (l − 1)τ )v, X (t + (l − 1)τ )v. Since X (t + (l − 1)τ ) =

l−1 

A(t + jτ ), l ≥ 0,

j=0

we have (16). The lemma is proved. Lemma 2 The following estimates hold: l−1  

h(t, l)v, v ≤ 1− j=0

 1

h(t, 0)v, v, l ≥ 1, H (t + jτ )

for any v ∈ C n . Proof By the definition (15), we have

(17)

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2 l−2  = h(t, l − 1)v, v, l ≥ 1.

h(t, l)v, v + A(t + jτ )v j=0 By Lemma 1, the inequality holds 2 l−2  , l ≥ 1.

h(t, l − 1)v, v ≤ H (t + (l − 1)τ ) A(t + jτ )v j=0 Since the matrix H (t + (l − 1)τ ) is positive definite, then  1

h(t, l − 1)v, v.

h(t, l)v, v ≤ 1 − H (t + (l − 1)τ ) 

By arbitrariness of l ≥ 1, we obtain 

h(t, l)v, v ≤ 1 −

   1 1 ... 1 −

h(t, 0)v, v. H (t + (l − 1)τ ) H (t)

The lemma is proved. Let us prove (14). As mentioned above, the solution x(t + (k − 1)τ ) to (7) has the form x(t + (k − 1)τ ) = X (t + (k − 1)τ )ϕ(t − τ ) =

k−1 

A(t + jτ )ϕ(t − τ ),

j=0

t ∈ [0, τ ), k = 1, 2, . . . . By (15), we have 2 2  k−1 ∞  i−1  ≤ x(t + (k − 1)τ )2 = A(t + jτ )ϕ(t − τ ) A(t + jτ )ϕ(t − τ ) j=0 i=k j=0

=

∞  i=k

⎞∗ ⎛ ⎞ 

⎛ i−1 i−1   ⎝ A(t + jτ )⎠ ⎝ A(t + jτ )⎠ ϕ(t − τ ), ϕ(t − τ ) j=0

j=0

= h(t, k)ϕ(t − τ ), ϕ(t − τ ). Taking into account (17), we obtain

On Estimates of Solutions to One Class of Functional Difference Equations …

x(t + (k − 1)τ )2 ≤

k−1   1− j=0

109

 1

h(t, 0)ϕ(t − τ ), ϕ(t − τ ). H (t + jτ )

Since H (t) = h(t, 0), then we have (14). The theorem is proved. Acknowledgements The authors are supported by the Russian Foundation for Basic Research (project no. 19-01-00754).

References 1. Daleckii, Ju.L., Krein, M.G.: Stability of Solutions of Differential Equations in Banach Space. American Mathematical Society, Providence (1974) 2. Agarwal, R.P.: Difference Equations and Inequalities. Theory, Methods and Applications. Marcel Dekker, New York(1992) 3. Kocic, V.L., Ladas, G.: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Kluwer Academic Publishers, Dordrecht (1993) 4. Elaydi, S.N.: An Introduction to Difference Equations. Springer, New York (1999) 5. Wilkinson, J.H.: The Algebraic Eigenvalue Problem. The Clarendon Press, Oxford University Press, New York (1988) 6. Akin, O., Bulgak, H.: Linear Difference Equations and Stability Theory. Selcuk University, Konya (1998) 7. Godunov, S.K.: Modern Aspects of Linear Algebra. American Mathematical Society, Providence (1998) 8. Aidyn, K., Bulgak, H., Demidenko, G.V.: Numeric characteristics for asymptotic stability of solutions to linear difference equations with periodic coefficients. Sib. Math. J. 41(6), 1005– 1014 (2000) 9. Aidyn, K., Bulgak, H., Demidenko, G.V.: Asymptotic stability of solutions to perturbed linear difference equations with periodic coefficients. Sib. Math. J. 43(3), 389–401 (2002) 10. Aidyn, K., Bulgak, H., Demidenko, G.V.: An estimate for the attraction domains of difference equations with periodic linear terms. Sib. Math. J. 45(6), 983–991 (2004) 11. Demidenko, G.V.: Matrix Equations. Novosibirsk State University, Novosibirsk (2009) 12. Demidenko, G.V.: Stability of solutions to difference equations with periodic coefficients in linear terms. J. Comput. Math. Optim. 6(1), 1–12 (2010) 13. Demidenko, G.V., Matveeva, I.I.: Stability of solutions to delay differential equations with periodic coefficients of linear terms. Sib. Math. J. 48(5), 824–836 (2007) 14. Demidenko, G.V., Matveeva, I.I.: On estimates of solutions to systems of differential equations of neutral type with periodic coefficients. Sib. Math. J. 55(5), 866–881 (2014) 15. Demidenko, G.V., Matveeva, I.I.: Estimates for solutions to a class of time-delay systems of neutral type with periodic coefficients and several delays. Electron. J. Qual. Theory Differ. Equ. 2015(83), 1–22 (2015) 16. Matveeva, I.I.: On exponential stability of solutions to periodic neutral-type systems. Sib. Math. J. 58(2), 264–270 (2017) 17. Matveeva, I.I.: On the exponential stability of solutions of periodic systems of the neutral type with several delays. Diff. Equ. 53(6), 725–735 (2017) 18. Demidenko, G.V., Matveeva, I.I., Skvortsova, M.A.: Estimates for solutions to neutral differential equations with periodic coefficients of linear terms. Sib. Math. J. 60(5), 828–841 (2019) 19. Matveeva, I.I.: Estimates of the exponential decay of solutions to linear systems of neutral type with periodic coefficients. J. Appl. Ind. Math. 13(3), 511–518 (2019)

Integral Operators at Settings and Investigations of Tensor Tomography Problems Evgeny Derevtsov, Yuriy Volkov and Thomas Schuster

Abstract Generalizations of attenuated ray transform (ART) and back-projection operators of tomography are suggested. The operators of ART of order m contain a complex-valued absorption, the weights of more general form, and the internal sources depending on time. Connections between ART of various orders are established, and corresponding differential equations and uniqueness theorems are obtained. We define the angular moments of ART of order m, representing generalization of back-projection operator, and establish connections between the moments of different orders. The generalized ART and angular moment operators have applications in integral geometry and tomography.

Introduction Impressive achievements of computer, emission, and MRI-tomography in biology and medicine [1, 2], progress of vector and tensor tomography lead to further development of mathematical models for tomography and their tools [3–6]. We suggest generalizations of the operators of attenuated ray transform (ART) and back projection. A generalization of the ART operator is realized at three directions. At first, the absorption coefficient ε and the attenuation function generated by it become complexvalued similar to those arising in inverse scattering problem at Rytov’s approach, and then within diffraction tomography, for instance [2]. Second, we take into account a concept of generalized ray transform (integral moments of generalized tensor fields) E. Derevtsov · Y. Volkov (B) Sobolev Institute of Mathematics, Novosibirsk, Russia e-mail: [email protected] E. Derevtsov e-mail: [email protected] Novosibirsk State University, Novosibirsk, Russia T. Schuster Saarland University, Saarbrücken, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_15

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considered in [3] in particular. A third direction of generalization is connected with settings of a dynamic tomography and consists in consideration depending upon time internal sources [7]. Connections between ART of various orders are established, and their differential equations and uniqueness theorems are obtained. Angular moments of ART of order m are defined and connections between the moments of different orders are established. It should be remarked that there exist certain relationships between the generalized operators, and integral geometry and tomography.

1 Main Equations and Uniqueness Let R 3 be Euclidean space with inner product x, y of elements x = (x 1 , x 2 , x 3 ), y = (y 1 , y 2 , y 3 ) ∈ R 3 . We use notations S R = R 3 × S 2 = {(x, ξ) | x ∈ R 3 , |ξ| = 1}, Sx2 = {(x, ξ) ∈ S R | x ∈ R 3 fixed} for sets of pairs (x, ξ). Functions f , ε ≥ 0, ρ are finite and bounded, α(x, ξ) := ε(x, ξ) + iρ(x, ξ), f describes a distribution of internal sources, and ε is absorption coefficient. The attenuated ray transform (ART) of order m, m integer, m ≥ 0, is defined as  u m (x, ξ) =

∞

  s  s m exp − α(x − σξ, ξ) dσ f (x − sξ)ds.

0

(1)

0

For a function f depending on time t, the definition of u m (t, x, ξ) has a form 

∞

u m (t, x, ξ) =





s exp − m

0

s

 α(x − σξ, ξ) dσ f (x − sξ, t − s)ds

(2)

0

and called the nonstationary ART of order m. An operator H is defined as (Hψ)(x, ξ) =

 d ψ(x + τ ξ, ξ)τ =0 = ξ, ∇ ψ = div(ψξ) dτ

in invariant and coordinate forms. Suppose ε, ρ ∈ C 1 (S R), f ∈ C 1 (R 3 ). Then a direct verification leads to relations (H + α)u m = m u m−1 , m ≥ 1,

(H + α)u m = f, m = 0.

(3)

With usage of (3), we obtain Theorem 1 For ε ≥ 0, ρ ∈ C m+1 (S R), f ∈ C m+1 (R 3 ), u m (x, ξ) defined by (1), the equation 1 (H + α)m+1 u m = f (4) m! for integer m, m ≥ 0, is performed.

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Suppose ε ≥ 0, ρ ∈ C 1 (S R), f (t, x) ∈ C 1 (R × R 3 ). Then for u m (t, x, ξ) given by (2), m ≥ 0, the relations, ∂  + H + α u 0 = f, ∂t

∂  + H + α u m = m u m−1 , m ≥ 1, ∂t

(5)

are derived by direct calculations. By analogy with how it was done above for stationary case, we use formulas (5) and have the following. Theorem 2 For ε ≥ 0, ρ ∈ C m+1 (S R), f ∈ C m+1 (R × R 3 ), the equation m+1 1 ∂ +H+α um = f m! ∂t for m integer, m ≥ 0, u m given by (2), is valid. Let D be a bounded convex domain in R 3 with smooth boundary ∂ D. A distribution f of sources has finite support, supp f ⊂ D. Theorem 3 Assume that for ε ≥ 0, ρ ∈ C m (D × Sx2 ), a function ϕ(x, ξ) ∈ C m+1 (D × Sx2 ) satisfies Eq. (4) with zero right-hand part, 1 (H + α)m+1 ϕ = 0, m! and boundary-value conditions ϕ(x, ξ) = (Hϕ)(x, ξ) = · · · = (Hm ϕ)(x, ξ) = 0, x ∈ ∂ D, n x , ξ < 0, where n x is outer normal to the boundary ∂ D at a point x. Then ϕ(x, ξ) = 0 for all x ∈ D, ξ ∈ Sx2 . Considering the case of time dependence of a function ϕ, we get the following. Theorem 4 Suppose that for ε ≥ 0, ρ ∈ C m (D × Sx2 ), a function ϕ(t, x, ξ) ∈ C m+1 (R+ × D × Sx2 ) is a solution of the equation m+1 1 ∂ +H+α ϕ = 0, m! ∂t which satisfies initial ϕ(0, x, ξ) =

∂ϕ ∂m ϕ (0, x, ξ) = · · · = m (0, x, ξ) = 0, ∂t ∂t

and boundary ϕ(t, x, ξ) = (Hϕ)(t, x, ξ) = · · · = (Hm ϕ)(t, x, X ) = 0

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conditions for x ∈ ∂ D, n x , ξ < 0, t > 0, n x to be outer normal to the boundary ∂ D at a point x. Then ϕ(t, x, ξ) = 0 for t > 0, x ∈ D, ξ ∈ Sx2 .

2 Angular Moments of Generalized ART Let on S R a finite smooth function f , integer p, p ≥ 0, and ART u m (x, ξ) of order m be given (see (1)). A contravariant p-tensor field Emp (x) is defined as follows:  i1 ...i p Emp ( f ) (x) =

 Sx2

u m (x, ξ)  p dλx (ξ),

p ≥ 0.

(6)

Here ξ ik is i k component of vector ξ, i k = 1, 2, 3,  p is short denotation for ξ i1 . . . ξ i p , and dλx (ξ) is the angular measure on the sphere Sx2 . The Einstein rule of summation is applied in (6) and below.  i1 ...i p ( f ) , defined by Eq. (6), is called the fields of anTensor field Emp ( f ) = Emp gular p-moments of ART of order m. We use denotation Emp for tensor field Emp ( f ) below. An operator of divergence, acting on a symmetric tensor field w of rank m + 1, is defined as ∂ w j ... j l δlk , (δ w) j1 ... jm = ∂x k 1 m where δlk is Kronecker symbol. Applying the definition of divergence δ and formula (3), we derive the relation  j1 ... j p−1 (δ Emp ) j1 ... j p−1 (x) = m E(m−1)( (x) − α(x, ξ) u m (x, ξ)  p−1 dλx (ξ). (7) p−1) Sx2

For the function α(x, ξ) of a special form α(x, ξ) = Q rj1 ... jr (x)ξ j1 . . . ξ jr , where Q r is covariant r -tensor field, (r ≥ 1), we obtain δ Emp = m E(m−1)( p−1) − Q r ∗ Em( p+r −1) ,

(8)

where Q r ∗ Em( p+r −1) is a convolution of tensor fields Q r and Em( p+r −1) . For m = 0, the connections (7) have another form,  j1 ... j p−1 (δ E0 p ) j1 ... j p−1 (x) = f p−1 (x) − α(x, ξ) u 0 (x, ξ)  p−1 dλx (ξ), (9) Sx2

where designation f p−1 means a field of angular ( p − 1)-moments of function f . The components of f p−1 looks like f

j1 ... j p−1

 (x) =

Sx2

f (x, ξ)  p−1 dλ(ξ).

(10)

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For α(x, ξ) = Q rj1 ... jr (x) r , the formula (9) has a form δ E0 p = f p−1 − Q r ∗ E0( p+r −1) .

(11)

The formulas (7)–(11) can be used for obtaining additional relations by multiple application of operator δ. Theorem 5 Let m, n, p be integer, n ≥ 1, m ≥ 0, p ≥ 0, and E(m+n)( p+n) be a field of angular ( p + n)-moments of ART u m+n of order (m + n), α(x, ξ) = ε + ik, ε = const, k = const. Then δ n E(m+n)( p+n) =

n

j=0

(−1) j

n! (m + n)! αj E(m+ j) p . j!(n − j)! (m + j)!

(12)

Multiple application of the operator δ to tensor fields of the type E0 p leads to another equation. Corollary 1 Let n, p be integer, n ≥ 1, p ≥ 0, E0( p+n) be a field of angular ( p + n)moments of ART u 0 of order 0, α(x, ξ) = ε + ik, ε = const, k = const. Then δ n E0( p+n) =

n

(−1) j j=0

n! α j (Hn− j−1 f ) p + (−1)n αn E0 p , j!(n − j)!

(13)

where (Hn− j−1 f ) p is the angular p-moment of the function Hn− j−1 f , (H0 f ) p = f p . Equations (12), (13) point out that tensor fields Emp , p ≥ 0, m ≥ 0 may be expressed through a direct formula by means of iterated divergence δ n of the fields Enq with more than p index q, with usage of field angular moments of ART with more than m orders n, and angular moments of function Hk− j f . The angular moments of function Hk− j f can be treated physically as multipole sources. Let in R+ × S R a field u m (t, x, ξ) of order m be given (see (2)). The generalizations of Eq. (6), defining the nonstationary fields Emp (t, x) of angular p-moments of ART of order m, are implemented and for the nonstationary sources. With usage of definition of the operator δ and the relations (5), we obtain the formulas similar to (7)–(13) and for nonstationary case. Theorem 6 Let m, n, p be integer, n ≥ 1, m ≥ 0, p ≥ 0, and E(m+n)( p+n) be a field of angular ( p + n)-moments of ART u m+n (t, x, ξ) of order (m + n), α(x, ξ) = ε + ik, ε = const, k = const. Then δ n E(m+n)( p+n) =

n

j=0

(−1) j

j (m + n)!  ∂ n! + α E(m+ j) p , m ≥ 1, j!(n − j)! (m + j)! ∂t

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(−1) j j=0

∂ ∂ j n n! + α Hn− j−1 f p + (−1)n + α E0 p j!(n − j)! ∂t ∂t

for m = 0. Here (Hn− j−1 f ) p is an angular p-moment of a function Hn− j−1 f , (H0 f ) p = f p .

3 Conclusion We describe the generalizations of the ART and the back-projection operators here. The operators of ART u m (x, ξ) of order m may be treated as the integral moments of a source distribution f with a weight generated by exponential function. Connections between ART of different orders are established, and differential equations whose solutions are the images of specified operators are derived. Uniqueness theorems for boundary-value and initial boundary-value problems in stationary and nonstationary cases are proved. We suggest notions of the angular moments of ART of order m and derive the differential relations between them and angular moments of function f . The operator of ART of order m is connected with various transforms of tomographic type. In terms of computerized tomography, the operator (1), for m = 0, ρ = 0, ε = 0, may be treated as fan-beam, cone-beam, or Radon transform. In mathematical model for emission tomography, the operator (1) is standard attenuated ray transform, and certain natural generalization of the integrand leads to a notion of longitudinal ray transform of symmetric tensor fields and to integral moments of generalized tensor fields [3]. Thus, it arises naturally the settings of inverse problems of determination of a function f by its known ART of order m. This inverse problem can be treated and as the inverse problem for generalized transport equation by determining its right-hand part. The results obtained here may be considered as the first step of investigations toward this direction, and have good potential for further generalization and settings in framework of a mathematical models for dynamic refractive tensor tomography. The operators of angular moments can be useful as additional information is applied in iterative numerical methods for solving nonlinear tomography problems. Acknowledgements The reported study was funded partially by RFBR and DFG according to the research project 19-51-12008.

References 1. Ell, P.J., Holman, B.L. (eds.): Computed emission tomography. Oxford University Press, Oxford (1982) 2. Natterer, F.: The Mathematics of Computerized Tomography. Wiley, Chichester (1986) 3. Sharafutdinov, V.A.: Integral Geometry of Tensor Fields. VSP, Utrecht (1994)

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4. Svetov, I.E., Derevtsov, E.Yu., Volkov, Yu.S., Schuster, T.: A numerical solver based on Bsplines for 2D vector field tomography in a refracting medium. Math. Comput. Simul. 97, 207–223 (2014) 5. Derevtsov, E.Yu., Svetov, I.E.: Tomography of tensor fields in the plane. Eurasian J. Math. Comput. Appl. 3(2), 24–68 (2015) 6. Derevtsov, E.Yu., Maltseva, S.V.: Reconstruction of the singular support of a tensor field given in a refracting medium by its ray transform. J. Appl. Ind. Math. 9(4), 447–460 (2015) 7. Hahn, B., Louis, A.K.: Reconstruction in the three-dimensional parallel scanning geometry with application in synchrotron-based X-ray tomography. Inverse Probl. 28, 045013 (2012)

Lagrangian Godunov Schemes Bruno Després

Abstract At the occasion of the 90th anniversary of S.K. Godunov, this contribution focuses on the structure of one-dimensional Lagrangian Godunov solvers, as introduced in the seminal S.K. Godunov contributions (Godunov in Math Sb (N.S.) 47(89):277–306, 1959) [4], (Godounov et al. in Résolution numérique des problémes multidimensionnels de la dynamique des gaz, 1979) [1], and on their modern evolution in the context of multidimensional Lagrangian solvers with constraints. It also tries to reflect the profound influence of the work of S.K. Godunov in France.

1 Seminal S.K. Godunov Ideas About Lagrangian Solvers The seminal ideas of S.K. Godunov about the mathematical and numerical analysis of the system of compressible fluid dynamics ⎧ ∂t ρ + ∇ · (ρu) = 0, ⎪ ⎪ ⎨ ∂t (ρu) + ∇ · (ρu2 + pI) = 0, ∂t (ρe) + ∇ · ( pu) = 0, ⎪ ⎪ ⎩ ∂t (ρ S) + ∇ · (ρ Su) ≥ 0,

(1)

penetrated very soon in France, thanks to early translation of important textbooks in French by the edition MIR. This was, in particular, the case of Résolution numérique des problèmes multidimensionnels de la dynamique des gaz [4] which was translated and edited in 1979 from a publication in Russian by the same authors in Izdat. Nauka, Moscow, 1976: the idea of the decomposition of discontinuities in dimension one (with shock waves, contact discontinuities, rarefaction fans, and vacuum regions) was introduced and implemented in iterative procedures.

B. Després (B) Laboratoire Jacques-Louis Lions, UMR 7598, UPMC Univ Paris 06, Sorbonne Universités, 75005 Paris, France e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_16

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With S.K. Godunov, let us call U and P the continuous velocity and pressure at the contact discontinuity. Page 113 of the above reference, one finds that U and P can be calculated from u I , p I and u I I , p I I which are the velocities and pressures on both sides P − pI P − pI I U − uI + = 0, U − u I I − = 0, (2) aI aI I and the coefficients a I , a I I which are aI =

γ −1 ρI cI 2γ

p0 p0 1 − pP+ 1 − pP+ γ −1 I + p0 I I + p0 ρ , a = c I I I I I I  γ2γ−1  γ2γ−1 .   2γ P+ p0 P+ p0 1 − p I + p0 1 − p I I + p0

(3)

Equations (2) can be interpreted as a system of two quasi-linear relations, where the nonlinearity is due to the coefficients. An immediate fixed point procedure is proposed to solve this system, by reevaluating the coefficients iteratively. The first step of the iteration is written as a linear system U − uI +

P − pI P − pI I = 0, U − u I I − = 0. ρI cI ρI I cI I

(4)

In dimension one, these solvers can be used either in Eulerian calculations (the grid is fixed) in any dimension, or in Lagrangian calculations in dimension one. In this case, the grid interface points move with the velocity U . In a next chapter in [4], the authors explained the difficulties encountered to extend these ideas for Lagrangian calculations in dimension two and more: this was due to geometrical issues which cannot be avoided on moving multidimensional grids. Since the beginning of my carrier at CEA (Commissariat à l’Energie Atomique) in the early 90’, many colleagues (P.A. Raviart, J. Ovadia, A. Bourgeade, H. Jourdren, and many more) told me it was a kind of paradox. Why mathematical ideas which revealed so powerful in dimension one cannot be used in such a simple way in higher dimensions? Not only it is a restriction for applications, but it shows that perhaps something was still not understood about multidimensional Lagrangian solvers.

2 Multidimensional Version 2.1 Position of the Problem Renewal started in ≈2000 with the Ph.D. thesis of Loubere [12] where some interesting ideas were discussed. Next, working with another Ph.D. student C. Mazeran, we propose a way to obtain the compatibility of Godunov solvers like (2)–(4) with multidimensional Lagrangian grids. The first publications were [7, 8] and the key idea was to use a nodal-based solver, and no more an edge-based solver as in the

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famous Godunov method. However, the geometrical structure of the method was not completely at the center of construction, it was more a specific discretization procedure for the system of non-viscous compressible gas dynamics equations in initial Lagrangian variables which writes ⎧ ∂t (ρ J ) = 0, ⎪ ⎪ ⎨ ∂t (ρ J u) + ∂ X ( pM) + ∂Y (− p B) = 0, ∂t (ρ J v) + ∂ X (− pL) + ∂Y ( p A) = 0, ⎪ ⎪ ⎩ ∂t (ρ J e) + ∂ X ( pu M − pvL) + ∂Y ( pv A − pu B) = 0, with additional Piola identities (coming from ∂t x = u and x(0) = X = (X, Y ) ) ⎧ ⎨ ∂t J − ∂ X (u M − vL) − ∂Y (v A − u B) = 0, ∂ X M − ∂Y B = 0, ⎩ −∂ X L + ∂Y A = 0. This system is only weakly hyperbolic [9]: it explains a posteriori why the theory of nonlinear hyperbolic systems of conservation laws was not able to provide a satisfactory framework for the development of fully conservative Lagrangian numerical methods. Its complexity also explains the technical difficulties encountered for the derivation of Lagrangian multidimensional solvers. Then, still in dimension two, a variant of the scheme [7, 8] was soon produced by Maire [11] with emphasis on simpler geometrical notations: it was another progress. Finally, a version in all dimensions of the initial scheme [7, 8] was published in [6]. The mathematical structure of this algorithm also has a pedagogical virtue and it is explained in the next section.

2.2 Discrete Objects The description of the method starts from purely discrete objects. The notations are: j for cells, n for time steps, r for nodes. Moving Lagrangian cells have constant mass M nj = M j for all time steps n and all cells j. From the knowledge of nodes, the volume of a cell is provided by the function (x1 , . . . , xr , . . .) → V j . For a moving cell, one has V j (t) = V j (x1 (t), . . . , xr (t), . . .). An important discrete object in the numerical construction is the corner vector C jr = ∇xr V j ∈ Rd . One notices that C jr = ∇xr V j = O(h d−1 ) is nonzero only when the node r is a corner of the cell j. Some abstract properties can be derived as follows: (a) Chain rule, dtd V j =   1   1 ∇x V j , x = r C jr , ur ; (b) Homogeneity,

V j = d ∇x V j , x = d r C jr , xr ; (c) Translation invariance for all cell j, r ∈nodes C jr = 0; (d) For all interior

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node

xr , j∈cells C jr = 0; a more general formula related to rotational invariance, j∈cells C jr ⊗ xr = V j Id . Assume a one-dimensional Godounov-type relation (4) in corner direction, and assume the sum of forces vanishes around node xr . One gets the linear system 

C jr p ∗jr = 0,  C p ∗jr − p j + ρ j c j |C jr | , ur∗ − u j = 0, ∀ j ∈ cells. j∈cells

(5)

jr

The system (5) is a generalization in all dimension of the structure (4). Under natural conditions on the mesh, it is solvable. Define the nodal matrix (non-singular under

C ⊗C standard mesh conditions) Ar = j ρ j c j |jrC | jr ∈ Rd×d and the nodal right-hand jr   



C jr side br = j C jr p j + j ρ j c j |C | , u j ∈ Rd . One notes the property jr Ar = Art > 0.

(6)

The simplest algorithm which uses this nodal solver is written as follows: (1) Compute geometrical vectors Cnjr . (2) Update nodal quantities ur∗ = Ar−1 br and p ∗jr .

un+1 −un (3) For all cell, update velocity M j j t j = − r ∈nodes Cnjr p ∗jr and total energy  

en+1 −en M j j t j = − r ∈nodes Cnjr , ur∗ p ∗jr .

(4) Move nodes xrn+1 = xrn + t ur∗ . M = V n+1j . (5) Update density ρ n+1 j j

It is an exercise to check that this loop is a natural first-order time discretization of the integral form on Lagrangian moving meshes of the equations of compressible fluids. This simple scheme has many good properties. It is conservative in mass, momentum, and total energy. It produces entropy and so is dissipative at shocks. A fundamental property is the weak consistency of the discrete scheme with Eulerian form of Eq. (1): a proof is available in [9]. An application to Inertial Confinement Flow calculations is in [10].

3 Constraints A recent evolution takes the advantage of the positivity (6) of the nodal symmetric matrix Ar to add inequality constraints. It also provides a completely different mathematical interpretation of the linear solver (4). Indeed consider the function Jr (v) = 21 (Ar v, v) − (br , v). Then Ar ur∗ = br is equivalent to Jr (ur∗ ) ≤ Jr (v) for all v ∈ Rd . More generally for V = (vr )r consider

Lagrangian Godunov Schemes

J (V) =

123

 1 (Ar vr , vr ) − (br , vr ) ∈ R. 2 r ∈nodes

Define the set of admissible solutions K = {V ∈ R N ×d , F(V) ≤ 0}, where F(V) = (F1 (V), F2 (V), . . . ) are nodal constraints. This set of constraints may come from the modeling of boundary conditions, internal sliding conditions, or contact problems also described in the seminal work already quoted [4]. New algorithm: determine the nodal velocities as a minimization problem U∗ = argmin J (V),

(7)

V∈K

where other ingredients are the same. Some properties are: (a) K non-empty and closed =⇒ existence of U∗ = argmin J (V); (b) K strictly convex =⇒ uniqueness V∈K

of the minimum; (c) momentum is preserved if K + T = K, for all V ∈ K and Ta = (a, a, . . . , a) ∈ T where a ∈ Rd is the (arbitrary) vector of the translation, then V + Ta ∈ K; (d) total energy is preserved if K is a cone, V ∈ K ⇒ ∀λ > 0, λV ∈ K. If K = ∅ and the geometry is planar (dimension one), then the minimum (7) recovers exactly the Godunov Lagrangian solver (4). As a simple illustration of the method, consider in dimension one 2 material with the same pressure law p = (γ − 1)ρ(e − 21 u 2 ) − p0 , separated by vacuum. Initial data is u ini = ±1 and pini = 0. The non-penetration condition (it is a contact problem) defines the set Kn ,   n  ∗ ∗ x n − x R+1 , u R , u R+1 ∈ Kn = V | v R − v R+1 ≤ − R t where R is the last node in the left part and R + 1 is the first node in the right part. This set Kn satisfies the above conditions (a), (b), and (c). The condition (d) is not x n −x n satisfied, but if u ∗R − u ∗R+1 ≤ 0 ≤ − R tR+1 which is the separation/bounce condition then the total energy is still preserved because the cone property is satisfied for the minimum. One can check that the scheme may loose a small amount of total energy at the time of contact. Acknowledgements The author acknowledges the support of CEA and thanks his colleagues for passionate discussion over years on Lagrangian Godunov solvers.

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References 1. Godounov, S.K., Zabrodin, A., Ivanov, M., Kraiko, A., Prokopov, G.: Résolution numérique des problémes multidimensionnels de la dynamique des gaz. (French) Translated from the Russian by Valeri Platonov. Mir edition (1979) 2. Godounov, S.K.: Lois de conservation de intégrales d’énergie des équations hyperboliques. (French) (Conservation laws and energy integrals for hyperbolic equations) Nonlinear hyperbolic problems (St. Etienne, 1986), vol. 1270, Lecture Notes in Mathematics, pp. 135–149. Springer, Berlin (1987) 3. Godunov, S.K., Ryabenkii, V.S.: Difference Schemes: An Introduction to the Underlying Theory. Studies in Mathematics and its Applications, vol. 19. North-Holland (1987). (Translated from the Russian by E.M. Gelbard) 4. Godunov, S.K.: A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Math. Sb. (N.S.) 47(89), 271–306 (1959) 5. Godunov, S.K., Romenskii, E.I.: Elements of Continuum Mechanics and Conservation Laws. Kluwer Academic/Plenum Publishers, Dordrecht (2003). (Translated from the 1998 Russian edition) 6. Carré, G., Del Pino, S., Després, B., Labourasse, E.: A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension. J. Comput. Phys. 228(14), 5160–5183 (2009) 7. Després, B., Mazeran, C.: Lagrangian gas dynamics in two dimensions and Lagrangian systems. Arch. Ration. Mech. Anal. 178(3), 327–372 (2005) 8. Després, B., Mazeran, C.: Symmetrization of Lagrangian gas dynamics and multidimensional solvers, C. R. Académie des Sciences Paris. Mécanique 331(7), 475–480 (2003) 9. Després, B.: Numerical Methods for Eulerian and Lagrangian Conservation Laws. Frontiers in Mathematics. Birkhauser/Springer, Basel (2017) 10. Breil, J., Galera, S., Maire, P.H.: Multi-material ALE computation in inertial confinement fusion code CHIC. Comput. Fluids 46, 161–167 (2011) 11. Maire, P.H., Abgrall, R., Breil, J., Ovadia, J.: A cell-centered Lagrangian scheme for twodimensional compressible flow problems. SIAM J. Sci. Comput. 29(4), 1781–1824 (2007) 12. Loubère, R., Ovadia, J., Abgrall, R.: A Lagrangian discontinuous Galerkin-type method on unstructured meshes to solve hydrodynamics problems. Int. J. Numer. Methods Fluids 44(6), 645–663 (2004)

On Numerical Methods for Hyperbolic PDE with Curl Involutions M. Dumbser, S. Chiocchetti and I. Peshkov

Abstract In this paper, we present three different numerical approaches to account for curl-type involution constraints in hyperbolic partial differential equations for continuum physics. All approaches have a direct analogy to existing and wellknown divergence-preserving schemes for the Maxwell and MHD equations. The first method consists in a generalization of the Godunov–Powell terms, which means adding suitable multiples of the involution constraints to the PDE system in order to achieve the symmetric Godunov form. The second method is an extension of the generalized Lagrangian multiplier (GLM) approach of Munz et al., where the numerical errors in the involution constraint are propagated away via an augmented PDE system. The last method is an exactly involution-preserving discretization, similar to the exactly divergence-free schemes for the Maxwell and MHD equations, making use of appropriately staggered meshes. We present some numerical results that allow to compare all three approaches with each other.

1 Introduction Very recently, several novel hyperbolic PDE systems were proposed for the description of dynamic processes in continuum physics that are endowed with curl-type involutions, i.e., where the curl of a certain set of variables either has to vanish or has to assume a prescribed value. The most prominent examples are the system of nonlinear hyperelasticity of Godunov, Peshkov, and Romenski (GPR model) [13, 19, 24, 27] written in terms of the distortion field A, the conservative compressible multi-phase flow model of Romenski et al. [27, 29], the new hyperbolic model for M. Dumbser (B) · S. Chiocchetti · I. Peshkov Laboratory of Applied Mathematics, University of Trento, Via Mesiano 77, 38123 Trento, Italy e-mail: [email protected] S. Chiocchetti e-mail: [email protected] I. Peshkov e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_17

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surface tension, and the recent hyperbolic reformulation of the Schrödinger equation of Gavrilyuk and Favrie et al. [12, 30], as well as first-order reductions of the Einstein field equations, such as those proposed, e.g., in [1, 8, 14, 15]. Many, but not all, of the aforementioned mathematical models fall into the larger class of symmetric hyperbolic and thermodynamically compatible (SHTC) systems, studied by Godunov and Romenski et al. in [17, 20, 25, 27]. Involution constraints, in general, are stationary differential equations that are satisfied by the governing PDE system for all times if they are satisfied by the initial data. The most famous involution is the divergence-free condition of the magnetic field in the Maxwell and magnetohydrodynamics (MHD) equations. As a consequence, a lot of research has been dedicated in the past to the appropriate numerical discretization of PDE with divergence constraints. However, much less is known on curl-preserving numerical schemes for PDE with curl involutions. In the context of the Maxwell and MHD equations, the most common involution-preserving numerical schemes fall into the following three categories: 1. Exactly divergence-free schemes, such as those proposed in [2, 4–7, 11, 16, 32], which make use of the definition of the electromagnetic quantities on appropriately staggered grids. To the best knowledge of the authors, the only extensions to curl-type involutions are those presented in [21, 22, 31] so far. 2. The formally nonconservative Godunov–Powell terms, which go back to a numerical implementation by Powell [26] of the symmetrizing terms of the MHD equations found by Godunov in [18], and which consist in adding suitable multiples of the divergence-free condition to the induction, momentum, and energy equations. Note that at the analytical level, all these terms are exactly zero, but they are, in general, nonzero for certain numerical discretizations of the equations that are not in the class of exactly divergence-free schemes. These nonconservative terms which formally correspond to zero were nevertheless needed in order to symmetrize the MHD system and to make it at the same time thermodynamically compatible, i.e., to give it the aforementioned SHTC structure. The obvious disadvantage of this approach is that it only works for the MHD equations, where a velocity vector is available, since all Godunov–Powell terms are proportional to the velocity. Hence, for the vacuum Maxwell equations, where such a velocity vector does not exist, the approach is not suitable. 3. The generalized Lagrangian multiplier (GLM) approach forwarded by Munz et al. in [10, 23] for the Maxwell and MHD equations. The main idea here consists in solving an augmented evolution system, where an artificial scalar cleaning variable ϕ is added and coupled to the induction equation, so that divergence errors in the magnetic field cannot accumulate, but rather propagate away via acoustic-type waves. The advantage of this approach is that it works for MHD as well as for the Maxwell equations and that it does not add any nonconservative terms to the governing equations. At this point, we recall the hyperbolic GLM approach of Munz et al. [10, 23] in more detail. In this paper, we make use of the Einstein summation convention, which implies summation over two repeated indices. We furthermore use the abbreviations ∂t = ∂/∂t, ∂k = ∂/∂ xk . The fully antisymmetric Levi-Civita symbol is denoted by

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εi jk . The induction equation in electrodynamics is well known and reads ∂t Bk + εki j ∂i E j = 0.

(1)

Here, Bk and E j denote the magnetic and the electric field, respectively. An immediate consequence of the induction equation is the involution constraint I = ∂m Bm = 0,

(2)

which states that the magnetic field will remain divergence-free for all times, if it was initially divergence-free. As already mentioned above, a classical way to preserve a divergence-free magnetic field within a numerical scheme is the use of an exactly divergence-free discretization on appropriately staggered meshes, see, e.g., [3, 6, 7, 11, 16, 32]. The very popular GLM method proposed by Munz et al. in [10, 23] is an alternative to exactly constraint-preserving schemes and requires only small changes at the PDE level. Instead of the original induction equation, the following augmented induction equation is solved: ∂t Bk + εki j ∂i E j + ∂k ϕ = 0, ∂t ϕ + ad2 ∂m Bm = −d ϕ.

(3) (4)

Here, ϕ is the new cleaning scalar, ad is an artificial cleaning speed, and d is a small damping parameter. For convenience, the new terms in the augmented PDE system (3) and (4) with respect to the original induction equation (1) are highlighted in red. It is easy to see that for ad → ∞ Eq. (4) leads to ∂m Bm → 0, i.e., in the asymptotic limit the involution constraint (2) will be preserved. In the remaining part of this paper, we will show the natural extensions of the exactly divergence-free schemes, the Godunov–Powell terms and the GLM cleaning to curl-type involutions. We will show computational results for the new hyperbolic surface tension model [30] and close with some concluding remarks and an outlook to future work.

2 Model Problem and Different Approaches to Account for the Curl Involution We illustrate the basic ideas on the following simple toy model, in order to ease notation and to facilitate the understanding of the underlying concepts. Consider the following evolution system for one scalar ρ and two vector fields vk and Jk : ∂t ρ + ∂i (ρvi ) = 0,   2 ∂t (ρvk ) + ∂i ρvi vk + ρ E ρ δik + ρ Jk E Ji = 0, ∂t Jk + ∂k (vm Jm ) + vm (∂m Jk − ∂k Jm ) = 0.

(5) (6) (7)

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Here, E = E(ρ, vk , Jk ) is a specific total energy potential and E ρ and E Jk are the derivatives of the energy potential with respect to the state variables ρ and Jk . In particular, p = ρ 2 E ρ is the fluid pressure. The above system satisfies the additional energy conservation law    ∂t (ρ E) + ∂k vk (ρ E) + vi ρ 2 E ρ δik + ρ Ji E Jk = 0.

(8)

It is easy to see that the PDE (7) is endowed with the linear involution constraint Imk = ∂m Jk − ∂k Jm = 0, i.e., if the curl of Jk is zero for the initial data, then it will remain zero for all times. For a general-purpose numerical method applied to (7), it is very hard to guarantee Imk = 0 at the discrete level. We stress that for smooth solutions, at the continuous level all the following reformulations of the PDE system are completely equivalent. The main differences arise at the discrete level.

2.1 SHTC Structure and Godunov–Powell Terms for Curl Involutions The system (5)–(7) can be written in symmetric hyperbolic form [17, 27] by adding the term ρ E Jk (∂i Jk − ∂k Ji ) = 0 to the momentum equation. Note that the term vm (∂m Jk − ∂k Jm ) proportional to the velocity field and to the curl of Jk is already contained in the evolution equation for Jk in order to make the system Galilean invariant. The modified system then reads ∂t ρ + ∂i (ρvi ) = 0,   ∂t (ρvk ) + ∂i ρvi vk + ρ 2 E ρ δik + ρ Jk E Ji + ρ E Jk (∂i Jk − ∂k Ji ) = 0,

(10)

∂t Jk + ∂k (vm Jm ) + vm (∂m Jk − ∂k Jm ) = 0,

(11)

(9)

where we have highlighted the additional symmetrizing term in red. Introducing the notation E = ρ E, m i = ρvi , r = Eρ , vi = Em i , ηi = E Ji , and the Legendre transform L(p) of the potential E(q) as L(p) = q · Eq − E = ρEρ + m i Em i + Ji E Ji − E(q),

(12)

  with the vector of conservative variables q = L p = L r , L vi , L ηi = (ρ,  m i , Ji ) and the vector of thermodynamic dual variables p = Eq = Eρ , Em i , E Ji = (r, vi , ηi ), one can write the above system (9)–(11) in the symmetric Godunov form   ∂t L r + ∂k (vk L)r = 0,   ∂t L vi + ∂k (vk L)vi + L ηi ∂k ηk − L ηk ∂i ηk = 0,   ∂t L ηi + ∂k (vk L)ηi − L ηi ∂k vk + L ηk ∂i vk = 0.

(13) (14) (15)

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The modified system (9)–(11) is not only symmetric hyperbolic for convex potentials L, but is also numerically much better behaved concerning the curl involution on Jk when solved with a general-purpose scheme.

2.2 GLM Curl Cleaning As already mentioned before, the main advantage of the GLM approach of Munz et al. [10, 23] for divergence constraints is its ease of implementation and the fact that it does not necessarily require a velocity field, since the transport of the divergence errors is achieved via acoustic-type waves. Here, in the case of curl involutions, we add a Maxwell-type subsystem, i.e., curl errors propagate away via electromagnetictype waves. The disadvantage of GLM curl cleaning is the need to add a rather large number of auxiliary evolution quantities to the system. The GLM curl cleaning proposed in [9, 15] can be explained on the toy system (5)–(7) as follows. The original governing PDE system (5)–(7) is simply replaced by the following augmented system that accounts for the curl constraint on Jk : ∂t ρ + ∂i (ρvi ) = 0,   ∂t (ρvk ) + ∂i ρvi vk + ρc02 Ji Jk = 0,

(16)

∂t Jk + ∂k (vm Jm ) + vm (∂m Jk − ∂k Jm ) + εklm ∂l ψm = 0,

(18)

∂t ψk −

(19)

ac2 εklm ∂l Jm +∂k ϕ

= −c ψk ,

∂t ϕ + ad2 ∂m ψm = −d ϕ,

(17)

(20)

where ac is a new cleaning speed associated with the curl cleaning. The new terms associated with the curl cleaning are highlighted in blue, for convenience, while the terms of the original PDE (7) are written in black. Since the evolution equation for the cleaning vector field ψk has formally the same structure as the induction equation (1) of the Maxwell equations, it is again endowed with the divergence-free constraint ∂m ψm = 0, which is taken into account via the classical GLM method (red terms). It is easy to see that from (19) for ac → ∞ we obtain klm ∂l Jm → 0 in the limit, thus satisfying the involution in the sense Imk → 0. The augmented system (16)– (20) can now be solved with any standard numerical method for nonlinear systems of hyperbolic partial differential equations. The main advantage over the Godunov– Powell terms proposed in the previous section is the fact that the GLM curl cleaning does not destroy conservation of momentum and it also works in absence of a physical velocity field vk .

2.3 An Exactly Curl-Free Discretization Here we present a compatible discretization that satisfies the curl constraint exactly at the discrete level. For this purpose, we use an appropriately staggered mesh,

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with the field Jk defined in the vertices of the main grid and the scalar field φ = vm Jm defined in the barycenters of the primary control volumes. To avoid confusion between tensor indices and discretization indices, throughout this paper, we will use the subscripts i, j, k, l, m for tensor indices and the superscripts n, p, q, r, s for the discretization indices in time and space, respectively. The discrete spatial coordinates will be denoted by x p and y q , while the set of discrete times will be denoted by t n . h h,n The z-component  of the discrete curl ∇ × of a discrete vector field J is denoted h h,n · ez and its degrees of freedom are naturally defined as by ∇ × J 

∇

p,q

×J

h,n



p+ 21 ,q+ 21 ,n

1 J2 · ez = 2

p+ 21 ,q+ 21 ,n

1 J1 2

p+ 21 ,q− 21 ,n

+ J2

p− 21 ,q+ 21 ,n

+ J1

p− 21 ,q+ 21 ,n

− J2 x

p+ 21 ,q− 21 ,n

− J1 y

p− 21 ,q− 21 ,n

− J2

−

p− 21 ,q− 21 ,n

− J2

(21)

making use of the vertex-based staggered values of the field Jh,n , see the right panel in Fig. 1. In Eq. (21), the symbol i jk is the usual Levi-Civita tensor. Equation (21) defines a discrete curl on the control volume p,q via a discrete form of the Stokes theorem based on the trapezoidal rule for the computation of the integrals along each edge of p,q . Last but not least, we need to define a discrete gradient operator that is compatible with the discrete curl, so that the continuous identity ∇ × ∇φ = 0

(22)

also holds on the discrete level. If we define a scalar field in the barycenters of the control volumes p,q as φ p,q,n = φ(x p , y q , t n ) then the corner gradient generates a natural discrete gradient operator ∇ h of the discrete scalar field φ h,n that defines a discrete gradient in all vertices of the mesh. The corresponding degrees of freedom generated by ∇ h φ h,n read ⎛ p+ 21 ,q+ 21

∇ p+ 2 ,q+ 2 φ h,n = ∂k 1

1

⎜ φ h,n = ⎜ ⎝

1 φ p+1,q+1,n +φ p+1,q,n −φ p,q+1,n −φ p,q,n 2 x 1 φ p+1,q+1,n +φ p,q+1,n −φ p+1,q,n −φ p,q,n 2 y

⎞ ⎟ ⎟, ⎠

(23)

0 see the left panel of Fig. 1. It is then straightforward to verify that an immediate consequence of (21) and (23) is ∇ h × ∇ h φ h,n = 0,

(24)

i.e., one obtains a discrete analog of (22). With this compatible discretization, Eq. (7), which contains a gradient and a curl operator, can be discretized so that Jk remains curl-free for all times.

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Fig. 1 Left: stencil of the discrete gradient operator, which computes the corner gradient of a scalar field defined in the barycenters. Right: stencil of the discrete curl operator, defining a curl inside the barycenter using the vector field in the corners

Fig. 2 Temporal evolution of the curl errors for different numerical methods applied to the hyperbolic surface tension model proposed in [30]

2.4 Numerical Results Here we present some numerical results obtained with the three approaches mentioned above, applied to the hyperbolic surface tension model of Gavrilyuk et al. [30]. It is important to note that the original model [30] is only weakly hyperbolic and thus not suitable for a stable numerical discretization with a general-purpose scheme. In Fig. 2, we compare the numerical results obtained for the original weakly hyperbolic model, for the nonconservative Godunov–Powell terms, for the GLM curl cleaning and for the exactly curl-free discretization. The results clearly show that the weakly hyperbolic system becomes unstable with a general-purpose scheme, while the nonconservative Godunov–Powell terms allow a stable discretization. Even better results are obtained for the conservative GLM curl cleaning approach. The best

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results are obtained by the exactly curl-free (structure-preserving) scheme, which can be even directly applied to the weakly hyperbolic system, thus emphasizing the important role of the curl involution at the continuous and discrete level.

3 Conclusions We have outlined three possible extensions of divergence-free schemes to hyperbolic PDE systems with curl-type involution constraints, namely, (i) the classical Godunov–Powell approach based on the symmetrization of the governing PDE system, (ii) the hyperbolic GLM cleaning approach that accounts for the involution constraint via an augmented PDE system and in which the numerical errors of the involution are transported away via a Maxwell-type subsystem, and (iii) an exactly curl-free scheme based on appropriately staggered meshes. Future work will consist in an extension of the exactly curl-free approach to higher order of accuracy and the application to other PDE systems with curl-type involutions, such as those presented in [12, 28–30]. First preliminary results of the authors indicate that the use of exactly curl-free schemes for hyperbolic PDE systems with curl involutions is by far superior in performance and accuracy compared to the Godunov–Powell terms and compared to the GLM cleaning approach. Acknowledgements The research presented in this paper has been funded by the European Union’s Horizon 2020 Research and Innovation Programme under the project ExaHyPE, grant no. 671698 and by the Deutsche Forschungsgemeinschaft (DFG) under the project DROPIT, grant no. GRK 2160/1. MD also acknowledges financial support from the Italian Ministry of Education, University and Research (MIUR) via the Departments of Excellence Initiative 2018–2022 attributed to DICAM of the University of Trento (grant L. 232/2016) and via the PRIN 2017 project.

References 1. Alic, D., Bona, C., Bona-Casas, C.: Towards a gauge-polyvalent numerical relativity code. Phys. Rev. D 79(4), 044026 (2009) 2. Balsara, D.S.: Divergence-free adaptive mesh refinement for magnetohydrodynamics. J. Comput. Phys. 174(2), 614–648 (2001) 3. Balsara, D.S.: Second-order accurate schemes for magnetohydrodynamics with divergencefree reconstruction. Astrophys. J. Suppl. Ser. 151, 149–184 (2004) 4. Balsara, D.S.: Multidimensional HLLE Riemann solver: application to Euler and magnetohydrodynamic flows. J. Comput. Phys. 229, 1970–1993 (2010) 5. Balsara, D.S.: Three dimensional HLL Riemann solver for conservation laws on structured meshes; application to Euler and magnetohydrodynamic flows. J. Comput. Phys. 295, 1–23 (2015) 6. Balsara, D.S., Dumbser, M.: Divergence-free MHD on unstructured meshes using high order finite volume schemes based on multidimensional Riemann solvers. J. Comput. Phys. 299, 687–715 (2015)

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7. Balsara, D.S., Spicer, D.: A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations. J. Comput. Phys. 149, 270–292 (1999) 8. Brown, J.D., Diener, P., Field, S.E., Hesthaven, J.S., Herrmann, F., Mroué, A.H., Sarbach, O., Schnetter, E., Tiglio, M., Wagman, M.: Numerical simulations with a first-order BSSN formulation of Einstein’s field equations. Phys. Rev. D 85(8), 084004 (2012) 9. Chiocchetti, S., Peshkov, I., Gavrilyuk, S., Dumbser, M.: High order ADER schemes and GLM curl cleaning for a first order hyperbolic formulation of compressible flow with surface tension. J. Comput. Phys. (2020), submitted 10. Dedner, A., Kemm, F., Kröner, D., Munz, C.D., Schnitzer, T., Wesenberg, M.: Hyperbolic divergence cleaning for the MHD equations. J. Comput. Phys. 175, 645–673 (2002) 11. DeVore, C.R.: Flux-corrected transport techniques for multidimensional compressible magnetohydrodynamics. J. Comput. Phys. 92, 142–160 (1991) 12. Dhaouadi, F., Favrie, N., Gavrilyuk, S.: Extended Lagrangian approach for the defocusing nonlinear Schrödinger equation. Stud. Appl. Math. 1–20 (2018) 13. Dumbser, M., Peshkov, I., Romenski, E., Zanotti, O.: High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: viscous heat-conducting fluids and elastic solids. J. Comput. Phys. 314, 824–862 (2016) 14. Dumbser, M., Guercilena, F., Köppel, S., Rezzolla, L., Zanotti, O.: Conformal and covariant Z4 formulation of the Einstein equations: strongly hyperbolic first-order reduction and solution with discontinuous Galerkin schemes. Phys. Rev. D 97, 084053 (2018) 15. Dumbser, M., Fambri, F., Gaburro, E., Reinarz, A.: On GLM curl cleaning for a first order reduction of the CCZ4 formulation of the Einstein field equations. J. Comput. Phys. 404, 109088 (2020). https://doi.org/10.1016/j.jcp.2019.109088 16. Gardiner, T.A., Stone, J.M.: An unsplit Godunov method for ideal MHD via constrained transport. J. Comput. Phys. 205, 509–539 (2005) 17. Godunov, S.K.: An interesting class of quasilinear systems. Dokl. Akad. Nauk SSSR 139(3), 521–523 (1961) 18. Godunov, S.K.: Symmetric form of the magnetohydrodynamic equation. Numer. Methods Mech. Contin. Medium 3(1), 26–34 (1972) 19. Godunov, S.K., Romenski, E.I.: Nonstationary equations of the nonlinear theory of elasticity in Euler coordinates. J. Appl. Mech. Tech. Phys. 13, 868–885 (1972) 20. Godunov, S.K., Romenski, E.I.: Elements of Continuum Mechanics and Conservation Laws. Kluwer Academic/Plenum Publishers, New York (2003) 21. Hyman, J.M., Shashkov, M.: Natural discretizations for the divergence, gradient, and curl on logically rectangular grids. Comput. Math. Appl. 33, 81–104 (1997) 22. Jeltsch, R., Torrilhon, M.: On curl-preserving finite volume discretizations for shallow water equations. BIT Numer. Math. 46, S35–S53 (2006) 23. Munz, C.D., Omnes, P., Schneider, R., Sonnendrücker, E., Voss, U.: Divergence correction techniques for Maxwell solvers based on a hyperbolic model. J. Comput. Phys. 161, 484–511 (2000) 24. Peshkov, I., Romenski, E.: A hyperbolic model for viscous Newtonian flows. Contin. Mech. Thermodyn. 28, 85–104 (2016) 25. Peshkov, I., Pavelka, M., Romenski, E., Grmela, M.: Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations. Contin. Mech. Thermodyn. 30(6), 1343–1378 (2018) 26. Powell, K.G.: An approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension). Technical Report ICASE-Report 94-24 (NASA CR-194902), NASA Langley Research Center, Hampton, VA (1994) 27. Romenski, E.I.: Hyperbolic systems of thermodynamically compatible conservation laws in continuum mechanics. Math. Comput. Model. 28(10), 115–130 (1998) 28. Romenski, E., Resnyansky, A.D., Toro, E.F.: Conservative hyperbolic formulation for compressible two-phase flow with different phase pressures and temperatures. Q. Appl. Math. 65, 259–279 (2007)

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Modeling Sodium Combustion with Liquid Water Damien Furfaro, Richard Saurel, Lucas David and François Beauchamp

Abstract Solid and liquid sodium combustion with liquid water occurs through a thin gas layer where exothermic reactions happen with sodium and water vapors. It thus involves multiple interfaces separating liquid and gas in the presence of surface tension, phase transition, and surface reactions. The gas-phase reaction involves compressible effects resulting in possible shock wave appearance in both gas and liquid phases. To understand and predict phenomena occurring during sodium combustion with water, a diffuse interface flow model is built. This formulation enables flow resolution in multidimension in the presence of complex motion, such as, for example, Leidenfrost-type thermochemical flow. More precisely sodium drop autonomous motion on the liquid surface is computed. Various modeling and numerical issues are present and addressed in the present contribution. In the author’s knowledge, the first computed results of such type of combustion phenomenon in multidimensions are presented in this paper thanks to the diffuse interface approach. Explosion phenomenon is addressed as well and is reproduced at least qualitatively thanks to extra ingredients such as turbulent mixing of sodium and water vapors in the gas film and finite rate chemistry. Shock wave emission from the thermochemical Leidenfrost-type flow is observed as well as sodium projection, as reported in related experiments.

D. Furfaro · R. Saurel RS2N SAS, Saint Zacharie, France e-mail: [email protected] R. Saurel (B) · L. David LMA, CNRS, Centrale Marseille, Aix-Marseille University, Marseille, France e-mail: [email protected] L. David e-mail: [email protected] L. David · F. Beauchamp CEA Cadarache, LTPS, Saint-Paul-lez-Durance, France e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_18

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Fig. 1 Schematic representation of a floating sodium drop on liquid water surface, separated by a gas layer. The gas layer thickness is controlled by gas ejection, density difference between the liquids, diffusion fluxes in the gas layer, phase change at interfaces, and exothermicity of the reactions, at sodium surface and in the gas mixture

1 Introduction Sodium Fast Reactors (SFR) as well as other engineering systems use sodium (Na) as coolant fluid. Sodium presents excellent physical properties regarding heat transfer efficiency as well as its ability to maintain kinetic energy of fast neutrons. However, it reacts exothermically with both air and water. In the limit, explosion may occur resulting in shock wave propagation in the liquid and surrounding media. When a liquid or solid sodium drop is set on a liquid water surface surprising phenomenon occurs. A reaction appears rapidly resulting in autonomous drop motion on the liquid surface. It seems that the drop is separated from liquid water by a small gas layer where combustion occurs both in the gas phase and at sodium surface (Fig. 1). The phenomenon is reminiscent of the Leidenfrost effect except that the heat needed to vaporize sodium and water comes from the combustion of themselves and their vapors. This combustion induces liquid water evaporation and heating of the sodium drop. After some delay, typically a few seconds, explosion occurs. These complex events are qualitatively reported on many videos available on the web, such as www. youtube.com/watch?v=ODf_sPexS2Q for example. They clearly illustrate complexity of the physics and chemistry in presence. Quantitative analytic experiments have been carried out at CEA Cadarache, France, in the facilities SOCRATE, DINAMO, VIPERE, and LAVINO [1–3]. In this context, gas layer thickness determination is very important as it controls heat and mass diffusion fluxes between sodium and water, responsible for reactants gas mixing and reaction. As illustrated in Fig. 1, self-selection of the gas layer thickness is directly linked to multidimensional effects. The present approach is based on diffuse interface formulation [4]. In this frame, interface deformations can be arbitrarily large as they are captured, as shocks and contact surfaces are captured in conventional gas dynamics computations. Moreover, diffuse interface approaches enable consideration of complex physics and chemistry needed to model sodium–water reaction. Indeed, the relevant phenomena to address are as follows: • Fluid compressibility and possible presence of shock waves;

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• Presence of interfaces with complex physics, such as surface tension, phase transition, chemical species, and thermal diffusion as well as surface reactions; • Intense motion, the interfaces being mobile and deformable; and • Chemical reactions in the gas phase.

2 Multiphase Model and Properties With diffuse interface approaches, the entire domain is considered as a multiphase mixture and interfaces correspond to zones where mixture density, mass, and volume fractions become discontinuous. The model considered hereafter is an extension of [5] to model phase change at interfaces. Five chemical species are present in the sodium–water reaction context, present in two thermodynamic phases: liquid and vapor water, liquid and vapor sodium, liquid soda, hydrogen, and nitrogen. In the present model, hydrogen combustion is not considered as sodium–water explosions have been observed in the absence of such reaction [1] where the atmosphere was made of argon. The corresponding flow model reads ⎧     ∂ρYHl 2 O ⎪ g l ⎪ ⎪ + ∇ · ρYHl 2 O u = ρνH2 O gH2 O − gH ⎪ 2O ⎪ ∂t ⎪ ⎪ ⎨ l     ∂ρYNa g l l + ∇ · ρYNa − ω˙ SR u = ρνNa gNa − gNa (liquid phase mass balance) ⎪ ⎪ ∂t ⎪ ⎪ ⎪ l     ⎪ ⎪ ∂ρYNa g l l ⎩ + ∇ · ρYNaOH + ϕNaOH (ω˙ SR + ω˙ GR ) u = ρνNaOH gNaOH − gNaOH ∂t ⎧ g     ∂ρYH2 O ⎪ g g g ⎪ l ⎪ + ∇ · ρYH2 O u + αg FH2 O = −ρνH2 O gH2 O − gH − ϕH2 O (ω˙ SR + ω˙ GR ) ⎪ 2O ⎪ ∂t ⎪ ⎪ ⎪ g ⎪   ⎪  g ∂ρYNa ⎪ g g l ⎪ + ∇ · ρYNa u + αg FNa = −ρνNa gNa − gNa − ω˙ GR ⎪ ⎪ ⎪ ∂t ⎪ ⎨  ∂ρYH2 + ∇ · ρYH2 u + αg FH2 = ϕH2 (ω˙ SR + ω˙ GR ) (gas-phase mass balance) ⎪ ⎪ ∂t ⎪ ⎪ g ⎪   ⎪  g ⎪ ∂ρYNaOH g g l ⎪ ⎪ + ϕNaOH (ω˙ SR + ω˙ GR ) + ∇ · ρYNaOH u + αg FNaOH = −ρνNaOH gNaOH − gNaOH ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪  ⎪ ⎩ ∂ρYair + ∇ · ρY g u + αg Fair = 0 air ∂t

(1)

  ∂ρu l +σ l + ∇ · ρu ⊗ u + p I = ρg + σNa κNa ∇αNa H2 O κH2 O ∇αH2 O ∂t

 ∂ρ E l · u + σl l l + ∇ · (ρ E + P)u + αg qM + qT = ρg · u + σNa κNa ∇αNa H2 O κH2 O ∇αH2 O · u ∂t

.

The flow model being quite sophisticated, some details are needed. Let us consider first mass balance equations. The mass balance equation of water vapor is a relevant candidate as the various considered effects are present:

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    ∂ρYH O g g g l 2 + ∇ · ρYH O u + αg FH O = −ρνH2 O gH O − gH − ϕH2 O (ω˙ SR + ω˙ GR ) , O 2 2 2 2 ∂t

  g N • ρ = k=1 ρYk with N = 7 denotes the two-phase mixture density. YH2 O represents the mass fraction of water vapor with respect to the two-phase mixture; • u represents the mixture center of mass velocity; N Yk , • αg represents the gas volume fraction. It is defined by αg = ρ k=4 ρk ( p,T ) where p and T represent the mixture pressure and temperature, respectively. The relation between ρk , p, and T is detailed in Sect. 3; g • FH2 O represents the molecular diffusion flux of water vapor. For a given chemical species k present in the gas phase, it reads Fk = C

1 (yk ∇ p − ∇ pk ) , p

where C represents the diffusion coefficient, yk the mass fraction of species k l l − YNaOH the gas mixture in the gas mixture (yk = YYgk , with Yg = 1 − YHl 2 O − YNa mass fraction with respect to the two-phase mixture), and pk the partial pressure. g As molecular diffusion is considered in the gas phase only, the diffusion flux FH2 O is weighted fraction αg in the mass balance equation;  g by the volume • ρνH2 O gH2 O − gHl 2 O represents water liquid-vapor mass transfer, evaporation or condensation. The relaxation parameter νH2 O controls the rate at which thermodynamic equilibrium is reached, i.e., when the Gibbs-free energies become equal: g gH2 O = gHl 2 O (gk = h k − T sk , where g, h, and s denote, respectively, the specific Gibbs-free energy, the enthalpy, and entropy of species k). In the present computations, the various relaxation parameters νk related to phase change are assumed to tend to infinity, meaning that local thermodynamic equilibrium is assumed; • ω˙ SR and ω˙ GR represent the chemical production rates, respectively, related to gasphase reaction (GR) and surface one (SR). Their modeling is detailed in [6]. ϕH2 O represents a weight factor. Let us now examine the momentum equation:   ∂ρu l + ∇ · ρu ⊗ u + p I = ρg + σNa κNa ∇αNa + σH2 O κH2 O ∇αHl 2 O ∂t

.

In addition to the conventional Euler equation of compressible fluids, gravity effects have been added through ρg where g denotes the gravity acceleration. Capillary forces have been added as well and are present at the liquid water–gas interface through the term σHl 2 O κHl 2 O ∇αHl 2 O and at the liquid sodium–gas interface l , with σHl 2 O and σNa the surface tension coefficients. The volume fractions σNa κNa ∇αNa are defined by αHl 2 O = are denoted by

κHl 2 O

ρYHl 2 O

ρHl 2 O ( p,T )

and κNa .

l and αNa =

l ρYNa . l ρNa ( p,T )

The local interface curvatures

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The last equation of System (1) corresponds to the balance energy for two the N Yk ek phase mixture. The total energy is denoted by E = e + 21 u2 , with e = k=1 the mixture internal energy. In addition to conventional terms in compressible fluids, the right-hand side involves the power of the forces appearing in the right-hand side of the mixture momentum equation. The  energy flux is augmented heat dif 3 ¯ fusion effects qT = − k=1 αk λk + αg λg ∇T , where λk represents the thermal conductivity of the phase or species   k. The gas-phase thermal conductivity is defined 1 1 ¯ , where xk represents the molar fraction of the by λ = 2 k>3 x k λk + x k λk k>3

k , and Wk represents the molar mass of species k in the gas mixture, xk = yk /W i>3 yi /Wi species k. Moreover, the energy flux is also by the heat diffusion due  augmented to mass diffusion in the gas phase qM = k>3 h k Fk weighted by the gas volume fraction αg . The model satisfies the fundamental principles of physics such as mixture mass, mixture momentum, and mixture energy conservation, a conservative formulation being given in [5]. The model is also thermodynamically consistent, i.e., it satisfies the second law of thermodynamics,

 2  2 ⎛ ⎞ l v v − gl ρνNa gNa ρνH2 O gH  ∂ρs λ Na O − gH2 O c 2 + ∇ · ⎝ρsu + αg ∇T ⎠ = sk Fk − + ∂t T T T k>3

αg P  λc |ω˙ SR | |ω˙ GR | G 0SR − G 0GR ≥ 0, + vk Fk2 + 2 (∇T )2 − CT T T T k>3

N where s = k=1 Yk sk is the mixture entropy and G 0 represents reactions-free energy (for both SR and GR, G 0 < 0). In the absence of diffusive effects, source, and capillary terms, the system is hyperbolic with wave speeds u, u − c, and u + c. The mixture sound speed definition is given in [5]. The present model dealing with mixtures in mechanical and thermal equilibrium is suitable for the computation of interfaces when heat conduction is present and related boundary layers resolved, as will be done in the computations.

3 Thermodynamic Closure The mixture equation of state results of the following mixture rules: T = Tk , ∀k ∈ {1, .., N }

p = pk , ∀k ∈ {1, .., N }

N N   1 = e= Yk vk ( p, T ) Yk ek ( p, T ) (2) ρ k=1 k=1 In this system, fluids are in pressure and temperature equilibrium, but each one occupies its own volume or its own volume fraction. Each fluid thermodynamics is

v=

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considered through the “Noble-Abel–Stiffened-Gas” (NASG) EOS [7]. Its caloric formulation reads, pk (vk , ek ) =

(γk − 1)(ek − qk ) − γk p∞,k . (vk − bk )

The term (γk − 1)(ek − qk ) represents thermal agitation while (vk − bk ) represents short-range repulsive effects. The term γk p∞,k , present in liquids and solids, corresponds to the attractive effects responsible of condensed matter cohesion. This equation of state is convex as well as the mixture one resulting of mixture rules (2).

4 Flow Solver The flow model (1) consists in a hyperbolic system with diffusion and relaxation terms, both stiff and finite rate. As interfaces are present and captured, numerical diffusion at corresponding discontinuities must be lowered. In this direction, the Overbee limiter of [8] is used for volume fraction transport in the frame of higher order [9] type schemes. Computation of stiff phase transition at interfaces is achieved by the stiff source terms limiter method given in [10]. Gravity and surface tension effects need attention in the Riemann problem resolution. Indeed, gravity is responsible for buoyancy effects while surface tension maintains spherical shape of sodium drop. A modified HLLC solver [11] is built to maintain mechanical equilibrium and prevent parasitic currents. The HLLC solver is modified through the following intermediate wave speed estimate: SM

 = p R − p L + ρ L u L (SL − u L ) − ρ R u R (S R − u R ) + ρ R g(y R − y¯ )−

ρ L g(y L − y¯ ) − σ κ(αl,R

   − αl,L ) / ρ L (SL − u L ) − ρ R (S R − u R ) ,

where y¯ denotes the face center vertical coordinate. The method has been validated on various test problems, such as Laplace law and transport of a spherical drop in a flow in mechanical equilibrium, showing excellent agreement.

5 Computational Example 2D axisymmetric configuration is considered in the computations shown in Fig. 2. Six relevant stages appear during time evolution.

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(a) During the first instants, the immersed liquid sodium drop reacts with the water vapor through surface reaction. Pressure and temperature increase setting to motion the various liquid–gas interfaces and increasing gas pocket volume. (b) The hot gas pocket reaches the liquid water surface and induces liquid film breakup. Surface waves propagate to the left and right of the liquid–gas interface while a liquid water fragment is projected vertically. The sodium drop moves l continues to increase as a consequence of surface up and its temperature TNa l = 735 K). reaction combined to heat diffusion (TNa (c) When the liquid water fragment falls, thickness of the gas film separating the two liquids decreases significantly and surface reaction intensifies. Large amount of sodium vapor locally appears and mixes with water vapor. When the liquid water fragment is very close to the sodium drop, the temperature of the thin gas layer separating both liquids sharply increases and reaches the threshold temperature (1250 K) of gas-phase reactions. Premixed vapors then strongly react, and a shock is emitted. It also leads to significant deformation of the sodium drop that fragments. The resulting small sodium drop is ejected vertically. (d) Later, the residual sodium drop reaches the water surface and floats, still separated from the liquid water by a gas layer. Thermochemical Leidenfrost-type effect happens. At time t = 0.81 s, sodium temperature reaches 879 K. l = 910 K), a new shock wave is emitted and the explosion (e) At time t = 1.06 s (TNa projects the sodium drop outside the domain, at velocity about 2 ms−1 on this example. This explosion results of premixed vapors inside the gas film located under sodium drop that have reached the ignition temperature of gas-phase reactions. (f) After sodium drop ejection, the water surface returns to its equilibrium state under the effect of gravity. It therefore appears that explosion can be qualitatively reproduced by the present model and is a consequence of gas premixing and finite rate chemistry in the gas phase.

6 Conclusion A diffuse interface approach, consisting in a flow model in temperature and pressure equilibrium, has been used as a basis to model contact and reaction between sodium and liquid water through a gas film. Thanks to this modeling, two main effects have been qualitatively reproduced: • The thermochemical Leidenfrost-type effect responsible for autonomous drop motion, as shown in Fig. 2, in accordance with experimental observations and • Explosion effect with shock wave emission and sodium ejection (Fig. 2). It seems that the present work provides the first qualitative reproduction of these events with a numerical approach. From the present computations, it seems that

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Fig. 2 2D axisymmetric computed results in the presence of compressible fluid motion, heat diffusion, phase transition of sodium, water and soda, mass diffusion in the gas phase, surface reaction, gas reaction with delayed ignition, gravity, and capillary effects. Turbulent mixing of sodium and water vapors has also been considered in the gas film. The geometric configuration being 2D axisymmetric, the calculation domain is the half of the one represented. The associated mesh is made of 25 725 square cells. Results are shown at several times. The liquid water is shown in black through its volume fraction, whereas the liquid sodium drop appears in dark gray. Elevated temperature zones are shown in shades of gray around the drop

explosion is a result of turbulent mixing of the various vapors, resulting in a premixed flame and not a diffusion one, as speculated formerly. Many perspectives emerge to improve present qualitative observations: • Determination of the chemical kinetics of sodium vapor with water vapor; • Use fine enough space resolution to capture small-scale physics such as bubbles dynamics in the vicinity of the gas film, responsible for turbulent mixing; and • Improve knowledge of mass diffusion coefficients. A key point relies on the numerical treatment of surface tension effects. Two regularizations have been used in the present study, with the color function and curvature computation. Efforts must be done to improve related computations. Another numerical issue is related to the consideration of mass diffusion in the context of diffuse interface formulations.

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References 1. Carnevali, S., Proust, C., Soucille, M.: Unsteady aspects of sodium-water-air reaction. Chem. Eng. Res. Des. 91(4), 633–639 (2013) 2. Daudin, K., Beauchamp, F., Proust, C.: Phenomenological study of the pre-mixing step of sodium-water explosive interaction. Exp. Therm. Fluid Sci. 91, 1–8 (2018) 3. David, L., Beauchamp, F., Allou, A., Saurel, R., Guiffard, P., Daudin, K.: A small scale experiment and a simplified model to investigate the runaway of sodium-water reaction. Int. J. Heat and Mass Transf. in press 4. Saurel, R., Pantano, C.: Diffuse-interface capturing methods for compressible two-phase flows. Annu. Rev. Fluid Mech. 50, 105–130 (2018) 5. Le Martelot, S., Saurel, R., Nkonga, B.: Towards the direct numerical simulation of nucleate boiling flows. Int. J. Multiph. Flow 66, 62–78 (2014) 6. Furfaro, D., Saurel, R., David, L., Beauchamp, F.: Towards sodium combustion modelling with liquid water. J. Comput. Phys. 403, 109060 7. Le Métayer, O., Saurel, R.: The noble-abel stiffened-gas equation of state. Phys. Fluids 28(4), 046102 (2016) 8. Chiapolino, A., Saurel, R., Nkonga, B.: Sharpening diffuse interfaces with compressible fluids on unstructured meshes. J. Comput. Phys. 340, 389–417 (2017) 9. Godunov, S.K.: A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat. Sb. 89(3), 271–306 (1959) 10. Chiapolino, A., Boivin, P., Saurel, R.: A simple and fast phase transition relaxation solver for compressible multicomponent two-phase flows. Comput. Fluids 150, 31–45 (2017) 11. Toro, E.F., Spruce, M., Speares, W.: Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 4(1), 25–34 (1994)

Geodesic Mesh MHD: A New Paradigm for Computational Astrophysics and Space Physics Applied to Spherical Systems Sudip K. Garain, Dinshaw S. Balsara and Vladimir Florinski

Abstract A majority of astrophysics and space physics systems tend to be spherical. The logically Cartesian r-θ -φ meshes that are often used in simulations of such systems suffer from a few major deficiencies such as nonuniform coverage of space, loss of accuracy close to the pole, and severe limitation on time stepping. Additionally, it becomes increasingly important to have higher orders of accuracy in order to reduce the dissipation and dispersion. In this work, we describe a new finite volume implementation of MHD equations on a triangular geodesic mesh that is accurate up to fourth order. To achieve the accuracy, we use a combination of: (a) WENO methods, (b) A re-formulation to support higher order divergence-free reconstruction of magnetic field, (c) multidimensional Riemann solvers, (d) higher order timestepping to match the spatial accuracy. A few representative test problems are presented here to highlight the utility of our present approach.

1 Introduction Several astrophysics and space physics problems require numerical simulations of neutral or ionized fluid on spherical meshes. Simulations of such systems tend to be done on logically Cartesian meshes. The r-θ -φ meshes that are often used have a few major deficiencies: (a) they do not cover the sphere uniformly, (b) the coordinate singularity on-axis results in loss of accuracy, and (c) the timestep is seriously diminished. As astrophysicists and space physicists actively take on the issues of MHD S. K. Garain Korea Astronomy and Space Science Institute, 776, Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea e-mail: [email protected] D. S. Balsara (B) Physics and ACMS Departments, University of Notre Dame, Notre Dame, IN 46545, USA e-mail: [email protected] V. Florinski Space Science Department, University of Huntsville, Huntsville, AL 35899, USA © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_19

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turbulence, it becomes increasingly important to have higher orders of accuracy. High accuracy schemes are the only way of reducing the dissipation and dispersion that should be held down in the turbulence simulations. In this work, we show that an excellent alternative is available. Here we describe a new finite volume implementation of MHD equations on a triangular geodesic mesh that is accurate up to fourth order. To achieve the accuracy, we use a combination of: (a) WENO methods, (b) a re-formulation to support higher order divergence-free reconstruction of magnetic field, (c) multidimensional Riemann solvers, and (d) higher order timestepping to match the spatial accuracy. This article is arranged in the following way. In Sect. 2, we briefly outline the mesh construction and our strategy to represent spherical geometry using iso-parametric mapping. In Sect. 3, we outline the algorithms that have been used for the present purpose. In Sect. 4, we present results of a few representative test problems to highlight the utility of our present approach. In Sect. 5, we show evidence of petascale performance of this scheme. Finally, in Sect. 6, we provide a discussion and our conclusion. Because of the limited space, majority of the technical details are omitted here. Instead our goal in this paper is to provide an outline of the numerical scheme. Interested readers are strongly encouraged to go through references [3–5] for all the algorithmic details.

2 Mesh Construction and Representing Spherical Geometry Figure 1 shows the construction of a geodesic mesh. Mesh construction starts by inscribing an icosahedron inside a sphere and projecting its edges on the surface of a sphere. This produces a division 0 mesh. Since an icosahedron has 20 triangular surfaces, a division 0 mesh also has 20 triangular faces (Fig. 1a and b). Each triangular face can be further subdivided into four triangles by connecting the centers of each side and the resulting division 1 mesh is generated (Fig. 1c). Following this procedure, a mesh of desired division can be produced on a spherical surface. As an example, Fig. 1d shows a division four mesh. Once a mesh of desired division is produced on the spherical surface, it can be extruded radially to generate a three-dimensional mesh. Each zone of such a mesh has a shape which is closest to a three-dimensional triangular prism and is called a frustum. In order to achieve a high-order accurate line, surface, or volume integrals inside a frustum, it is found advantageous to use an iso-parametric mapping from a reference zone. In this case, the reference zone is an equilateral triangular prism. Iso-parametric mappings of various orders have been discussed in Chap. 8 of [8]. We have found that iso-parametric mapping based on quadratic and cubic polynomials (termed as thirdand fourth-order mappings, respectively) reproduces the curvature of a frustum with remarkable accuracy.

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Fig. 1 Has been taken from [4]. This shows the construction of a geodesic mesh. See text for details

3 Algorithm The challenging task in designing the high-order accurate algorithm is to extend the well-known algorithms on the rectangular meshes to unstructured meshes with curved boundaries. Any loss of accuracy in the geometric representation will immediately manifest itself as a loss of accuracy in the overall algorithm. The iso-parametric mapping has also to be integrated into all requisite steps of the algorithm. The first step is to make a TVD or WENO spatial reconstruction. As in the structured mesh RIEMANN code, the primal variables in this code are also the zone averaged density, three components of momentum, energy, and the facially averaged three components of the magnetic fields. The reconstruction algorithms that are used here are all based on WENO algorithms for the higher order spatial reconstruction [1, 2, 7]. Taylor series basis, with appropriate modification, has been used for polynomial representation of the primal variables inside the zones. Additional innovations

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are incorporated to ensure that a curved iso-parametric mapping is fully integrated into the reconstruction strategy. We also accurately preserve the analogy between the constrained update of magnetic field on rectangular meshes with the update on frustums. Interested readers are encouraged to see [3, 4] for efficient implementation of reconstruction of zone-centered variables as well as divergence-free reconstruction of magnetic field which extend to iso-parametrically mapped meshes with curved boundaries. WENO reconstruction inside a zone is performed using a set of stencils that include zones in a certain proximity of that zone. The size of a stencil depends on the desired accuracy. A detailed discussion on the stencil construction with a particular focus for such geodesic mesh is provided in Sect. 6 of [5]. The code is based on a predictor–corrector philosophy. To enable efficient parallel processing with a minimum of messaging, we also use ADER timestepping algorithms, which only require one exchange of ghost zone data per timestep. The technical- and implementation-related details of the ADER step are provided in Sect. 5 of [4]. The ADER provides the predictor step. The corrector step is provided by one-dimensional and multidimensional Riemann solvers. The one-dimensional Riemann solver sits at the zone boundary between two zones and uses the states from either side of the zone boundary. They are used at iso-parametrically mapped quadrature points on the faces of the frustums to compute the upwinded flux, and subsequently facially averaged fluxes are computed using the quadrature weights. Multidimensional MuSIC Riemann solvers are called at the curved edges to compute the edge averaged, multidimensionally upwinded line integrals of the electric fields for updating the edge-centered magnetic field variables.

4 Results In this section, we present results of a few truly multidimensional test problems. In the first example, we demonstrate that our scheme achieves the desired accuracies for both hydrodynamic and magnetized flows. In the second example, we demonstrate the performance of our code for a strong MHD blast problem.

4.1 Accuracy Tests For accuracy performance tests, we conducted two experiments. For the first problem, we simulated the Parker solar wind problem on a computational domain of radial extent [2.0 : 3.5] and compared the numerical result with a steady-state solution for computing the errors. The second test problem is the manufactured solution on spherical mesh adopted from [6]. Figure 2 shows the L1 errors for the two problems. More detailed results for both these problems are presented in [3, 4].

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Fig. 2 Shows the L1 errors for (a) density variable for solar wind test problem and (b) x-magnetic field variable for manufactured solution

4.2 MHD Blast Problem In this test problem, we show the performance of our code against a stringent, threedimensional MHD blast problem. The detailed setup and the results of this test problem are also presented in [4]. The entire computational domain is filled up with the initially static, uniform density plasma and immersed in a uniform magnetic field pointing along a unit vector. A high-pressure spherical explosion zone of unit radius is set up at the center of the computational domain. The magnitudes of the pressure and the magnetic field are chosen such that the initial plasma beta of the explosion zone is 2.51 and the ambient is 0.000251. Figure 3 shows the (a) density, (b) pressure, (c) total velocity, and (d) magnitude of magnetic field at a final time of 0.06. A third-order accurate ADER-WENO scheme is used for this result.

5 Parallelism and Achieving Petascale Performance The domain decomposition is achieved both on the spherical surface and in the radial direction. By recursive subdivision, we can decompose the entire spherical surface into several sectors. Additionally, the radial zones are subdivided into several radial slabs. A three-dimensional block consisting of a sector and a radial slab is called a patch. Each such patch can be thought of as a mesh patch that can be processed on an individual processor. Each patch has 38 neighbors and well-designed message packing and unpacking strategies have been built to ensure efficient exchange of data across the processors. We have performed weak scalability study on Blue Waters supercomputer at NCSA. Figure 4 shows that our scheme scales quite well.

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Fig. 3 Has been taken from [1]. This shows the results of the MHD blast test problem: (a) density, (b) pressure, (c) total velocity, and (d) magnitude of magnetic field are shown at a final time of 0.06

Fig. 4 Has been taken from [4]. This shows the scalability study for (a) second-order and (b) third-order ADER-WENO scheme

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6 Discussion and Conclusion In this article, we outline an up to fourth-order accurate, novel algorithm for carrying out the divergence-free MHD simulations on a geodesic mesh. Several innovations have been achieved. The iso-parametric mapping is found to accurately represent the spherical geometry. The reconstruction of the fluid variables is carried out using WENO algorithms that integrate the mesh geometry. Divergence-free reconstruction for magnetic fields and the ADER predictor step are efficiently adapted to isoparametrically mapped meshes. The use of ADER allows one-step update that only requires one messaging step per complete timestep and hence helps in achieving petascale performance.

References 1. Balsara, D.S., Shu, C.-W.: Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160(2), 405–452 (2000) 2. Balsara, D.S., Garain, S., Shu, C.-W.: An efficient class of WENO schemes with adaptive order. J. Comput. Phys. 326, 780–804 (2016) 3. Balsara, D.S., Garain, S., Florinski, V., Boscheri W.: An efficient class of WENO schemes with adaptive order for unstructured meshes. J. Comput. Phys. 404, 109062–109094 (2020) 4. Balsara, D.S., Florinski, V., Garain, S., Subramanian, S., Gurski, K.: F: Efficient, divergencefree, high-order MHD on 3D spherical meshes with optimal geodesic meshing. MNRAS 487(1), 1283–1314 (2019) 5. Florinski, V., Balsara, D.S., Garain, S., Gurski, K.F.: Technologies for supporting high-order geodesic mesh frameworks for computational astrophysics and space sciences. Comput. Astrophys. Cosmol. (2020) 6. Ivan, L., Sterck, H., De Susanto, A., Groth, C.P.T.: High-order central ENO finite-volume scheme for hyperbolic conservation laws on three-dimensional cubed-sphere grids. J. Comput. Phys. 282, 157–182 (2015) 7. Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996) 8. Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method. Volume I: The Basis, 5th edn. Butterworth Hienemann Publishers, Oxford (2000)

The Works of the S. K. Godunov School on Hyperbolic Equations Valeri˘ı M. Gordienko

To Sergey Konstatinovich Godunov on his 90th Birthday

Abstract In the mid-1970s, at Novosibirsk State University, the S. K. Godunov seminar on hyperbolic equations started its work. The article describes the works of the participants on hyperbolic equations. The main interest was concentrated around two problems. The first is the reduction of a high-order Petrovski˘ı hyperbolic equation to a first-order Friedrichs hyperbolic symmetric system. The second problem is that if a boundary-value problem is posed for a hyperbolic equation then it is required to reduce it to a symmetric system so that the posed boundary condition be dissipative.

Hyperbolic equations and systems describe wave processes and therefore play an important role in mathematical physics. The definition of hyperbolic equations and systems was given in 1937 by I. G. Petrovski˘ı. His definition is based on the properties of the roots of the characteristic polynomial. He proved the well-posedness of the Cauchy problem by using the Fourier method. In 1953, J. Leray proposed simpler proofs of Petrovski˘ı’s results, paying more attention to topological questions. In 1954, K. O. Friedrichs gave another definition of hyperbolic systems, which was based on matrix symmetry and did not require the calculation of the roots of polynomials. Friedrichs systems admit multiple characteristics. K. O. Friedrichs bases his proof of the well-posedness of the Cauchy problem for his systems on an energy integral identity which easily follows from the symmetry of the matrices and the a priori estimates resulting from this identity. Transfer to variable coefficients and to the lower V. M. Gordienko (B) Sobolev Institute of Mathematics, pr. Koptyuga 4, 630090 Novosibirsk, Russia e-mail: [email protected] Novosibirsk State University, ul. Pirogova 2, 630090 Novosibirsk, Russia © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_20

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terms in Friedrichs systems is much simpler than in systems hyperbolic according to Petrovski˘ı. Friedrichs systems also turned out to be very convenient in computational mathematics. In the mid-1970s, at Novosibirsk State University, the S. K. Godunov seminar on hyperbolic equations started its work. The following question was posed: Can a high-order Petrovski˘ı hyperbolic equation always be reduced to a first-order Friedrichs hyperbolic symmetric system (see [1]). By that time, such a reduction was known for any hyperbolic equation with one space variable and for a second-order hyperbolic equation with an arbitrary number of variables. The proof of the fact that a hyperbolic equation with two space variables is reduced to a symmetric system turned out to be nontrivial. It was given by S. K. Godunov and V. I. Kostin (see [2]) and based on a nontrivial theorem by Rosenblatt on the decomposition of polynomials, which is a generalization of the well-known Fejér–Riesz theorem. Later Kostin [3] and Mikha˘ılova [4] showed that rotation-invariant hyperbolic equations and all equations sufficiently close to them are reduced to a symmetric system. But in general, the answer to the question about the equivalence of the two definitions of hyperbolicity turned out to be negative. In [5], V. V. Ivanov constructed a fourth-order hyperbolic equation with four space variables that was not reduced to a first-order Friedrichs symmetric hyperbolic system. V. V. Ivanov’s example is connected with Hilbert’s assertion of the existence of homogeneous positive polynomials not decomposable into a sum of squares. However, it is still not known whether the definitions are equivalent in the case of three space variables. If the definitions are not equivalent in the three-dimensional case then it is interesting to distinguish a class of hyperbolic equations reducible to Friedrichs systems. Perhaps, nonreducible hyperbolic equations do not occur in applications? Parallel to general questions, the participants of the S. K. Godunov seminar were engaged in the research of the well-posedness of boundary-value problems. The fact is that if a hyperbolic equation was reduced to a symmetric system then it was always in a nonunique way; this nonuniqueness was described with the use of Young diagrams by A. V. Tyshchenko in [6]. It was proposed to use the nonuniqueness of the reduction of an equation to a system in the theory of boundary-value problems. It was required to reduce a hyperbolic equation with a boundary condition to a symmetric system for which the posed boundary condition would be dissipative. The first work of the S. K. Godunov seminar was the paper [7], which focused on a mixed problem for the real wave equation with two and three space variables. Formulate the results for the two-dimensional case (in the three-dimensional case, the results are similar). The statement of the problem is as follows: ⎧ ⎪ ⎨u tt − u x x − u yy = 0, t > 0, x > 0; u t + bu x + cu y = 0, x = 0; ⎪ ⎩ t = 0. u = ϕ, u t = ψ,

(1)

If (b, c) belong to the half-disk {b2 + c2 < 1, b > 0} or lie on the straight line {b = 1}, the Hadamard ill-posedness example is constructed for the problem. In the remain-

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ing cases, the mixed problem (1) was reduced to a symmetric hyperbolic system with dissipative boundary condition. Two cases were distinguished. If the parameters of the boundary condition (b, c) lay in the half-plane {|c| ≤ 1, b < 0} then the symmetric system to which the wave equation was reduced contained all the first derivatives, and in other cases, some (but not all) of the second derivatives had to be added. Thus, a loss in smoothness occurred. This loss in smoothness was related to the plane waves found in this case, which propagated along the boundary faster than the speed of sound. In 1976, when S. K. Godunov spoke on this work at a conference held in Alma-Ata, O. A. Ole˘ınik, who attended his talk, drew attention to the works by Japanese mathematicians (in particular, by M. Ikava and S. Miataki) which involved a stronger constraint on the boundary condition—the uniform Lopatinski˘ı condition. If the Lopatinski˘ı condition is the absence of Hadamard ill-posedness examples for the problem then the uniform Lopatinski˘ı condition is the absence of Hadamard examples for the problem and for all problems with sufficiently close coefficients in the boundary condition. Moreover, also sufficiently close complex (b, c) must be considered. It turned out that (in contrast to the Lopatinski˘ı condition) the uniform Lopatinski˘ı condition ensures the preservation of smoothness and the absence of waves running along the boundary faster than the speed of sound. In problem (1), the uniform Lopatinski˘ı condition is fulfilled only in the half-strip {|c| ≤ 1, b < 0}. Later we always required the fulfillment of the uniform Lopatinski˘ı condition. In addition, we did not confine ourselves to formulating well-posedness in terms of the roots of polynomials but strove for formulating it in terms of the coefficient of the boundary condition, i.e., solved the corresponding Routh–Hurwitz problem. In [8], V. M. Gordienko considered a generalization of problem (1)—a mixed problem for a vector complex-valued wave equation with two space variables and a complex boundary condition: ⎧ ⎪ t > 0, x > 0; ⎨Utt − Ux x − U yy = 0, AUt + BUx + CU y = 0, x = 0; ⎪ ⎩ t = 0. U = Φ, Ut = Ψ,

(2)

Here U is a column of n complex-valued functions (u 1 , u 2 , . . . , u n ); A, B, C are complex square matrices. It is not easy to find out whether the uniform Lopatinski˘ı condition is fulfilled for this problem (in terms of the coefficients of the boundary condition). Therefore, it is very important that the question of the uniform Lopatinski˘ı condition for problem (2) was reduced to the question of the asymptotic stability of the following system of ordinary differential equations of order 2n: dW = Λ2n W, ds

(3)

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 O I , Γ = −(A + C)−1 (A − C), Δ = 2(A + C)−1 B. Γ Δ Here I is the identity matrix and O is the zero matrix of size n × n. As is known, for studying the asymptotic stability of system (3), one can use the matrix Lyapunov condition (4) Λ∗2n H2n + H2n Λ2n = −G 2n .

where Λ2n =

Namely, choose arbitrarily a Hermitian positive definite matrix G 2n of order 2n and try to solve the matrix equation (4) with respect to H2n . If we succeed and the matrix H2n turns out to be positive definite then, in this case (and only in this case), system (3) is asymptotically stable, and hence the uniform Lopatinski˘ı condition is fulfilled for problem (2). If the uniform Lopatinskii condition is fulfilled then the mixed problem (2) is reduced to a mixed problem for a symmetric hyperbolic system with dissipative boundary condition. The reduction is based on the same Lyapunov equation (4) which is involved in the study of well-posedness. Describe the construction of this reduction. It is easy to prove that if a vector function U satisfies problem (2) then the composed vector function of size 3n of the first derivatives   U1 U2 U3 = Ut U x U y satisfies the problem ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎧⎡ K L M U1 L K iN U1 M −i N K U1 ⎪ ⎪ ⎪ ⎢ ⎥∂⎢ ⎥ ⎢ ⎥∂⎢ ⎥ ⎢ ⎥∂⎢ ⎥ ⎪ ⎪ − − L K i N U U U K L M i N −M L ⎣ ⎦ ⎣ ⎦ ⎣ ⎣ ⎦ ⎣ ⎣ ⎦ ⎦ ⎪ 2 2 2 ⎦= 0, ∂t ∂x ∂y ⎪ ⎪ ⎨ M −i N K U3 U3 U3 −i N M −L K L M ⎪ t > 0, x > 0; ⎪ ⎪ ⎪ ⎪ ⎪ x = 0; ⎪ AU1 + BU2 + CU3 = 0, ⎪ ⎩ t = 0. U1 = Φx , U2 = Φ y , Ut = Ψ, (5) Here K , L, M, N are (now) arbitrary matrices of rank n × n. If the matrices K , L, M, N are Hermitian then the composed matrices in system (5) are also Hermitian. If, moreover, the composed matrix at ∂t∂ is positive definite, ⎡

⎤ K L M ⎣ L K i N ⎦ > 0, M −i N K

(6)

then system (5) is Friedrichs symmetric t-hyperbolic. The boundary condition in (5) is dissipative if ⎛⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎞ L K iN U1 U1 ⎝⎣ K L M ⎦ ⎣U2 ⎦ , ⎣U2 ⎦⎠ > 0 (7) −i N M −L U3 U3

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for AU1 + BU2 + CU3 = 0, ||U1 || + ||U2 || + ||U3 || = 0. Thus, the mixed problem for the wave equation (2) will be reduced to a mixed problem for a symmetric hyperbolic system with dissipative boundary condition (5) if there are Hermitian matrices K , L, M, N such that conditions (6) and (7) are fulfilled. It turns out that such a choice of the matrices K , L, M, N is always possible if problem (2) satisfies the uniform Lopatinski˘ı condition. Let us formulate this rule of choice. Suppose that problem (2) satisfies the uniform Lopatinski˘ı condition. As we said, this is equivalent to the asymptotic stability of system (3), which is in turn equivalent to the existence of a matrix H2n > 0—a positive definite solution to equation (4) (for any G 2n > 0). We must define the matrices K , L, M, N so that 

 K − M −L − i N = H2n . −L + i N K + M

(8)

It is always possible to do that in a unique way, and the matrices K , L, M, N will be Hermitian. Moreover, condition (6) will follow from the fact that H2n > 0, whereas condition (7) will stem from the fact that G 2n > 0. Thus, the mixed problem (2) either does not satisfy the uniform Lopatinski˘ı condition or is reduced to a symmetric hyperbolic system with dissipative boundary condition. Moreover, the study of the question of the fulfillment of the uniform Lopatinski˘ı condition and the construction of the necessary symmetric hyperbolic system are reduced to solving the same Lyapunov matrix equation (4). This article was published in Comptes Rendus de l’Académie des Sciences (communicated by J. Leray), see [8]. The found relationship of the mixed problem for the wave equation with the asymptotic stability of some system of ordinary differential equations has not yet been understood. A particular case of [8]—n = 1—is exposed in [9]. Let us make some more observations. As we already said, if H2n > 0 and the matrices K , L, M, N are defined from (8) then condition (6) holds, which guarantees the hyperbolicity of the system on (5). It turns out that the converse fails. Condition (6) does not imply the positive definiteness of matrix (8). Give an example: n = 2,         10 l 0 0 m 00 K = , L= , M= , N= . 01 0 −l m 0 00 It is easy to check that ⎡

⎤ K L M ⎣ L K i N ⎦ > 0 for m 2 + l 2 < 1; M −i N K this is the unit disk in the plane (m, l), and

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 H2n =

 K − M −L − i N > 0 for |m| + |l| < 1; −L + i N K + M

this is the interior of a square inscribed in this unit disk. Thus, the condition H2n > 0 is stronger than (6), it ensures not only the hyperbolicity of the system in (5) but, as was shown by N. G. Marchuk in [10], also the coincidence of the Hamilton–Jacobi cones of the vector wave equation with two spacial variables (2) and the system in (5) to which it is reduced. In other words, the condition H2n > 0 guarantees not only the hyperbolicity of the system in (5) (the positive definiteness of the matrix for ∂t∂ ) but also the coincidence of the uniqueness domain of the system in (5) with the uniqueness domain of the initial wave equation. The following remark is concerned with the proof of an existence theorem for problem (2). Two approaches are possible. The first approach is as follows: using the reduction of the mixed problem (2) to the mixed problem (5) obtain a dissipative energy integral; using it, get a priori estimates, and with the use of these obtain an existence theorem in a standard manner (as, for example, in S. K. Godunov’s textbook [11]). Another approach was proposed by N. G. Marchuk in [10, 12]. He proved the possibility of inverse passage from problem (5) to problem (2). Here substantial was the condition H2n > 0 and not only the fact that the system in (5) is symmetric hyperbolic. The problem for the vector wave equation (2) is interesting in its own right but it also made it possible to consider the mixed problem for the scalar wave equation with a high-order boundary condition ⎧ ⎪ ⎨ u tt − u x x − u yy = 0, t > 0, x > 0; pn (∂t , ∂x , ∂ y )u = 0, x = 0; ⎪ ⎩ t = 0. u = ϕ, u t = ψ,

(9)

Here pn (τ , ξ, η) is a homogeneous polynomial of degree n, possibly, with complex coefficients. Introduce the polynomial of one variable f (z) = pn (z 2 + 1, −2z, z 2 − 1). It was proved in [13] that the mixed problem (9) satisfies the uniform Lopatinski˘ı condition if and only if the coefficient at the higher degree of the polynomial f (z) is nonzero and all its 2n roots lie in the left half-plane Re z k < 0. The condition that all the roots of the polynomial f (z) lie in the left half-plane can be written down with the use of Hermite’s theorem as the condition of the positive definiteness of a real symmetric matrix of order 2n with entries expressed explicitly in terms of the coefficients of f (z). In the case of the fulfillment of the uniform Lopatinski˘ı condition, in [13], we constructed a symmetric hyperbolic system with dissipative boundary condition satisfied by the vector composed of all derivatives up to order n of a function u satisfying the mixed problem (9). A generalization of problems (2) and (9) was considered in [13–15]. Instead of the wave equation, a second-order hyperbolic equation with two space variables, lower terms, and variable coefficients was considered. In all the cases, the check

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of the uniform Lopatinski˘ı condition amounts to the positive definiteness of a matrix explicitly expressed in terms of the coefficients of the equation and the boundary condition. If the condition is fulfilled then the original problem is reduced to a symmetric hyperbolic system with dissipative boundary condition. In [16], A. N. Malyshev considered a mixed problem for a second-order hyperbolic equation with many space variables (a first-order boundary condition). Using the Lorentz transform, he reduced this problem to the case of two space variables. For the class of Friedrichs hyperbolic systems to which the wave equation with three space variables can be reduced, a relation along an additional characteristic was constructed. With the use of this relation, an assertion on the preservation of the rotor was proved. Conditions were formulated under which solutions to Friedrichs hyperbolic systems of the mentioned type are solutions to the wave equation (see [17]). In [18], we considered a mixed problem for the three-dimensional wave equation satisfying the uniform Lopatinski˘ı condition and described all the reductions to a symmetric hyperbolic system with dissipative boundary condition. It turned out that the set of all ways of reduction is a four-parameter set—a second-order solid cone. We explicitly expressed the geometric sizes and disposition of this cone in terms of the coefficients of the boundary condition. In [19], these results were extended to the multidimensional case. In [20], for the real- or complex-valued wave equation, we described all the Friedrichs symmetric hyperbolic systems to which the wave equation can be reduced. It turned out that the set of such systems constitutes a eight-parameter family and the reduction itself of the wave equation to a Friedrichs system is connected with the properties of the quaternions. The main properties of such systems were established. Systems were separated for which the propagation speeds of the perturbations coincide with the propagation speeds for the initial wave equation. The rules for transforming the systems under Lorentz transforms were found.

References 1. Godunov, S.K.: The symmetrization problem for hyperpolic operators, Uspekhi Mat. Nauk 38(5(233)), 169 (1983) 2. Godunov, S.K., Kostin, V.I.: Reduction of a hyperbolic equation to a symmetric hyperbolic system in the case of two space variables. Sib. Math. J. 21(6), 755–768 (1980) 3. Kostin, V.I.: On symmetrization of hyperbolic operators, In: Correct Boundary Value Problems for Non-Classical Equations of Mathematical Physics, Collect. Artic., pp. 112–116, Novosibirsk (1980) 4. Mikhajlova, T.Yu.: Symmetrization of invariant hyperbolic equations, Sov. Math. Dokl. 27, 644–648 (1983) 5. Ivanov, V.V.: Strictly hyperbolic operators not admitting hyperbolic symmetrization, In: Partial Differential Equations, Proc. Int. Conf., Novosibirsk 1983, pp. 84–93, Nauka, Novosibirsk (1986) 6. Tyshchenko, A.V.: A basis of solutions of the homogeneous Hörmander identity. Sib. Math. J. 26(1), 117–124 (1985)

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7. Godunov, S.K., Gordienko, V.M.: A mixed problem for the wave equation, In.: Partial Differential Equations. Tr. Semin. S.L. Soboleva 2, pp. 5–31, Inst. Mat. SO AN SSSR, Novosibirsk (1977) 8. Gordienko, V.M.: Un problème mixte pour l’équation vectorielle des ondes : Cas de dissipation de l’énergie ; Cas mal posés . C. R. Acad. Sci., Paris, Sér. A 288, 547–550 (1979) 9. Gordienko, V.M.: On well-posedness of a mixed problem for the wave equation. Sib. Èlektron. Mat. Izv. 7, 130–138 (2010) 10. Marchuk, N.G.: On the existence of solutions of a mixed problem for the vector wave equation, Sov. Math., Dokl. 21, 785–788 (1980) 11. Godunov, S.K.: Equations of Mathematical Physics. Nauka, Moscow (1979). [in Russian] 12. Marchuk, N.G.: The correctness of the formulation of a mixed problem for the vector wave equation. Differ. Equations 20, 627–631 (1984) 13. Gordienko, V.M.: Symmetrization of a mixed problem for a hyperbolic equation of second order with two spatial variables. Sib. Math. J. 22(2), 231–248 (1981) 14. Marchuk, N.G.: Existence of a solution of a mixed problem for a hyperbolic vector equation of second order. Sib. Math. J. 23(1), 53–64 (1982) 15. Bondarenko, O.A., Gordienko, V.M.: Symmetrization of second-order vector hyperbolic equation with two space variables. Sib. Math. J. 27(3), 302–312 (1986) 16. Malyshev, A.N.: Mixed problem for second-order hyperbolic equation with first-order complex boundary condition. Sib. Math. J. 24(6), 906–923 (1983) 17. Gordienko, V.M.: Hyperbolic systems equivalent to the wave equation. Sib. Mat. Zh. 50(1), 14–21 (2009) 18. Gordienko, V.M.: Dissipativity of boundary condition in a mixed problem for the threedimensional wave equation. Sib. Èlektron. Mat. Izv. 10, 311–323 (2013) 19. Gordienko, V.M.: Dissipative energy integrals in a mixed problem for the wave equation. Mat. Zamet. YAGU 20(2), 23–33 (2013) 20. Gordienko, V.M.: Friedrichs systems for the three-dimensional wave equation. Sib. Math. J. 51(6), 1013–1027 (2010)

On Two-Flow Instability of Dynamical Equilibrium States of Vlasov–Poisson Plasma Yuriy G. Gubarev

Abstract The problem on linear stability of the subclass of one-dimensional (1D) states of dynamic equilibrium boundless electrically neutral collisionless plasma in electrostatic approximation (the Vlasov–Poisson plasma) is studied. By the direct Lyapunov method, we prove that these equilibrium states are absolutely unstable relative to small 1D perturbations when the Vlasov–Poisson plasma contains electrons and the only grade of ions with stationary distribution functions which are isotropic on the physical space but dependent on velocity. We state sufficient conditions for linear practical instability; for small 1D perturbations growing over time, we construct the a priori exponential lower estimate and describe the initial data. In addition, we construct the analytical example of the studied 1D states of dynamic equilibrium and their small perturbations of the same type of symmetry growing over time according to the obtained estimate.

1 Formulation of Problem in Kinetic Description Hereinafter, we study the 1D case of the Vlasov–Poisson plasma [1, 2] which is simpler than the general case but still informative, ∂f + ∂f + nq ∂ϕ ∂ f + +v − = 0, ∂t ∂x m + ∂ x ∂v ∂f − ∂f − q ∂ϕ ∂ f − ∂ 2ϕ = − 4πq +v + = 0, ∂t ∂x m − ∂ x ∂v ∂x2

(1.1)

+∞  +  n f − f − dv; −∞

Y. G. Gubarev (B) Lavrentyev Institute for Hydrodynamics SB RAS, Lavrentyev Av., 15, Novosibirsk, Russia Novosibirsk National Research State University, Pirogova Str., 1, Novosibirsk 630090, Russian Federation e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_21

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f ± = f ± (x, v, t) ≥ 0; f ± (x, v, 0) = f 0± (x, v) . Here f + and f − denote the distribution functions of ions and electrons, respectively, t denotes time, x and v are coordinates and velocities of ions and electrons, n is a natural number determined by the charge of an ion, q denotes the modulus of the charge of an electron, m + and m − are masses of an ion and an electron, respectively, ϕ(x, t) denotes the potential of self-consistent electric field, π is a constant equal to the ratio of the length of a circle to its diameter, and f 0± denote the initial data for the functions f ± . We suppose that the functions f ± approach zero as |v| → ∞, and they are periodic on x or approach zero as |x| → ∞. There are exact stationary solutions to the initial-boundary value problem (1.1) in the form ϕ = ϕ ≡ const, f 0

±

= f

±0

+∞ (v) s.t. n

f

+0

−∞

+∞ dv =

f −0 dv,

(1.2)

−∞

where f ±0 are non-negative functions on the independent variable v. In [3–5], it was shown that the inequalities v

d f ±0 ≤0 dv

(1.3)

are the sufficient conditions for linear stability of the states (1.2) of dynamic equilibrium of the Vlasov–Poisson plasma.

2 Reformulation of Problem in Hydrodynamic Description To clarify the area where linear stability of dynamic equilibrium states (1.2) of the Vlasov–Poisson plasma follows from the sufficient condition (1.3), we apply the change of variables [6–8] in the form [9] 

∂u ± (x, λ, t) f (x, v, t) = ρ (x, λ, t) ∂λ ±

±

λ ∈ (−∞, +∞),

−1

, v = u ± (x, λ, t);

∂u ± dλ ≡ 0, (x, λ, t) = 0. dt ∂λ

Thus, the mixed problem (1.1) can be rewritten in terms of the new variables as follows: ∂u + nq ∂ϕ ∂u − q ∂ϕ ∂u + ∂u − + u+ =− , + u− = , (2.1) ∂t ∂x m+ ∂ x ∂t ∂x m− ∂ x

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    +∞  +  ∂ u+ρ+ ∂ u−ρ− ∂ρ − ∂ 2ϕ ∂ρ + + =0, + = 0, = − 4πq nρ − ρ − dλ; 2 ∂t ∂x ∂t ∂x ∂x −∞

± ± u ± (x, λ, 0) = u ± 0 (x, λ) , ρ (x, λ, 0) = ρ0 (x, λ) .

Here λ denotes the Lagrangian coordinates of ions/electrons, u ± and ρ ± denote velocities and densities of ions and electrons, respectively, with the initial data u ± 0 and ρ0± . We suppose that the functions u ± and ρ ± approach zero as |λ| → ∞, and they are periodic on x or approach zero as |x| → ∞. There are exact stationary solutions to the initial-boundary value problem (2.1) in the form ±

ϕ = ϕ ≡ const, u = u 0

±0

±

(λ), ρ = ρ

±0

+∞ +∞ +0 (λ) s.t. n ρ dλ = ρ −0 dλ, −∞

−∞

(2.2) where u ±0 are some increasing functions, and ρ ±0 are some negative functions on the independent variable λ. To find out if the solutions (2.2) are stable relative to small 1D perturbations u ± (x, λ, t), ρ ± (x, λ, t) and ϕ (x, t), we linearize the mixed problem (2.1) near the solutions (2.2).

3 A Priori Exponential Lower Estimate for Small 1D Perturbations Growing over Time We may describe these perturbations by means of the Lagrangian displacements fields ξ ± = ξ ± (x, λ, t) [10] determined as follows: ∂ξ ± ∂ξ ± = u ± − u ±0 . ∂t ∂x

(3.1)

Applying (3.1), we obtain the following initial-boundary value problem: 2 + 2 + + nq ∂ϕ ∂ 2ξ + +0 ∂ ξ +02 ∂ ξ + +0 ∂ξ + u , ρ , + 2u = − = −ρ ∂t 2 ∂ x∂t ∂x2 m+ ∂ x ∂x 2 − 2 − − ∂ 2ξ − q ∂ϕ −0 ∂ ξ −02 ∂ ξ − −0 ∂ξ + u , ρ , + 2u = = −ρ ∂t 2 ∂ x∂t ∂x2 m− ∂ x ∂x

∂ 2ϕ = −4πq ∂x2

+∞ + − +0 ∂ξ −0 ∂ξ −nρ +ρ dλ; ∂x ∂x

−∞

(3.2)

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ξ ± (x, λ, 0) = ξ0± (x, λ),

∂ξ ± (x, λ, 0) = ∂t



∂ξ ± ∂t

 (x, λ). 0

Let us introduce the Lyapunov functional M [11], +∞ +∞   M≡ m + ρ +0 ξ +2 + m − ρ −0 ξ −2 dλd x ≥ 0.

(3.3)

−∞ −∞

By (3.1), (3.2), the following differential inequality for M (3.3) holds [12–17]: dM d2 M + 2(ν 2 + ω)M ≥ 0. − 2ν dt 2 dt

(3.4)

Here ν is a constant, and ω is the known positive constant. The inequality (3.4) together with the countable set of the conditions

−1 M(kt∗ ) > 0; k = 0, 1, 2, . . . ; t∗ ≡ π 2 ν 2 + 2ω ;

(3.5)

dM ω M (kt∗ ) ; M (kt∗ ) ≡ M(0) exp(νkt∗ ), (kt∗ ) ≥ 2 ν + dt ν ω dM dM dM (0) exp(νkt∗ ); M(0) > 0, (0) ≥ 2 ν + M(0) (kt∗ ) ≡ dt dt dt ν imply the a priori exponential lower estimate M(t) ≥ C exp(νt),

(3.6)

where C is a known positive constant, for the growth over time of small 1D perturbations ξ ± (x, λ, t) (3.1), (3.2) of the exact stationary solutions (2.2) to the mixed problem (2.1). Since we have obtained the lower estimate (3.6) without any restrictions on the exact stationary solutions (2.2) to the initial-boundary value problem (2.1), so, these solutions are absolutely unstable with respect to small 1D perturbations (3.1), (3.2), (3.5). Then the inequalities (1.3) play the role of the criterion for linear stability of the exact stationary solutions (1.2) to the mixed problem (1.1) with respect to small 1D perturbations u ± (x, λ, t), ρ ± (x, λ, t) and ϕ (x, t) from the incomplete and unclosed subclass +∞ −∞

 u ±0 κ ±0 u ± 2

λ→+∞

 − u ±0 κ ±0 u ± 2

λ→−∞



d x → 0; κ ±0 ≡ ρ ±0



du ±0 dλ

−1 .

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Note that the first two inequalities of (3.5) are the desired sufficient conditions for practical instability of the exact stationary solutions (2.2) to the initial-boundary value problem (2.1) with respect to small 1D perturbations (3.1), (3.2), (3.5). The same inequalities play the role of the criterion for linear practical instability of the exact stationary solutions (2.2) to the mixed problem (2.1) with respect to 1D perturbations ξ ± (x, λ, t) (3.1), (3.2), (3.5) presented in the form of normal waves. In the next section, we construct an example of the exact stationary solutions (2.2) to the initial-boundary value problem (2.1) joint with small 1D perturbations (3.1), (3.2), (3.5) growing over time while the criterion (1.3), estimate (3.6), and the first two inequalities of (3.5) are satisfied. For simplicity, we study the case n = 1, i.e., the hydrogen model of the Vlasov–Poisson plasma (see the mixed problem (1.1)).

4 Example Namely, we choose the exact stationary solution (2.2) to the initial-boundary value problem (2.1) in the form u ± = u ±0 (λ) = λ, ρ ± = ρ ±0 (λ) =

1 + λ2 , ϕ = ϕ 0 ≡ const. exp(λ2 )

(4.1)

Let us explain the choice of the solution (4.1). The inverse change of variables (x, λ, t) → (x, v, t) maps u ±0 to v and ρ ±0 to f ±0 . Thus, the exact stationary solution (2.2) to the mixed problem (2.1) in the form (4.1) is also the dynamic equilibrium state (1.2) of the Vlasov–Poisson plasma with n = 1. It is easy to check that the inequalities (1.3) for the solution (1.2) to the initial-boundary value problem (1.1) in the form (4.1) are fulfilled, since dv

dρ ±0 f ±0 =λ = − 2λ4 exp(−λ2 ) ≤ 0. dv dλ

So, the criterion (1.3) holds for the corresponding state (1.2) of dynamic equilibrium of the hydrogen Vlasov–Poisson plasma. However, growing over time small 1D perturbations (3.1), (3.2), (3.5) in the form of normal waves for the exact stationary solution (4.1) to the mixed problem (2.1) exist. Indeed, let small 1D perturbations (3.1), (3.2) have a form of normal waves, ξ ± (x, λ, t) = ξ1± (λ) exp(αt + i x).

(4.2)

Here ξ1± are some functions of λ, α = 0 is some real constant, i is the imaginary unit. Then the dispersion expression for the exact stationary solution (4.1) to the initialboundary value problem (2.1) with q = m − ≡ 1, m + ≈ 1837 has the following form:

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   √ √ g(α) ≡ 1 + 4.0022 π π 1 − 2α 2 + 2 π α 3 exp α 2 [1 − erf(α)] ≈ 0. (4.3)

We do not know how to solve the equation (4.3) analytically, so we have solved it graphically. From the plot of the function g, we have a family of positive roots 5.5 < α < 6 for the Eq. (4.3). These roots correspond to growing over time small 1D perturbations ξ ± (x, λ, t) (3.1), (3.2), (3.5) in the form (4.2) of normal waves for the exact stationary solution (4.1) to the mixed problem (2.1). Moreover, choosing the constant ν (3.4) equal to the minimal positive root of (4.3), we have that the inequality (3.4), the conditions (3.5), and the estimate (3.6) are satisfied for growing over time small 1D perturbations (3.1), (3.2), (3.5) in the form (4.2) of normal waves for the exact stationary solution (4.1) to the initial-boundary value problem (2.1). As a result, the example is constructed. Note that this example is in fact a counterexample for the spectral Newcomb–Gardner theorem [3, 4] as well as for the spectral Penrose criterion [18].

Conclusion Since the differential inequality (3.4) is quite general, we expect that this inequality can be useful in the study of other mathematical models for liquid, gas, and plasma. Besides, the algorithm for the construction of the growing over time Lyapunov functional will be useful in problems of both linear theoretical and practical stability of either hydrodynamic or kinetic type. Acknowledgements Author is sincerely grateful to G.V. Demidenko and K.V. Lotov for formulating the problem and for useful advice. Also, author thanks V.Yu. Gubarev for the help with translation from Russian to English.

References 1. Holm, D.D., Marsden, J.E., Ratiu, T., Weinstein, A.: Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123, 1–116 (1985) 2. Pokhozhaev, S.I.: On stationary solutions of the Vlasov-Poisson equations. Differ. Equ. 46, 530–537 (2010) 3. Bernstein, I.B.: Waves in a plasma in a magnetic field. Phys. Rev. 109, 10–21 (1958) 4. Gardner, C.S.: Bound on the energy available from a plasma. Phys. Fluids 6, 839–840 (1963) 5. Rosenbluth, M.N.: Topics in microinstabilities. In: Rosenbluth, M.N. (ed.) Advanced Plasma Theory, pp. 137–158. Academic Press, New York (1964) 6. Zakharov, V.E., Kuznetsov, E.A.: Hamiltonian formalism for nonlinear waves. Phys.-Uspekhi 40, 1087–1116 (1997) 7. Zakharov, V.E.: Benney’s equations and quasiclassical approximation in the inverse problem method. Funct. Anal. Its Appl. 14, 89–98 (1980) 8. Zakharov, V.E., Kuznetsov, E.A.: Hamiltonian Formalism for Hydrodynamic Type Systems, vol. 186. IA&E SB AS USSR, Novosibirsk (1982) [in Russian]

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9. Gubarev, Y.G.: On an analogy between the Benney’s equations and the Vlasov-Poisson’s equations. In: Kedrinskiy, V.K. (ed.) Acoustics of Heterogeneous Media. Dinamika Sploshn. Sredy, vol. 110, pp. 78–90. LIH SB RAS, Novosibirsk (1995) [in Russian] 10. Chandrasekhar, S.: Ellipsoidal Figures of Equilibrium. Yale University Press, New Haven (1969) 11. Gubarev, Y.G.: On two-flow instability of states of dynamic equilibrium of Vlasov-Poisson plasma. In: Fomin, V.M., Zapryagaev, V.I. (eds.) Jet, Separated and Unsteady Flows, pp. 50–51. Parallel , Novosibirsk (2015) [in Russian] 12. Gubarev, Y.G.: The Direct Lyapunov Method. The Stability of Quiescent States and SteadyState Flows of Fluids and Gases. Palmarium Academic Publishing, Saarbrucken (2012) [in Russian] 13. Gubarev, Y.G.: Linear stability criterion for steady screw magnetohydrodynamic flows of ideal fluid. Thermophys. Aeromechanics 16, 407–418 (2009) 14. Gavrilieva, A.A., Gubarev, Y.G.: Stability of steady-state plane-parallel shear flows of an ideal stratified fluid in the gravity field. Vestnik of NEFU 9, 15–21 (2012) [in Russian] 15. Gubarev, Y.G.: The problem of adequate mathematical modeling for liquids fluidity. Am. J. Fluid Dyn. 3, 67–74 (2013) 16. Gavrilieva, A.A., Gubarev, Y.G., Lebedev, M.P.: Rapid approach to resolving the adequacy problem of mathematical modeling of physical phenomena by the example of solving one problem of hydrodynamic instability. Int. J. Theor. Math. Phys. 3, 123–129 (2013) 17. Gubarev, Y.G.: On stability of steady-state three-dimensional flows of an ideal incompressible fluid. J. Math. Sci. Appl. 3, 12–21 (2015) 18. Penrose, O.: Electrostatic instabilities of a uniform non-Maxwellian plasma. Phys. Fluids 3, 258–264 (1960)

Analytical Results for the Riemann Problem for a Weakly Hyperbolic Two-Phase Flow Model of a Dispersed Phase in a Carrier Fluid Maren Hantke, Christoph Matern and Gerald Warnecke

1 Introduction We study the two-phase flow model proposed by Hantke et al. [1]. 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, see, e.g., Drew and Passman [2], and it is completely in divergence form. For the mathematical analysis here, we will consider one space dimension only, neglect the phase exchange terms handling phase transitions, and assume isothermal flow. In this form, we may also consider the case of droplets in vapor or solid particles in a fluid. It turns out that, due to the equations of state for the carrier phase, droplets or solid particles in a gas are a mathematically simpler case than gas bubbles or solids in a liquid. We will show the existence of self-similar solutions to the Riemann problem that are uniquely determined by the initial data of the problems. These use the wellknown self-similarity of conservation laws. Our system (1) below has some partial similarities to the Euler equations for which details on the solutions to Riemann problems can be found in Smoller [3]. In our case, the solutions include classical waves such as rarefactions, shock waves as well as contact discontinuities. Interestingly, in

M. Hantke Institute for Mathematics, Martin-Luther-University Halle-Wittenberg, Universitätsplatz 10, 06108 Halle (Saale), Germany e-mail: [email protected] C. Matern · G. Warnecke (B) 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] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_22

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some cases it is possible to obtain solutions involving vaporless states as well as the non-classical delta-shock waves. Another important mathematical feature of multi-phase mixture conservation laws is that they have weakly hyperbolic states. By this, we mean that all eigenvalues are real but there is at least one multiple eigenvalue that does not have a full set of eigenvectors. Existence results for hyperbolic systems of conservation laws in one space dimension require strict hyperbolicity, i.e., a full set of distinct real eigenvalues. Therefore, the existing theory does not apply to our system. In a gas, the equation of state (EOS) for isothermal flow yields the pressure as a linear function of the density. For a liquid, the simplest realistic assumptions lead to an affine function for the EOS. In the case of a liquid carrier phase, this leads to a considerable complication in the determination of the solutions to Riemann problems. This is another key point of this paper.

2 The Two-Phase Flow Model We study the two-phase flow model proposed by Hantke et al. [1]. In order to have a tractable model, we make the assumption that the fluids are isothermal and gravity is ignored. The terms modeling the phase transitions are dropped. Our variables are volume fraction c, pressure p, mass density ρ, mean particle radius density R, and velocity v. We will use boldface to represent vectors as well as the subscript C to distinguish the carrier phase from the dispersed phase, the latter one without a subscript. The Riemann problem we want to study is given by the system of conservation laws ∂ ∂c + (c v) = 0, ∂t ∂x ∂ ∂c ρ + (c ρ v) = 0, ∂t ∂x   ∂ ∂c ρ v + c ρ v2 = 0, ∂t ∂x ∂ ∂c R + (c R v) = 0, ∂t ∂x   ∂  ∂  (1 − c )ρC + (1 − c ) ρC vC = 0, ∂t ∂x    ∂  ∂  ∂  2 (1 − c )ρC vC + (1 − c ) ρC vC + (1 − c ) pC = 0, ∂t ∂x ∂x

(1)

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)

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In order to maintain the nature of the disperse phase in a given carrier phase, the dispersed phase volume fraction should not exceed a certain threshold which should be smaller than one. Otherwise, the assumptions made in deriving the model lose their validity and solutions may not have a valid physical interpretation. Note that the first four equations of the system (1) decouple from the last equations. Therefore, we will solve them separately from the rest of the system in the next section.

2.1 Equation 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. We denote the speed of sound of a fluid by a 2 = ∂∂ρp . We may assume an equation of state of the following form throughout the rest of this paper p = p(ρ) with p = a 2 ρ + d0 and a 2 = const.

(3)

Taking the vapor phase to be an ideal gas is a valid choice for an equation of state. One can use the Tait equation [4] or the stiffened gas equation [5, Sect. 7] for the liquid phase. Both descriptions are valid equations of state under our assumptions.

3 The Dispersed Phase Equations 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, we study as a first step the solution to the Riemann problem for this subsystem. We introduce the vector u of primitive variables defined by u = (c, ρ, v, R)T . For smooth solutions, we use the product rule of differentiation to obtain the quasi-linear form of our system ∂u ∂u + A(u) = 0. ∂t ∂x

(4)

The matrix A in (4) has the repeated eigenvalue λ = v of multiplicity four and only three linearly independent eigenvectors as well as one generalized eigenvector. The equation for the velocity is the Burgers equation and can be solved in conservative form completely independently of the other equations. For the then given v, the second and fourth equations are linear advection equations with a variable, possibly discontinuous coefficient. In case of a discontinuous velocity, we see that the first equation would involve a derivative of this velocity. This leads, as we will see, to solutions with singular measures.

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3.1 The Case v− = v+ = v We observe that in this case the velocity v remains constant while the equations for c and ρ become linear advection equations with constant speed v. Therefore, the solution for the Riemann problem (2) is given by the contact discontinuity  (c, ρ, v, R) =

(c− , ρ− , v, R− ) (c+ , ρ+ , v, R+ )

−∞ < x ≤ v t, v t ≤ x < +∞.

(5)

3.2 The Case v− < v+ Putting ρˆ = c ρ the second and third equations of (1) are equivalent to the zero pressure gas dynamics model ∂   ∂ ρˆ + ρv ˆ = 0, ∂t ∂x

∂   ∂  2 ρv ˆ + ρv ˆ = 0, ∂t ∂x

(6)

These models therefore have a similar structure of solutions. Following the results by Sheng and Zhang [6, p. 11] on the zero pressure gas dynamics model, we can construct a solution which consists of two contact discontinuities and a vaporless state, with c = 0, between two constant states. We introduce the radial variable ξ = x/t. After this transformation, the system corresponds to a two-point boundary value problem of first-order ordinary differential equations with boundary data at infinity. lim (c, ρ, v, R) (ξ ) = (c− , ρ− , v− , R− ) ,

ξ →−∞

lim (c, ρ, v, R) (ξ ) = (c+ , ρ+ , v+ , R+ ) .

ξ →∞

If we set c = 0 on the interval [v− , v+ ], all four equations are satisfied automatically. We choose the function v to be any continuous function satisfying v(v− ) = v− and v(v+ ) = v+ . The choice v(ξ ) = ξ is the simplest choice. The functions ρ and R are arbitrary. This gives the solution ⎧ ⎨ (c− , ρ− , v− , R− ) (c, ρ, v, R) = (0, ρ(ξ ), v(ξ ), R(ξ )) ⎩ (c+ , ρ+ , v+ , R+ )

−∞ < ξ ≤ v− , v− < ξ < v+ , v+ ≤ ξ < +∞,

with ρ and R arbitrary functions, v a continuous function satisfying ρ(v± ) = ρ± , v(v± ) = v± ,

R(v± ) = R± .

(7)

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3.3 The Case v− > v+ In this case, one can show the existence of blow-up solutions, even for continuous initial data. Due to the occurrence of the blow up of c, it is natural to seek solutions in the space of Borel measures. We follow [7] and construct generalized Rankine– Hugoniot relations. Now we propose to find a solution with a discontinuity at x = x(t), of the form ⎧ ⎨ (c  x < x(t),  − , ρ− , v− , R− ) x = x(t), w(t)δ(x − x(t)), ρδ (t), vδ (t), Rδ (t) (8) (c, ρ, v, R) (t, x) = ⎩ x > x(t), (c+ , ρ+ , v+ , R+ ) where x(t) ∈ C 1 and δ(x) is the standard Dirac measure with all mass at x ∈ R. Note that we use the notations ρδ , vδ and Rδ for the exceptional values taken by these physical states along the path of the singular measure. In order to obtain a physical relevant solution, the discontinuity must satisfy the Lax entropy condition λ(v+ ) < σ < λ(v− ) or equivalently v+ < vδ < v− . The Riemann problem is now reduced to solve the generalized Rankine–Hugoniot conditions with the initial data x(0) = 0, w(0) = 0, under the given entropy condition. Like Yang [7], we assume a delta shock of the form x(t) = σ t = vδ t, w(t) = cδ t, vδ (t) = vδ , ρδ (t) = ρδ and Rδ (t) = Rδ . The system then possesses the entropy solution vδ =



√ √ √ √ v− c− ρ− + v+ c+ ρ+ c+ c− ρ− + c− c+ ρ+ , c = (v− − v+ ) , (9) √ √ √ √ δ c− ρ− + c+ ρ+ c− ρ− + c+ ρ+

and √ √ √ √ c+ρ+ c−ρ− +c−ρ− c+ρ+ c+R+ c−ρ− +c−R− c+ρ+ , Rδ = . ρδ = √ √ √ √ c+ c−ρ− +c− c+ρ+ c+ c−ρ− +c− c+ρ+

(10)

4 The Carrier Phase Quantities Again, we introduce a vector u of primitive variables, this time defined as T  u = c, ρ, v, ρC , vC . We neglected the equation for the mean bubble radius R. The system (1) can then be written in quasi-linear form as ∂u ∂u + B(u) = 0. ∂t ∂x The eigenvalues of the matrix B are

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λ1 = vC − aC , λ2 = λ3 = λ4 = v, λ5 = vC + aC , but there are only four corresponding eigenvectors. The λ1 and λ5 characteristic fields are genuinely nonlinear. Therefore, solutions for the acoustic waves may contain rarefaction waves or shock waves. We use generalized Riemann invariants to obtain relations for the carrier phase quantities which hold across a rarefaction wave and Rankine–Hugoniot jump conditions to derive relations across shock waves or contact waves, see [8] for details. We have to solve this nonlinear system of equations to find the solution in the *-region. In the case of a contact wave in the dispersed phase, see Sect. 3.1, it is given by 0 = v[[(1 − c) ρC∗ ]] − [[(1 − c) ρC∗ vC∗ ]], ∗ ∗

∗ ∗2

(11) ∗

0 = v[[(1 − c) ρC vC ]] − [[(1 − c) ρC vC + (1 − c) pC ]], ⎧ ρ ∗ −ρ ⎨ aC √C±∗ C± ρ ∗ > ρC± (shock), C± ρC± ρC± ∗ vC± = vC± ± ∗ ⎩ a ln ρC± ρ ∗ ≤ ρ (rarefaction), C C± ρ C±

(12) (13)

C±

where the first two equations are the jump conditions over the contact wave and the last equation describes the relation over the acoustic waves. For the four unknown quantities of the carrier phase in the *-region, we have four  relations.  The ∗ ∗ ∗ = V vC− , ρˆC− and first two equations possess a unique subsonic solution vC+   ∗ ∗ ∗ ρˆC+ which we will not state here in detail. Rewriting the first ex= P vC− , ρˆC− pression in the form    ∗  ∗ ∗ ∗ ∗ ∗ = −vC+ , vC− , vC+ + V vC− , ρˆC− 0 = F ρˆC−  ∗  ∗ , obtained from (13) we can reduce the number of unand using vC± = W± ρˆC± ∗ only. A solution is then known quantities. Finally, the function F is a function of ρˆC− determined implicitly by  ∗    ∗  ∗   ∗ , W+ P ρˆC− . , W− ρˆC− , W− ρˆC− 0 = F ρˆC− To verify uniqueness of solutions, one checks the strict monotonicity of F, that is dF < 0. ∗ dρˆC−

(14)

The final result is a set of inequalities for the relative velocity between the two phases involved. These inequalities state that the relative velocity between the dispersed and the carrier phase has to be a certain amount smaller than the sound speed in the carrier phase. Its explicit value depends on the chosen equation of state and the parameters therein as well as the initial data used.

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5 Conclusion In this paper, we gave a brief overview of the analysis of the Riemann problem for the two-phase flow model proposed by Dreyer, Hantke and Warnecke [1]. We performed the linear analysis on the dispersed phase alone as well as the full twophase system of equation. The system is weakly hyperbolic, so the general theory for strictly hyperbolic systems in one space dimension cannot be applied. We extended the usual discussion of a linear equation of state to an affine linear one. The work therefore takes a first step from a linear equation of state toward more general ones, which is very important with regard to applications. Commonly used equations of state like the Tait equation or the stiffened gas equation are included in our analysis. It is remarkable how the small change in the equation of state complicates the analysis dramatically. Due to the complexity of the analysis, we could not present the analysis for all possible wave configurations here. We will present the detailed calculations for all possible cases as well as a complete discussion of vapor and liquid carrier phases. This work is already accepted for publication in Quarterly of Applied Mathematics [9]. We have started some first numerical investigations which are also not included here but will be published in a follow-up paper.

References 1. Dreyer, W., Hantke, M., Warnecke, G.: 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) 2. Drew, D.A., Passman, S.L.: Theory of Multicomponent Fluids. Springer, New York (1999) 3. Smoller, J.: A Shock Waves and Reaction-Diffusion Equations. Springer, New York (1994) 4. Dymond, J., Malhotra, R.: The Tait equation: 100 years on. Int. J. Thermodyn. 9(6), 941–951 (1988) 5. Menikoff, R., Plohr, B.J.: The Riemann problem for fluid flow of real materials. Rev. Mod. Phys. 61, 75–130 (1989) 6. Sheng, W., Zhang, T.: The Riemann Problem for the Transportation Equations in Gas Dynamics. Memoirs of the American Mathematical Society, vol. 654, p. 137 (1999) 7. Yang, H.: Riemann problems for a class of coupled hyperbolic systems of conservation laws. J. Differ. Equ. 159, 447–484 (1999) 8. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, Berlin (2009) 9. Hantke, M., Matern, C., Ssemaganda, V., Warnecke, G.: The Riemann problem for a weakly hyperbolic two-phase flow model of a dispersed phase in a carrier fluid. Q. Appl. Math.

A Numerical Method for Two Phase Flows with Phase Transition Including Phase Creation Maren Hantke and Ferdinand Thein

Abstract Two phase flows including phase transition, especially phase creation, with a sharp interface remain a challenging task for numerics. We consider the isothermal Euler equations with phase transition between a liquid and a vapour phase. The phase interface is modelled as a sharp interface, and the mass transfer across the phase boundary is modelled by a kinetic relation [6]. Existence and uniqueness results were proven in [2, 6]. We present a method to obtain the numerical solution for associated Riemann problems. In particular, we show how the cases of nucleation and cavitation may be treated. We will highlight the major difficulties and propose possible strategies to overcome these problems.

1 Problem Formulation and Main Challenges We study inviscid, compressible and isothermal two phase flows. The two phases are either the liquid or the vapour phase of one substance. The phases are distinguished by the mass density ρ and further described by the velocity u. Sometimes, it is convenient to use the specific volume v = 1/ρ instead of the mass density. The physical quantities depend on time t ∈ R≥0 and space x ∈ R. In regular points of the bulk phases, the fluid is described using the isothermal Euler equations ∂t ρ + ∂x (ρu) = 0,

(1)

∂t (ρu) + ∂x (ρu + p) = 0.

(2)

2

We consider Riemann initial data given by

M. Hantke Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), Germany e-mail: [email protected] F. Thein (B) Otto-von-Guericke-Universität Magdeburg, 39016 Magdeburg, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_23

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 ρ(x, 0) =



ρV , x < 0 ρL , x > 0

and u(x, 0) =

uV , x < 0 . uL, x > 0

(3)

The jump conditions across discontinuities are ρ(u − W ) = 0, ρ(u − W ) u +  p = 0.

(4) (5)

Furthermore, every discontinuity satisfies the entropy inequality   ρ(u − W ) g + ekin ≤ 0.

(6)

The quantity W is the speed of the discontinuity and Z = −ρ(u − W ) the mass flux where we will distinguish between a classical shock wave and the phase boundary (non-classical shock) 



Q, shock wave Z= z, phase boundary

and W =

S, shock wave w, phase boundary

.

The pressure p is linked to the mass density ρ via the EOS. We have for the mass density ρ ∈ ρ ⊆ (0, ∞). This domain can be split into the liquid, spinodal and vapour region, i.e. ρ = liq ∪ spin ∪ vap . Accordingly, the EOS consists of three corresponding parts, i.e. an EOS for the vapour phase, the liquid phase and an intermediate part. The (unphysical) intermediate part is characterized by the relation 

∂p ∂ρ

 < 0, T

and thus the considered Euler system becomes elliptic inside this region. The exact solution of the two-phase Riemann problem consists of three waves, separating four constant states. In particular, we are searching the values in the star region, i.e. ρV∗ , pV∗ , u ∗V in the vapour phase, ρ L∗ , p ∗L , u ∗L in the liquid phase and the velocity w of the phase boundary or the mass flux z. To solve the system numerically, we use a discretization of the domain and a finite volume approach where cell averages are used. The averaging inside a cell causes the first problem. Assume that at some time t n the position of the phase boundary coincides with a cell boundary. In the (most likely) case that the phase boundary moves with the velocity w = 0, the phase boundary will have travelled the distance wt after the time step t. Thus, the phase boundary will lie inside a cell at time t n+1 . Now averaging can lead to unphysical states ρ n+1 x = wtρVn+1 + (x − wt)ρ Ln+1 ∈ spin

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and hence this will result in a wrong solution. Thus, we ensure that the phase boundary will always coincide with a cell boundary. The second crucial point is the calculation of the quantities inside the star region. We would like to do this with sufficient accuracy, yet minimizing the computational costs. Finally, we need to determine the correct flux (since we have two different phases) using the solution of the Riemann problem.

2 Numerical Method In this section, we want to present a numerical method to solve the stated problem in one space dimension. One of the main benefits of this method is its ability to treat nucleation and cavitation. The key idea of our approach is to use Godunov’s method where the fluxes at cell interfaces between cells with equal phase may be calculated with any Riemann solver, see [3, 4, 8]. The flux at the phase boundary will be discussed later on.

2.1 Grid Alignment and Adaption We now want to comment on the first stated difficulty that we pointed out in the context of two phase flows with a sharp interface. For this reason, we assume that we have constant cell values Uin belonging to two different phases and that the phase boundaries are aligned with the cell boundaries. Let us focus on a single phase boundary moving at the velocity w = 0 located at a certain xi0 +1/2 . As explained in Sect. 1, the phase boundary will lie inside a cell at t n+1 = t n + t, and hence averaging will lead to unphysical phase states. In order to avoid this situation, we always align our grid to the phase boundary. This changes the size of the cells, and thus we have to deal with different cell sizes. This is done in the following way. We move the cell boundary according to the phase boundary, i.e. xi0 +1/2 = xi0 +1/2 + wt, see Fig. 1. Next we compare every cell to certain thresholds given by the average cell size xav := (b − a)/N and the two parameters 1 and 2 . Now we adjust the grid according to the following rules. If a cell fulfils xi > 1 xav , the cell is too large and we split it into two. If a cell fulfils xi < 2 xav , the cell is too small and we merge it with a neighbouring cell, if both belong to the same phase. The cell values have to be updated accordingly. In the case of nucleation and cavitation, we may encounter small cells that must not be merged with the neighbouring cells. How we deal with this situation will be explained later on. Since cell sizes may change in every time step, we need to modify Godunov’s method. The result reads

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Fig. 1 Align grid to the new position of the phase boundary

Uin+1 =

xin

Un − n+1 i

xi

t xin+1

  Fi+ 21 − Fi− 21 .

(7)

with xin and xin+1 being the cell sizes at the times t n and t n+1 . One immediately verifies that (7) reduces to the well-known Godunuv formula for constant cell sizes. Further, we calculate and update the time step as follows. The initial time step is obtained by using 0 Smax = max{|u − a|, |u + a|} and t = CC F L

x0 , 0 Smax

(8)

where the maximum is computed considering all cell values. As the computation evolves, we recalculate the time step as n xmin = min {xi | xi ≥ 2 xav } , Smax = max |S|, t = CC F L i=1,...,N

S

xmin . n Smax (9)

It remains to discuss the case if a small cell of one phase is surrounded by cells of the other phase, and thus must not be merged with its neighbours. This is the case when phase creation occurs. In this case, we proceed as follows. For cells with size xi < 2 xav , we apply a local time stepping method as properly discussed by Müller and Stiriba [5]. We only need a simple reduced version of the method presented in [5] and will briefly present the used method in the following. Let i 0 be the index of the small cell. We then consider the triplet {Ui0 −1 , Ui0 , Ui0 +1 } and the corresponding quantities. Now we perform a time evolution starting at t n with the small time steps τ until the final time t n+1 = t n + t. The small cell is updated according to formula (7) with the time step τ and the fluxes Fiν0 ±1/2 at the current time level τ ν = t n + ντ with ν = 0, . . . , n 0 and τ n 0 = t n+1 . We have

A Numerical Method for Two Phase Flows with Phase …

Uin,ν+1 = 0

xiν0 xiν+1 0

Uin,ν − 0

τ xiν+1 0

181

  Fiν0 + 1 − Fiν0 − 1 . 2

(10)

2

The values of the neighbouring cells remain unchanged throughout the time evolution. For the update of these (large) neighbouring cells, we have to take care of the fluxes at xi0 −1/2 and xi0 +1/2 . According to [5], this is done in the following way. At t n we calculate the fluxes Fi0 ±1/2 as usual. At every further time level, we update these fluxes as follows: Fiν0 ±1/2 = Fiν−1 + 0 ±1/2

τ ν , ν = 1, . . . , n 0 . F t i0 ±1/2

The fluxes obtained at the end of the evolution of the small cell may then be used in (7) to advance the neighbouring cells from t n to t n+1 .

2.2 Solution at the Interface In the following, we want to discuss the solution at the phase boundary. For details on the exact solution, we refer to [2, 6]. The pressure and the density inside each phase are not independent from each other due to the EOS given for the particular phase. The mass flux is given by the kinetic relation     kin . z = τ pV g + ekin = τ pV g L − gV + ekin L − eV

(11)

We know that the pressures at the interface are uniquely linked using the kinetic relation (11) and Eq. (5), see [6, Theorem 3.2.2]. Thus according to Theorem 3.2.2 in [6], we have to solve the following system for ( pV∗ , p ∗L ) to obtain the complete solution: 0=

G( pV∗ ,

p ∗L )

 =

  p + z 2 v . f V ( pV∗ , WV ) + f L ( p ∗L , W L ) + z v + u

(12)

The quantities in the second component are given as in Theorem 3.2.2 in [6], with the slight difference that we now explicitly use the liquid pressure. Once we have determined the pressures, we can calculate the densities via the corresponding EOS, the mass flux z via the kinetic relation, the velocities via the wave relations for the classical waves, and the velocity of the phase boundary using the definition of the mass flux z = −ρ(u − w). Since we always align the computational grid with the phase boundary, we have to use the corresponding flux when we apply (7). The flux at the phase boundary is determined using the jump conditions (4) and (5). These may be rewritten as

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ρV∗ (u ∗V − w) = ρ L∗ (u ∗L − w) = −z, −zu ∗V + pV∗ = −zu ∗L + p ∗L . Thus we directly see the flux across the phase boundary, i.e. FP B =

−z . −zu ∗ + p ∗

(13)

Here one has the freedom to choose either the vapour or the liquid star state values. So far we have discussed the case with phase transition. The case without phase transition is obtained using z = 0, which then implies  p = 0 and u = 0. Thus we have to solve the single nonlinear equation given in Theorem 3.3.5 in [6]. The flux is given by

0 = ∗ . p

FP B

(14)

Further, one immediately verifies that in this case the phase boundary is quite analogue to a contact discontinuity. Therefore, we use in our numerical simulations, for the case without phase transition, the wave speed estimates for the HLLC solver as given in [1, 8]. Given the (left) vapour state and the (right) liquid state, we proceed as follows: SL = u L + a L , SV = u V − a V , p L − pV + ρV u V (SV − u V ) − ρ L u L (SL − u L ) , w= ρV (SV − u V ) − ρ L (SL − u L ) SL − u L . ρ L∗ = ρ L SL − w

(15)

Here, SV and SL denote the velocities of the classical waves. The density is calculated according to the HLLC solver as presented in [7, 8]. The pressure p ∗ may then be calculated using the liquid EOS. This procedure gives satisfactory results.

3 Phase Creation The case of phase creation is a challenging issue since we not only have to detect when phase creation occurs. Further, we also have to deal with technical problems like the creation of new cells which in most cases are very small and of course with multiple phase boundaries.

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3.1 Cavitation and Nucleation When we encounter a liquid/liquid Riemann problem, we may observe cavitation, i.e. the creation of vapour. The detailed analysis is presented in Subsection 3.4.2 in [6]. In view of the results given there in Theorem 3.4.10, Definition 3.4.12 and Theorem 3.4.15 in [6], the main outline for the numerics is as follows. We solve the single phase Riemann problem in the liquid phase. If there is no solution to this problem, in particular when the liquid pressure in the star region is smaller than the predefined minimum liquid pressure, p ∗ < pmin , we have cavitation. In this case, we store the position of the involved cells and perform an extra calculation. Summarized performed steps are as follows: Solve the single phase Riemann problem, if p ∗ < pmin cavitation occurs. Solve the single phase Riemann problem according to Theorem 3.4.15. From the solution we obtain pV∗ , z, wle f t and wright . Create a vapour cell of size (wright − wle f t )t with the cell values ρV∗ , pV∗ and u ∗V . (v) The fluxes at the phase boundaries are given by

(i) (ii) (iii) (iv)

(le f t) FP B

=

z

−zu ∗V + pV∗

and

(right) FP B

−z . = −zu ∗V + pV∗

(16)

In the next time step, we then have two phase boundaries which are treated as discussed in Sect. 2. The case of nucleation is discussed analogously, see [6, Subsection 3.4.1, 4.4.2].

References 1. Batten, P., Clarke, N., Lambert, C., Causon, D.M.: On the choice of wavespeeds for the HLLC Riemann solver. SIAM J. Sci. Comput. 18(6), 1553–1570 (1997). https://doi.org/10.1137/ S1064827593260140 2. Hantke, M., Thein, F.: A general existence result for isothermal two-phase flows with phase transition. J. Hyperbolic Differ. Equ. (2017). arXiv:1703.09431 3. Kröner, D.: Numerical Schemes for Conservation Laws. Advances in Numerical Mathematics. Wiley, Hoboken (1997) 4. LeVeque, R.J.: Numerical Methods for Conservation Laws, vol. 132. Springer, Berlin (1992) 5. Müller, S., Stiriba, Y.: Fully adaptive multiscale schemes for conservation laws employing locally varying time stepping. J. Sci. Comput. 30(3), 493–531 (2007). https://doi.org/10.1007/s10915006-9102-z 6. Thein, F.: Results for two phase flows with phase transition. Ph.D. thesis, Otto-von-GuerickeUniversität Magdeburg (2018) 7. Toro, E., Spruce, M., Speares, W.: Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 4(1), 25–34 (1994). https://doi.org/10.1007/BF01414629 8. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, Berlin (2009)

An Optimization Method for Feedback Stabilization of Linear Delay Systems Guang-Da Hu and Renghong Hu

Abstract This paper is concerned with feedback stabilization of linear systems with multiple delays. Based on an integral performance of the state transition matrix of the linear delay systems, a numerical optimization algorithm is presented to seek feedback gain matrices to stabilize the delay systems. The effectiveness of the algorithm is interpreted by a theoretical result and numerical examples.

1 Introduction We are concerned with linear systems with multiple delays described by x(t) ˙ = A0 x(t) +

m 

A j x(t − τ j ) + Bu(t),

(1)

j=1

where constant matrices A0 , A j ∈ Rd×d , B ∈ Rd× p , state vector x(t) ∈ Rd , control vector u(t) ∈ R p , and τ j > 0 for j = 1, . . . , m. Stability of system (1) has been investigated in [2–4, 7]. In [3, 4], a necessary and sufficient condition for asymptotic stability of system (1) is derived by the argument principle. In this paper we investigate feedback stabilization of system (1) using the results in [3, 4].

G.-D. Hu (B) Department of Mathematics, Shanghai University, Shanghai, China e-mail: [email protected] R. Hu School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing, China e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_24

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2 Preliminaries The characteristic equation of system (1) is ⎡

⎤ m  P(s) = det ⎣s I − A0 − (A j exp(−τ j s)⎦ = 0.

(2)

j=1

Now we will review the results in [3] which are concerned with stability of system (1). The proofs of the lemmas are found in [3]. Lemma 1 ([3]) For system (1), let s be an unstable characteristic root of Eq. (2), then m  |s| ≤ r = A0  + A j . (3) j=1

Definition 1 Assume that the conditions of Lemma 1 hold. The set D is defined by D = {s : s ≥ 0 and |s| ≤ r }, where r is given by (3) in Lemma 1. The boundary of D is denoted by l. The boundary l = la ∪ lb , where la is the straight segment from point d1 = (0, ir ) to point d2 = (0, −ir ) which is on the imaginary axis, and lb is the half circumference on the right half plane defined by lb = {(ρ, θ ) : ρ = r, −π/2 ≤ θ ≤ π/2}.

(4)

Lemma 2 ([3]) System (1) is asymptotically stable if and only if P(s) = 0 for s ∈ l

(5)

l arg P(s) = 0

(6)

and hold, where arg P(s) stands for the argument of P(s) and l arg P(s) change of the argument of P(s) along the contour l. Lemma 3 ([3]) For system (1), assume that there are no characteristic roots on the imaginary axis. If P(s) = 0 for s ∈ l (7) and

1 l arg P(s) = Z , 2π

then the number of the unstable characteristic roots is Z for system (1).

(8)

An Optimization Method for Feedback Stabilization of Linear Delay Systems

187

Remark 1 Algorithm 1 is given in [3] to check the delay-dependent stability of system (1) due to Lemmas 2 and 3. Lemma 4 ([5]) Let g(t)  t : R → R be a uniformly continuous function on [0, ∞). Suppose that limt→∞ 0 g(σ )dσ exists and is finite. Then, g(t) → 0 as t → ∞.

3 Optimization Method for Feedback Stabilization From the viewpoint of ease of implementation, let controller structure be u(t) = K 0 x(t) +

m 

K j x(t − τ j ),

(9)

j=1

where feedback gain matrices K 0 , K j ∈ R p×d for j = 1 . . . , m. From (1) and (9), we have the closed-loop system x(t) ˙ = (A0 + B K 0 )x(t) +

m 

(A j + B K j )x(t − τ j ).

(10)

j=1

Our aim is to seek gain matrices such that the closed-loop system (10) is asymptotically stable. Now we review the concept of the state transition matrix (fundamental matrix) of linear delay systems and a stability result based on it (see, for example, [2, 7]). Let K stand for all the gain matrices K 0 and K j for j = 1, . . . , m. The vector K is formed by “stacking” the columns of the compound matrix [K 0 , K 1 , . . . , K m ]

(11)

into one long vector. The state transition matrix of the closed-loop system (10) is denoted by F[K, t] ∈ Rd×d , which is the solution of the matrix delay differential equation ˙ F[K, t] = (A0 + B0 K 0 )F[K, t] +

m  (A j + B K j )F[K, t − τ j ] for t > 0, j=1

(12) under the condition F[K, t] = 0 for t < 0 and F[K, 0] = I. A stability criterion via the state transition matrix is as follows.

(13)

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Lemma 5 ([2]) The closed-loop system (10) is asymptotically stable if and only if there exist positive constants γ and μ such that ||F[K, t]|| ≤ γ exp (−μt), for t > 0.

(14)

The solution of system (10) can be represented by means of the state transition matrix. Lemma 6 ([2, 7]) Let x(t) be the solution of system (10) with the condition x(t) = φ(t) for t ∈ [−τ, 0], where τ = max j {τ j } for j = 1, . . . , m. We have x(t) = F[K, t]x(0) +

m  

0

−τ j

j=1

F[K, t − τ j − θ ]A j x(θ )dθ for t > 0.

(15)

We now present the main result in this paper. Theorem 1 The closed-loop system (10) is asymptotically stable if and only if there exist feedback gain matrices K 0 and K j for j = 1, . . . , m, i.e., K such that 

T

lim

T →∞ 0

||F[K, σ ]||2 dσ + ||K||2

(16)

exists and is finite. Proof Suppose that the closed-loop system (10) is asymptotically stable. By means of T T γ2 Lemma 5, lim T →∞ 0 ||F[K, σ ]||2 dσ ≤ 2μ . This means that lim T →∞ 0 ||F[K, σ ]||2 dσ + ||K||2 exists and is finite. T Conversely, suppose that lim T →∞ 0 ||F[K, σ ]||2 dσ + ||K||2 exists and is finite. T This means that lim T →∞ 0 ||F[K, σ ]||2 dσ exists and is finite. The state transition matrix F[K, σ ] is absolutely continuous [2], thus it is uniformly continuous. According to Lemma 4, ||F[K, t]||

→

0 as t →

∞.

(17)

By means of (15) in Lemma 6 and (17), we obtain x(t) → 0 as t → ∞. Thus the closed-loop system (10) is asymptotically stable. The proof is completed. In order to satisfy the performance requirements of the closed-loop system, the setting (transient process) time must be not too long [6]. This means that the state transition matrix of the closed-loop system, F[K, t] as measured by some norm should become small after some finite time. On the other hand, in any engineering system, an upper bound is set on the magnitude of the gain matrices in the feedback controller by practical considerations [6]. It is necessary to restrict the magnitudes of the feedback gain matrices.

An Optimization Method for Feedback Stabilization of Linear Delay Systems

189

According to the above argument and Theorem 1, we formulate an optimization problem as follows. The performance index 

T

min K

||F[K, σ ]||2 dσ + ||K||2 ,

(18)

0

subject to the constraints, Eqs. (12) and (13). Here, T is a sufficient large positive constant. From the viewpoint of ease of implementation, the norm || · || is taken as the Fr obenius one. Assume that the numerical solution gives a sequence of approximated values {F1 (K), F2 (K), . . . , FL (K)} of {F[K, t1 ], F[K, t2 ], . . . , F[K, t L ]} of Eq. (12) on certain equidistant step-values {tn = nh} with the step-size h = T /L , where L is a positive integer. Let li = [τi h −1 ], δi = li − τi h −1 , 0 ≤ δi < 1, for i = 1, . . . , m, where [a] stand for the smallest integer that is greater than or equal to a ∈ R. By means of the simplest numerical integration, the performance index (18) reduces to min K

L 

||Fn (K)||2 h + ||K||2 .

(19)

n=1

Using Euler scheme combining with the linear interpolation for Eqs. (12) and (13), for n = 0, 1, . . . , L − 1, we obtain the constraints ⎡ ⎤ m  Fn+1 (K) = Fn (K) + h ⎣(A0 + B K 0 )Fn (K) + (A j + B K j )Fn−l j +δ j (K)⎦ , j=1

Fn−l j +δ j (K) = 0 for n − l j + δ j < 0,

(20) (21)

Fn−l j +δ j (K) = I for n − l j + δ j = 0,

(22)

and Fn−l j +δ j (K) = (1 − δ j )Fn−l j (K) + δ j Fn−l j +1 (K) for n − l j + δ j > 0.

(23)

It is necessary to combine a standard optimization technique [1] with the stability criterion, Algorithm 1 (Lemmas 2 and 3) in [3] for checking whether the gain matrices stabilize system (1). The algorithm for the optimization problem is as follows. Algorithm 2 Step 1. Choose a sufficient large constant T and a set of values for K 0 , K 1 , . . . , K m and get an initial value K0 . Step 2. Solve the optimization problem with the objective function (19) subjecting to Eqs. (20)–(23) using a standard optimization technique and obtain K, i.e., the gain matrices K 0 , K 1 , . . . , K m .

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Step 3. For the gain matrices K 0 , K 1 , . . . , K m obtained in step 2, check whether the closed-loop system (10) is asymptotically stable using Algorithm 1 (Lemmas 2 and 3). If the gain matrices K 0 , K 1 , . . . , K m can stabilize the system, then stop the algorithm. Otherwise, increase T or choose a new initial value K0 , and go to step 2. The following numerical example is given to demonstrate Algorithm 2. x(t) ˙ = A0 x(t) + A1 x(t − τ1 ) + A2 x(t − τ2 ) + Bu(t), ⎡

⎤ 0.1 −0.5 −0.3 0.3 ⎢ 0 0 0.2 −0.2 ⎥ ⎥ A0 = ⎢ ⎣ 0 0 −0.1 0 ⎦ , 0 0.4 0.3 −0.4 ⎡

0.1 ⎢ 0 A2 = ⎢ ⎣ 0 0

−0.1 −0.2 0.1 −0.3

0 0.1 0 0.1

⎤ 0.2 0 ⎥ ⎥, 0.3 ⎦ 0

⎡

−0.2 ⎢ 0 A1 = ⎢ ⎣ −0.1 0.1

0.3 −0.2 0.1 −0.6

(24)

⎤ 0.2 −0.5 −0.2 0.2 ⎥ ⎥, 0.1 0.1 ⎦ −0.6 0

⎡

⎤ 0.2 ⎢ 0 ⎥ ⎥ B=⎢ ⎣ −0.3 ⎦ , τ1 = 2 and τ2 = 3. 0.1

Let u(t) = K 0 x(t) + K 1 x(t − τ1 ). The closed-loop system is x(t) ˙ = (A0 + B K 0 )x(t) + (A1 + B K 1 )x(t − τ1 ) + A2 x(t − τ2 ).

(25)

By means of Algorithm 1 (Lemmas 2 and 3), we know that the uncontrolled system (24) has 3 unstable characteristic roots and is unstable. Let T = 100, h = 0.01  T and K0 = 0 0 0 0 0 0 0 0 . Using Algorithm 2, we obtain the gain matrices as follows.   K 0 = 0.2674 2.7938 1.7085 −0.7494 , K 1 = −0.7690 1.1718 0.5075 0.4501 .

The closed-loop system (25) with the controller is asymptotically stable.

References 1. Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms. Wiley, Hoboken (2006) 2. Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993) 3. Hu, G.D.: Delay-dependent stability of Runge-Kutta methods for linear neutral systems with multiple delays. Kybernetika 54, 718–735 (2018)

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4. Hu, G.D., Liu, M.: Stability criteria of linear neutral systems with multiple delays. IEEE Trans. Autom. Control 52, 720–724 (2007) 5. Khalil, H.K.: Nonlinear Sytems. Prentice-Hall, Upper Saddle River (2002) 6. Kirk, D.E.: Optimal Control Theory: An Introduction. Prentice-Hall, Englewood Cliffs (1970) 7. Kolmanovskii, V.B., Myshkis, A.: Introduction to Theory and Applications of Functional Differential Equations. Kluwer Academic Publishers, Dordrecht (1999)

Ten Good Reasons for Using Polyharmonic Spline Reconstruction in Particle Fluid Flow Simulations Armin Iske

Abstract We develop supporting arguments in favour of kernel-based reconstruction methods in the recovery step of particle simulations.We strongly recommend polyharmonic spline kernels, whose key features and advantages are briefly explained.

1 Introduction Particle methods provide flexible discretizations for the numerical simulation of multiscale phenomena in fluid flows. For time-dependent evolution processes, for instance, particle models are particularly well-suited to cope with rapid variation of domain geometries and anisotropic large-scale deformations. To discuss their governing equation, we remark that fluid flow simulations require the numerical solution of a hyperbolic conservation law of the form ∂u + ∇ f (u) = 0, ∂t

(1)

on given computational domain  ⊂ Rd , d ≥ 1, compact time interval [0, T ], T > 0, and flux tensor f (u) = ( f 1 (u), . . . , f d (u))T , where at time t = 0 initial conditions u(0, x) = u 0 (x)

for x ∈ 

(2)

are assumed. Nonlinear flux tensors f lead to discontinuous solutions u, shocks, which may develop spontaneously, even at smooth initial data u 0 in (2). Therefore, nonlinear flow simulation requires flexible computational methods to solve (1), (2). For an introduction to numerical methods for fluid dynamics we refer to [10, 12, 13].

A. Iske (B) Department of Mathematics, University of Hamburg, Hamburg, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_25

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2 The Basic Reconstruction Problem of Particle Flow Simulation To compute the numerical solution u : [0, T ] ×  −→ R of (1), (2), particle methods are popular tools, especially for their high flexibility. The Eulerian finite volume particle method (FVPM) [4] and the semi-Lagrangian particle method (SLPM) [8] are only two prototypes. For a comprehensive discussion on particle-based flow simulation, we refer to [8] and references therein. Although Eulerian and Lagrangian particle methods, such as FVPM and SLPM, are conceptually different, they rely on the solution of a specific reconstruction problem, which we can describe as follows. To this end, let  = {ξ1 , . . . , ξn } ⊂  be a stencil, i.e., a finite point set of particle positions. For any particle ξ ∈ , we assume a scalar value u(ξ ) ≡ u(ξ, t) ∈ R, which is used, at time t, to compute the numerical solution u of (1), (2). In the Eulerian FVPM, for instance, u(ξ, t) is the particle average for the influence area of particle ξ , cf. [7, 8] for further details. Now the generic formulation of the basic reconstruction problem is as follows. Problem 1 Compute from a given stencil  = {ξ1 , . . . , ξn } ⊂  of n ∈ N pairwise distinct particle positions in the domain  ⊂ Rd , d ∈ N, and a data vector u  = (u(ξ1 ), . . . , u(ξn ))T ∈ Rn a reconstruction s :  −→ R satisfying s = u  , i.e., s(ξ j ) = u(ξ j )

for all 1 ≤ j ≤ n.

(3)

Although particle-based simulation methods are so popular for their high flexibility, they may have severe drawbacks. Indeed, especially in relevant situations of anisotropic stencils , e.g. in WENO schemes [11], the numerical stability is very critical, due to the heterogenous distribution of particles in . In such cases, it is desirable to work with robust reconstruction schemes in order to better capture discontinuities (shocks fronts) and preference directions of the flow. Other problems of particle methods are moving boundaries and highly turbulent flows. Needless to say that high performance particle simulation methods essentially require high performance reconstruction schemes. For the solution of Problem 1, accurate, stable, efficient and flexible reconstruction algorithms are therefore of vital importance.

3 Kernel-Based Reconstruction Versus Polynomial Reconstruction Most of the particle-based simulation methods rely on polynomial reconstruction. However, as documented for WENO methods [1], polynomials may lead to severe numerical instabilities. Especially in situations of anisotropic stencils , polynomial reconstructions are often doomed to fail. In this paper, we propose to avoid

Ten Good Reasons for Using Polyharmonic Spline Reconstruction …

195

polynomial reconstruction methods. We rather give a strong recommendation in favour of kernel-based reconstructions, where polyharmonic splines are our favourite choice. Before we start our discussion on kernel-based reconstructions, we wish to provide only one argument against using polynomials, which should convince the numerical reader already now: Recalling polynomial interpolation from univariate data, where d = 1 and  ⊂ R, we understand from basic numerical analysis that polynomial interpolation (a) is numerically unstable, especially for large and unevenly distributed data; (b) is highly sensitive with respect to noisy input data; (c) may lead to high oscillations near the boundary of the domain (interval) . On the other hand, we understand from basic numerical analysis that splines, e.g. cubic splines, are powerful tools for univariate interpolation. In fact, spline reconstruction schemes avoid the well-known shortcomings (a)–(c) of polynomial interpolation. Thereby, spline reconstruction is far superior to polynomial reconstruction, already in the case of one dimension, d = 1, and so the following two leading questions are not undue: (Q1) Should we use polynomial reconstruction for the multivariate case? (Q2) Is there a generalization of univariate splines to higher dimensions? Our answer to (Q1) is clearly no, since polynomial reconstruction does not even work properly for the univariate case. Our answer to (Q2) is yes: polyharmonic splines extend univariate splines to higher dimensions d > 1, as we explain here.

4 Kernel-Based Reconstruction in Particle Fluid Flow Simulations To introduce basic ingredients of kernel-based reconstruction, let us first recall the conditions s = u  in Problem 1, for whose solution s :  −→ R we now assume the form n  c j ϕ(x − ξ j ) + p(x) for p ∈ Pdm (4) s(x) = j=1

with parameters (coefficients) c = (c1 , . . . , cn )T ∈ Rn . Moreover, ϕ :  −→ R is an even kernel function and Pdm denotes the linear space of all d-variate polynomials of order m ∈ N0 . The order m in (4) is determined by the choice of ϕ as follows. For the well-posedness of the reconstruction scheme, we require the kernel ϕ to be conditionally positive definite of order m on Rd (see [14, Chap. 8]), in short: ϕ ∈ CPD(m, Rd ). For the following discussion, it is sufficient to say that, for ϕ ∈ CPD(m, Rd ), we obtain under the moment conditions n  j=1

c j p(ξ j ) = 0

for all p ∈ Pdm

(5)

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a reconstruction s of the form (4), where s is unique, if the stencil  is Pdm -unisolvent, i.e., any polynomial p ∈ Pdm can uniquely be reconstructed from its values at , or, in other words, p = 0 implies p ≡ 0. T Therefore, the coefficients c = (c j )1≤ j≤n ∈ Rn and d = (dα )|α| d,

which is equipped with the semi-norm |u|2BLm =

 m D α u 2L 2 (Rd ) α |α|=m

for u ∈ BLm (Rd ).

In other words, the polyharmonic spline reconstruction s in (4), satisfying s = u  , minimizes the energy | · |BLm among all recovery functions u in BLm (Rd ), i.e.,

Ten Good Reasons for Using Polyharmonic Spline Reconstruction …

|s|BLm ≤ |u|BLm

for all u ∈ BLm (Rd ) with u  = s ,

197

(7)

where the semi-norm |s|BLm of the polyharmonic spline reconstruction s is given by the quadratic form n  |s|2BLm = c j ck φd,m ( ξk − ξ j ), (8) j,k=1

whose coefficient vector c = (c1 , . . . , cn )T ∈ Rn comes with the solution in (6).

6 Ten Good Reasons for Using Polyharmonic Spline Reconstruction We summarize the discussion of this contribution by giving ten good reasons in favour of using polyharmonic spline reconstruction in particle flow simulations. Reason 1: Well-posedness. Polyharmonic splines yield a well-posed reconstruction method, which guarantees the existence and uniqueness for a solution of Problem 1, for arbitrary stencils  ⊂ Rd and values u  , and for any dimension d ≥ 1. Indeed, due to Proposition 1, the system (6) has for φd,m ∈ CPD(m, Rd ) a unique solution. This is in contrast to polynomial reconstruction. In fact, due to the Mairhuber-Curtis theorem [5, Theorem 5.25] from approximation theory, a reconstruction scheme can, for d > 1, only be well-posed, if the reconstruction space depends on the stencil . Reason 2: Efficient implementation and preconditioning. The implementation of the polyharmonic spline reconstruction scheme merely requires solving square linear systems of the form (6), which can be set up very easily. To obtain efficient and numerically stable solutions of (6), we recommend our recent preconditioner, relying on hierarchical matrix approximation [6]. Reason 3: Stable and efficient evaluation. Polyharmonic splines allow for stable and efficient evaluations of their reconstructions. This is due to the scale-invariance of the reconstruction scheme’s Lagrange basis, see [7, Sect. 7.4] for more details. Reason 4: Stability by orthogonal projection. The polyharmonic spline reconstruction s, satisfying s = u  , is characterized by the (unique) orthogonal projection of u ∈ BLm (Rd ) onto the linear subspace of reconstructions of the form (4). This property is covered by approximation theory in Euclidean spaces, cf. [5, Chap. 4], which further implies that s is the best approximation to u ∈ BLm (Rd ) w.r.t. | · |BLm . Reason 5: Optimality by energy minimization. Polyharmonic spline reconstruction by φd,m is optimal in the Beppo-Levi space BLm (Rd ), by the energy minimization in (7). The latter already follows from the best approximation property of the polyharmonic spline reconstruction scheme in BLm (Rd ) w.r.t. | · |BLm (cf. Reason 3). Reason 6: Polynomial reproduction. If the stencil  is Pdm -unisolvent and if the input data u  is sampled from a polynomial u ∈ Pdm , then we have c = 0 for the major

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part of the reconstruction s in (4), satisfying s = u  . This implies the polynomial reproduction property s ≡ u, due the well-posedness of the reconstruction scheme. Reason 7: Arbitrary local approximation order. Polyharmonic spline reconstruction by φd,m has local approximation order m with respect to C m functions, i.e., |u(hx) − s h (hx)| = O(h m )

for h 0

for u ∈ C m ,

where s h is the (unique) polyharmonic spline reconstruction satisfying sh = u h . This result is due to the reproduction of polynomials from Pdm , cf. [7, Sect. 7.5]. Reason 8: Flexible stencil selection. According to the kernel-based reconstruction scheme of Sect. 4, we merely require n ≥ q for the number of particles n = ||, which allows us to work with variable stencil sizes. This is contrast to polynomial reconstruction, where the stencil size || is fixed a priori by the dimension of the polynomial space. The latter is often viewed as a severe restriction of polynomial WENO reconstructions (see [2]), where flexible stencils (of variable sizes) are desired. Reason 9: Natural oscillation indicator. To avoid oscillations of reconstructions s, satisfying s = u  , WENO schemes work with oscillation indicators. To this end, the polyharmonic spline reconstruction scheme provides a natural choice by the energy functional | · |BLm . This is further supported by the variational principle, according to which s minimizes | · |BLm among all recovery functions in BLm (Rd ), see (7). Moreover, the minimum |s|BLm is readily available by the quadratic form (8). Reason 10: Meshfree reconstruction and high flexibility in adaptive methods. The reconstruction scheme of polyharmonic splines is meshfree and therefore very flexible, especially when it comes to modifying the set of moving particles adaptively (cf. [9, Chap. 6]), which is particularly important for problems with solutions of rapid variation or singularities, or for problems with free or complicated boundaries. Finally, meshfree reconstruction does obviously not rely on sophisticated algorithms for the generation and maintenance of a computational mesh, unlike in FD, FV, FE, DG and other mesh-based methods. This gives polyharmonic spline reconstructions (and other meshfree methods) yet another advantage, in particular for high-dimensional problems, where mesh generation is prohibitively expensive.

References 1. Abgrall, R.: On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation. J. Comput. Phys. 144, 45–58 (1994) 2. Aboiyar, T., Georgoulis, E.H., Iske, A.: Adaptive ADER methods using kernel-based polyharmonic spline WENO reconstruction. SIAM J. Sci. Comput. 32(6), 3251–3277 (2010) 3. Duchon, J.: Splines minimizing rotation-invariant semi-norms in Sobolev spaces. In: Schempp, W., et al. (eds.) Constructive Theory of Functions of Several Variables, pp. 85–100. Springer, Berlin (1977) 4. Hietel, D., Steiner, K., Struckmeier, J.: A finite-volume particle method for compressible flows. Math. Model. Methods Appl. Sci. 10(9), 1363–1382 (2000)

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5. Iske, A.: Approximation Theory and Algorithms for Data Analysis. Texts in Applied Mathematics, vol. 68. Springer, Cham (2018) 6. Iske, A., Le Borne, S., Wende, M.: Hierarchical matrix approximation for kernel-based scattered data interpolation. SIAM J. Sci. Comput. 39(5), A2287–A2316 (2017) 7. Iske, A.: On the construction of kernel-based adaptive particle methods in numerical flow simulation. In: Ansorge, R., Bijl, H., Meister, A., Sonar, T. (eds.) Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation, pp. 197–221. Springer, Berlin (2013) 8. Iske, A.: Polyharmonic spline reconstruction in adaptive particle flow simulation. In: Iske, A., Levesley, J. (eds.) Algorithms for Approximation, pp. 83–102. Springer, Berlin (2007) 9. Iske, A.: Multiresolution Methods in Scattered Data Modelling. Lecture Notes in Computational Science and Engineering, vol. 37. Springer, Berlin (2004) 10. LeVeque, R.L.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002) 11. Liu, X., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212 (1994) 12. Morton, K.W., Sonar, T.: Finite volume methods for hyperbolic conservation laws. Acta Numer. 16, 155–238 (2007) 13. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, 3rd edn. Springer, Berlin (2009) 14. Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)

Generalized Solutions of Galileo-Invariant Thermodynamically Consistent Conservation Laws, Constructed Using Ideas of Fundamental Works by S. K. Godunov M. J. Ivanov

Abstract The analysis of Galileo-invariant thermodynamically consistent conservation laws admitting a class of generalized solutions is performed. The main feature of generalized solutions considered in the article is the description of total pressure losses accompanying dynamic process of heat supply. The fundamental properties of constructed generalized solutions are considered in Godunov’s works the Galileoinvariance and the thermodynamic consistency of original conservation laws in closed mathematical formulation of a thermal gas dynamic process.

1 Introduction It is well known that the class of generalized solutions in addition to traditional classical smooth solutions includes relations on discontinuities. In our case, in the presence of heat supply, the class of generalized solutions of quasi-linear equations of thermo gas dynamics expands. However, the problem statement is considered only for the simplest part of phenomenological thermodynamics (without chemical reactions, phase transitions, etc.). The determining factors of the analyzed generalized solutions will be the growth of entropy and the presence of already mentioned total pressure losses directly related to it. Following the classical works of Ladyzhenskaya [1], Oleynik [2], Gelfand [3] and Godunov [4], we call the solutions of quasilinear equations by generalized solutions, which can be obtained as the limits of solutions of extended equations with viscosity and thermal conductivity in the pursuit to zero coefficients of viscosity and thermal conductivity. In the cited works the conditions of convergence of smooth solutions for extended equations, including terms with derivatives of the second order, to disM. J. Ivanov (B) Central Institute of Aviation Motors (CIAM), Aviamotornaya Street 2, 111116 Moscow, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_26

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continuous solutions are studied in detail. The features of solutions we consider follow are, firstly, the use of the original closed mathematical formulation for a thermal gas process in the form of three balance conservation laws of mass, momentum and energy and, secondly, from the construction of final solutions, including smooth solutions characterized by the presence of total pressure losses. By that the fundamental properties of the constructed generalized solutions are considered in Godunov’s works [5, 6] the Galileo-invariance and the thermodynamic consistency of the original conservation laws in a closed mathematical formulation of thermal gas processes. Fulfillment of the Galilean invariance property to obtain the closed mathematical formulation is also required for the given boundary and initial conditions of the problem. This circumstance is related, in particular, to the input boundary parameters for simulated flows. As a visual example, we present the change in values of the total pressure and the total temperature of flow inputting into a simulated area. In the presence of a speed of a incoming flow, its total parameters increase and this should be taken into account in the thermodynamically coordinated formulation. The paper also formulates a refined thermodynamically consistent procedure for setting Galileo invariant initial conditions of an unsteady problem taking into account total pressure losses. When solving stationary problems by the method of a time resolution the thermodynamic consistency of the final solution is achieved in the process of establishing, through the integration of thermodynamically consistent equations. For the analyzed generalized solutions with total pressure losses, entropy is not a function of the state and there is no reason to apply the method of thermodynamic potentials. The known theoretical approaches of phenomenological thermodynamics, based on variation principles and Pfaff’s forms, as well as methods of statistical thermodynamics require a certain revision in connection with mentioned above. Here it should be also emphasized the modern theoretical physics has no answer on entropy growth nature for isolate thermo and fluid dynamic systems. In the famous 10-volume course on theoretical physics argues: “The question of the physical basis of monotonic increase of entropy thus remains open” (Landau and Lifshitz. Stat. Phys. 1996, p. 52, [7]). On the problem of increasing entropy soviet academician V.L. Ginzburg also puts in first place among the outstanding “three great challenges”: “First, we are talking about the increase of entropy, irreversibility and the “arrow of time”” [8].

2 Closed System of Quasi-linear Equations with Continuous Heat We start with equations of the first and the second laws of phenomenological thermodynamics, which here we write using generally accepted notation

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ds dε dv δQ =T· = + p· . δt dt dt dt

(1)

The ratio (1) describes quasi-static reversible processes and is the initial energy conservation ratio of conventional thermodynamics. About Eq. (1) Godunov [4] rightly points out that in the statements of thermodynamics “proof of the existence of universal multiplier integrating equation heat thermally homogeneous systems shall be conducted only based on the study of energy equations, do not represent a closed system”. It is also important to emphasize the principle issue, namely that when solving thermodynamic tasks the use of the ratio (1) doesn’t allow obtaining weak solutions of quasi-linear equations involving total pressure losses. The ratio (1) provides to the theorems of “alive force”, which in our case can be written in the form d dt



q2 2

 +

q grad p = 0. ρ

(2)

The theorem of “alive forces” (2) displays from the vector motion equation dq 1 + grad p = 0 dt ρ

(3)

by a simple scalar multiplication on vector velocity q. The vector equation (3) includes in the three-dimensional case the three scalar conservation laws of momentum, which are easily represented in divergence form. At the same time, the scalar equation (2) cannot lead to divergent differential or integral conservation law. From (2) it is possible to obtain a non-divergent equation of change in specific kinetic energy q 2 /2 (“living force”) along a stream line ∂ ∂t

 2   q q2 ρ + div ρ q + q · grad p = 0. 2 2

(4)

Application of theorem “alive forces” (2) or equation of specific kinetic energy changes (4), as well as the use of ratios (1) does not allow receiving thermodynamically generalized solutions for total pressure losses. The present paper addresses the closed system of quasi-linear equations for thermo gas dynamics, following from integral balance conservation laws of mass, momentum and energy for a fixed time volume ω with the boundary γ [9]

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∂ ∂t ∂ ∂t

 ω

γ

ρ q dω = −

γ

( p · n + ρ q(q · n)) dγ ,

ω

 

ρ q · n dγ ,





γ



∂ ∂t =−

 ρ dω = −

ω



(5)

  q2 dω ρ ε+ 2

q2 p+ρ ε+ 2



 q · n dγ +

ρ ω

δQ dω, δt

where n—the unit vector normal to the exterior boundaries. Determining thermodynamic process of this system is the presence in the volume specific heat supply with speed δ Q/δt. From integral system (5) we can get the differential divergent equations of continuity, momentum and energy ∂ρ + div (ρ q) = 0, ∂t ∂ ρ q + div  = 0, ∂t        q2 q2 ∂ δQ ρ ε+ + div p+ρ ε+ . q =ρ ∂t 2 2 δt

(6)

Systems of Eqs. (5) and (6) close the state equation ε = ε( p, ρ).

(7)

The differential system (6) in divergence form gives the generalized solutions of thermo gas dynamics equations in the presence of heat supply. Now we compare these mathematical formulations with the first and the second principles of thermodynamics (1). From the left part of the energy equation of the system (6) we can write       q2 q 2 ∂ρ q2 ∂ ε+ + ε+ + ρ q grad ε + ρ ∂t 2 2 ∂t 2   q2 δQ div (ρ q) + div ( p q) = ρ . + ε+ 2 δt Using continuity equation we arrive at the ratio

Generalized Solutions of Galileo-Invariant Thermodynamically …

d dt

  q2 1 δQ ε+ + div ( p q) = . 2 ρ δt

205

(8)

Such form there is accepted the differential law of energy conservation when using closed mathematical formulations of process. Further, taking into account the equation of continuity the ratio (8) comes dε d +p dt dt

    1 d q2 δQ q + + grad p = . ρ dt 2 ρ δt

(9)

The ratio (9) shows that the applied heat to medium volume ω moving with speed q, spent on increasing the specific internal energy (with the speed dε/dt), on changing of the work (quasi-static p d(1/ρ)/dt), on changing of the specific kinetic energy (with the speed d(q 2 /2)/dt) and the amount of extra power, produced by pressure forces q(grad p)/ρ, missing usually in the energy ratios of conventional thermodynamics. Equations (8) and (9) allow simulating the appearance of the total pressure losses at addition to heat moving gas stream [10]. It isn’t very complex to see the entropy growth closely connects with molecule mass decreasing [11]. The main conclusion of the carried out transformation is the fact that the process of heat supply to the moving gas flow describes by the ratio (9), differing from the first laws of thermodynamics for quasi-static reversible processes (1). Based on the ratio (9) we can give the response to outfit significant theoretical issues. The first question is in the execution conditions of additional conservation law of gas dynamics, namely the law of entropy conservation. With this major issue are other three important issues: the “mechanical” energy equation (theorem of “alive force”), and the loss of total pressure and on the arm of the famed “Riemann” errors in thermodynamics.

3 Thermodynamically Consistent Conservation Laws The paper demonstrates some detail of thermodynamically consistent conservation laws [12, 13] (the entropy growth with inextricably related total pressure losses). The analysis bases on conservation laws of mass, momentum and energy. Following by Ladyzhenskaya’s methodology [1] we can write for one dimensional unsteady case the energy conservation law          u2 ∂ u2 ∂ 4 ∂u ∂T ∂ ρ ε+ + p+ρ ε+ u = ηu +λ . ∂t 2 ∂x 2 ∂x 3 ∂x ∂x Here and below conventional variables are used. After differencing we lightly have d dε +p dt dt

      1 d u2 u ∂p 1 ∂ 4 ∂u ∂T + + = ηu +λ . ρ dt 2 ρ ∂x ρ ∂x 3 ∂x ∂x

(10)

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When η and λ → 0 the relation (10) gives dε d +p dt dt

    1 d u2 u ∂p + + = 0. ρ dt 2 ρ ∂x

(11)

For generalized solutions the Eq. (11) can be divided on two equations for the entropy growth   d 1 dε +p =σ >0 (12) dt dt ρ and the total pressure losses d dt



u2 2

 +

u ∂p = −σ < 0. ρ ∂x

(13)

We can see closed connection between the entropy growth (the relation (12)) and total pressure losses (the relation (13)). If σ = 0 applying the multiple 1/T we get from (12) the additional law of entropy conservation ds = 0. dt Although the method was shown here to deal with one dimensional problem it may be used to construct solutions of initial value problems for simulations in any number of space variables at aerodynamic heating. Now we can introduce aerodynamic heating as specific integral heat value δ Q in right part of energy conservation law (9) and considerate corresponded advance weak solutions of hydrodynamic equations. Dividing δ Q = δ Q 1 + δ Q 2 we have d dε +p dt dt

  1 δ Q1 = , ρ δt

d dt



q2 2

 +

u δ Q2 grad p = , ρ δt

that demonstrates the closed connection between entropy growth at presence δ Q 1 and inextricably related total pressure losses at presence δ Q 2 .

4 Conclusions Thermodynamically consistent solutions of quasilinear equations of thermo gas dynamics are described by a complete system of Galileo-invariant integral conservation laws or differential equations in divergent form. One of the main conditions for obtaining generalized solutions is the integration, along with the laws of conservation of mass and momentum, also the law of conservation of energy, describing the total change in internal and kinetic energies (ε + q 2 /2). The resulting solutions

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will contain the total pressure losses on the solution discontinuities or in the areas of smooth solutions in the presence of heat supply. In connection with above item generalized solutions should be called solutions that include classical smooth solutions and solutions with losses of total pressure (at shocks and in areas with heat). The additional entropy conservation law holds only in the domains of justice of classical smooth solutions. Only in these regions entropy can be a function of state. Outside these areas there is no reason to apply the method of thermodynamic potentials. The known theoretical approaches of phenomenological thermodynamics, based on variation principles and Pfaff forms, as well as methods of statistical thermodynamics require a certain revision. Traditional thermodynamics is actually for thermo-static reversible processes. The theorem of “live forces”, using a convolution of three scalar momentum equations to a single equation describing the change of the specific kinetic energy in the general case does not satisfy by all required boundary conditions for three scalar equations of momentum conservation.

References 1. Ladyzhenskaya, O.A.: On the construction of discontinuous solutions of quasilinear hyperbolic equations as limits of solutions of the corresponding parabolic equations when the “viscosity coefficient” tends to zero. Rep. Acad. Sci. USSR 111(2), 291–294 (1956) 2. Oleynik, O.A.: Discontinuity solutions of nonlinear differential equations. Adv. Math. Sci. XII 3(75), 3–73 (1957) 3. Gelfand, I.M.: Some problems of quasilinear equation theory. Adv. Math. Sci. XIV 14, 2(86), 87–158 (1959) 4. Godunov, S.K.: The concept of generalized solutions. Rep. Acad. Sci. USSR 134(6), 1279– 1282 (1960) 5. Godunov, S.K., Romenski, E.I.: Elements of continuum medium mechanics and conservation laws. Books. Springer, Novosibirsk (1998) 6. Godunov, S.K., Gordienko, V.M.: Simplest Galileo-invariant and thermodynamically consistent conservation laws. Appl. Mech. Tech. Phys. 43, 3–16 (2002) 7. Landau, L.D., Lifshitc, E.M.: Statistical Physics. Theoretical Physics. Nauka, Moscow (1996) 8. Ginzburg, V.M.: Adv. Phys. Sci. 189, 46–49 (1999) 9. Ovsiannikov, L.V.: Lectures on Gas Dynamics Bases. Nauka, Moscow (1981) 10. Air Breathing Engines Theory: In: Schliakhtenko, C.M. (ed.) Mashinostroenie, Moscow (1975) 11. Ivanov, M.J.: Space energy. In: Ahmed, A.Z. (ed.) Energy Conservation, pp. 3–56. In Tech, London (2012). https://doi.org/10.5772/52493 12. Ivanov, M.J.: Thermodynamically compatible conservation laws in the model of heat conduction radiating gas. Comp. Math. Math. Phys. 51(1), 133–142 (2011) 13. Ivanov, M.J., Mamaev, V.K., Zhestkov, G.B.: Thermodynamically compatible theory of high temperature turbojet engines. In: Proceedings of the 29th ICAS, 7–14 Sept., St.-Petersburg

Godunov-Type Schemes for Diffusion and Advection-Diffusion Steven Jöns and Claus-Dieter Munz

Abstract We study Godunov’s method for diffusion and advection-diffusion problems. The numerical fluxes for the finite volume scheme are based on an approximation of the generalized Riemann Problem. Hereby, approximate Riemann solvers are constructed, which approximate the solutions by a space-time discontinuous Galerkin approach. The implementation to a Godunov-type finite volume method and numerical results are discussed.

1 Introduction In recent years, Godunov-type finite volume schemes have been used for a variety of engineering applications, ranging from the Euler equations to the full Navier–Stokes equations. In the latter case, the conventional approximation of the diffusion fluxes usually departs from Godunov’s idea [11] and borrows ideas from finite difference methods, which assume a continuous approximation at the grid cell interface and applies central differencing or a least-squares method [4]. Other schemes combine Godunov-type finite volume methods with finite element discretizations [5]. Hence, advection and diffusion are typically handled differently. However, Godunov’s idea may be applied similarly to advection-diffusion which allows to avoid a splitting of the two fluxes. This is evident by reconsidering the finite volume method: u in+1 = u in +

 t  gi−1/2 − gi+1/2 . x

(1)

S. Jöns · C.-D. Munz (B) Institute of Aerodynamics and Gasdynamics, University of Stuttgart, Pfaffenwaldring 21, 70569 Stuttgart, Germany e-mail: [email protected] S. Jöns e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_27

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Hereby, u in approximates the mean in the ith cell and at the nth time step and the numerical flux gi−1/2 approximates the time average flux between the cells: gi−1/2 =

1 t



t

f (u(xi−1/2 , t))dt.

(2)

0

The idea of Godunov was to define the numerical flux by the solution of the corresponding Riemann problem at the cell interface: u t + f (u, u x )x = 0,  n u i−1 for x < xi−1/2 , u(x, 0) = for x > xi−1/2 . u in

(3a) (3b)

Note that in general the Riemann Problem (3) can be considered for an unsplitted advection-diffusion equation. This was done by Harabetian in [12]. He approximated the solution of the Riemann Problem by a diffuse traveling wave and used this solution to calculate the flux (2). The numerical results in [12] showed an improvement of the solution quality due to the coherent approach. However, its extension to the Navier– Stokes equation by Weekes [17] is cumbersome. Later, Toro and Montecinos used in [16] a hyperbolization of the equations within the ADER framework. In addition to the mentioned works on advection-diffusion, the Godunov method has been extended to diffusion equations by Gassner et al. [9]. They stated that for a diffusion term with a second order derivative, the diffusive generalized Riemann Problem (dGRP) with at least piecewise linear data has to be considered. The generalized Riemann problem (GRP) has been studied extensively for pure hyperbolic equations starting with the work of Ben-Artzi and Falcovitz [3]. An extensive overview of the activities about the GRP can be found in Balsara et al. [1]. The outline of the paper is as follows. First, we review shortly the dGRP flux in Sect. 2. Next, a novel approach to consider advection and diffusion simultaneously within Godunov-type schemes is discussed in Sect. 3. In Sect. 4 we discuss numerical results. Finally, we will give a short conclusion and outlook in Sect. 5.

2 Diffusive Generalized Riemann Problem For the linear, scalar diffusion equation u t = κu x x ,

(4)

a well-known solution for the initial value problem is given by the so-called heat kernel, which can be used to solve the Riemann problem and to construct a numerical flux. We assume that a high order reconstruction procedure is applied to obtain a polynomial, at least linear, in every grid cell. In our FV scheme we use a

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WENO-type reconstruction detailed in [2, 7]. Then we can define the diffusive generalized Riemann problem at the element boundaries as u t = κu x x ,  k cl x j for x < 0, u(x, 0) = kj=0 rj j for x > 0. j=0 c j x

(5a) (5b)

Note that we use an element local coordinate system in (5), centered at the interface. A solution is then retrieved by inserting the initial data into the heat kernel. Its derivative at the interface x = 0 for the diffusion flux can then be written as u x (0, t) =

k 

  j−1 C j crj − clj (−1) j t 2 .

(6)

j=0

This is inserted into the flux integral (2) to obtain g dG R P =

k 

  2κ j−1 t 2 , C j crj − clj (−1) j 1 + j j=0

(7)

with the coefficients C j = 2 jκC j−2 with C0 = √

1 4π κ

and C1 =

1 . 2

(8)

This linear, scalar case is relatively simple. However, extensions to diffusion systems and nonlinear diffusion can be found in [9, 10, 15].

3 Advection Diffusion Generalized Riemann Problem As depicted earlier, we apply the Godunov idea directly to advection-diffusion equations. Instead of the traveling wave approach of Harabetian in [12] we consider a space-time discontinuous Galerkin approximation. Mirroring the dGRP approach, we consider piecewise polynomial initial data u t + f a (u)x = κu x x ,  p u l φ(x) j for x < 0, u(x, 0) =  pj=0 rj j=0 u j φ(x) j for x > 0.

(9a) (9b)

Hereby, f a (u) denotes a convex nonlinear flux function, κ denotes the constant diffusion coefficient and φ j denotes the jth Lagrange polynomial. The domain of

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interest is  = [− x , x ] × [0, t]. We split  into to two space-time elements, 2 2 which are separated by a single wave:   x x , st × [0, t],  R = st, × [0, t]. L = − 2 2

(10)

The constant velocity s is defined by the Rankine–Hugoniot condition. In this situation we apply a discontinuous Galerkin (DG) approach and consider the weak formulation from [9] for the advection-diffusion equation in both space-time elements. The approximate solution consists of a polynomial in each element with the corresponding initial data distribution. Transformed onto a reference element ˜ = [−1, 1] × [−1, 1], we can apply the space-time DG methodology proposed in  [8, 14]. The polynomial solution of the Riemann problem in the reference space (ξ, τ ) is obtained by solving uˆ j (t) = uˆ j (0) −

p+1  wk

k=1

−

p+1  wk  k=1

wj

g˜

s,+

wj

(τk )(φ j )+ ξ

g˜ + (τk )φ +j − g˜ − (τk )φ −j

− g˜

s,−

(τk )(φ j )− ξ





p+1  wk wl + g(ξ ˜ l , τk )Dl, j . wj k,l=1

(11)

Hereby uˆ j depicts the polynomial coefficients of the solution, wi are the Gaussian integration weights, g˜ the cumulative flux in the reference element, p the polynomial degree of the solution and Di, j the differentiation matrix. The superscripts + and − denote the evaluation of the respective flux or basis function at the left or right boundary of the space-time cell. The additional surface flux g˜ s stems from the ultraweak formulation, for details see [9]. Similar to [8], the evaluation of the surface and volume fluxes are realized with the help of a predictor. In this paper, we use the local space-time DG predictor proposed by Dumbser et al. in [6]. The surface flux at the conjunction of the two space-time elements needs special care. Here we employ the splitting of the fluxes using the dGRP flux. For advection a simple upwinding for a shock wave and an arithmetic mean of the left and right states for a rarefaction is used. Equation (11) gives a piecewise polynomial solution of the advection-diffusion Riemann problem (9), which is used to calculate the flux integral (2). To evaluate the diffusion flux with derivatives, the recovery method of van Leer and Nomura [13] is applied. The numerical flux function finally yields  1  a w j f (q(0, tk )) − κr x (0, tk ) . 2 j=1 p+1

adG R P ≈ gi+1/2

(12)

Hereby q denotes the piecewise polynomial solution retrieved from Eq. (11) and r x (x, t) the derivative of the recovery polynomial.

Godunov-Type Schemes for Diffusion and Advection-Diffusion

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Fig. 1 Viscous shock profile calculated with the adGRP solver at N = 3, κ = 1 and t = 0.01

4 Numerical Results First we show results to document the approximate Riemann solution by considering  2 for x < 0, u t + uu x = κu x x , u(x, 0) = 1 for x > 0.

(13a)

Figure 1 shows the solution of the Riemann problem at t = 0.01 with κ = 1 and p = 3. The initial data generated a right moving viscous shock profile. The novel Riemann solver (adGRP) was able to capture the viscous shock profile very well, both in the piecewise polynomial representation and in the recovery polynomial. A second test case validates the use of the proposed Riemann solver in an ADERtype FV method. Here, we consider the following exact solution of the viscous Burgers equation obtained by Wood [18]: u(x, t) =

2κπ exp(−π 2 κt) sin(π x) . 2 + exp(−π 2 κt) cos(π x)

(14)

Using the flux calculation (12), a convergence analysis is performed for different polynomial degrees. The results can be seen in Fig. 2. With respect to diffusion the order of accuracy is identical to the degree p. This stems from the reconstruction because the derivatives are needed—for advection the order is p + 1. This aspect has already been observed by Gassner et al. in their construction of the dGRP solver [9].

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Fig. 2 Convergence plots of the adGRP Riemann solver for the sinewave problem of Wood

5 Conclusion We considered Godunov-type methods for advection-diffusion. The construction of the Riemann solver is based on a space-time DG approach. Numerical results for the finite volume scheme were shown and indicate a good quality of the approximation. In future work, the proposed Riemann solver will be used to solve Riemann problems with phase-transition, in which the heat flux plays an important role. Acknowledgements This work was supported by the German Research Foundation (DFG) through the collaborative research center SFB TRR 75 Droplet Dynamics Under Extreme Ambient Conditions and by the DFG under Germany’s Excellence Strategy - EXC2075 - 3g0740016.

References 1. Balsara, D.S., Li, J., Montecinos, G.I.: An efficient, second order accurate, universal generalized Riemann problem solver based on the HLLI Riemann solver. J. Comput. Phys. 375, 1238–1269 (2018) 2. Balsara, D.S., Rumpf, T., Dumbser, M., Munz, C.D.: Efficient, high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics. J. Comput. Phys. 228(7), 2480–2516 (2009) 3. Ben-Artzi, M., Falcovitz, J.: A second-order Godunov-type scheme for compressible fluid dynamics. J. Comput. Phys. 55(1), 1–32 (1984) 4. Bertolazzi, E., Manzini, G.: Least square-based finite volumes for solving the advection diffusion of contaminants in porous media. Appl. Numer. Math. 51(4 SPEC. ISS.), 451–461 (2004)

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5. Dawson, C.: Godunov-mixed methods for advection-diffusion equations in multidimensions. SIAM J. Numer. Anal. 30(5), 1315–1332 (1993) 6. Dumbser, M., Balsara, D.S., Toro, E.F., Munz, C.D.: A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes. J. Comput. Phys. 227(18), 8209–8253 (2008) 7. Dumbser, M., Käser, M.: Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. J. Comput. Phys. 221(2), 693–723 (2007) 8. Gassner, G., Dumbser, M., Hindenlang, F., Munz, C.D.: Explicit one-step time discretizations for discontinuous Galerkin and finite volume schemes based on local predictors. J. Comput. Phys. 230(11), 4232–4247 (2011) 9. Gassner, G., Lörcher, F., Munz, C.D.: A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes. J. Comput. Phys. 224(2), 1049–1063 (2007) 10. Gassner, G., Lörcher, F., Munz, C.D.: A discontinuous Galerkin scheme based on a space-time expansion II. Viscous flow equations in multi dimensions. J. Sci. Comput. 34(3), 260–286 (2008) 11. Godunov, S.: Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb. 47(89)(3), 271–306 (1959) 12. Harabetian, E.: A numerical method for viscous perturbations of hyperbolic conservation laws. SIAM J. Numer. Anal. 27(4), 870–884 (1990) 13. van Leer, B., Nomura, S.: Discontinuous Galerkin for Diffusion. In: 17th AIAA Computing Fluid Dynamic Conference American Institute of Aeronautics and Astronautics (2012) 14. Lörcher, F., Gassner, G., Munz, C.D.: A discontinuous Galerkin scheme based on a spacetime expansion. I. Inviscid compressible flow in one space dimension. J. Sci. Comput. 32(2), 175– 199 (2007) 15. Lörcher, F., Gassner, G., Munz, C.D.: An explicit discontinuous Galerkin scheme with local time-stepping for general unsteady diffusion equations. J. Comput. Phys. 227(11), 5649–5670 (2008) 16. Toro, E.F., Montecinos, G.I.: Advection-diffusion-reaction equations: hyperbolization and high-order ADER discretizations. SIAM J. Sci. Comput. 36(5), A2423–A2457 (2014) 17. Weekes, S.L.: The travellingwave scheme for the Navier-Stokes equations. SIAM J. Numer. Anal. 35(3), 1249–1270 (2003) 18. Wood, W.L.: An exact solution for Burger’s equation. Commun. Numer. Methods Eng. 22(7), 797–798 (2006)

Direct and Inverse Problems for Conservation Laws S. I. Kabanikhin, D. V. Klychinskiy, I. M. Kulikov, N. S. Novikov and M. A. Shishlenin

Abstract We investigate the mathematical modeling of the 2D acoustic waves propagation. We solve the hyperbolic first order system of partial differential equations by the method of S. K. Godunov. The inverse problem of reconstructing the density of the medium is solved by the gradient method. The gradient of the functional is obtained.

1 Introduction Let us consider the Cauchy inverse problem for the scalar or vector conservation law wt + f (w)x = 0,

x ∈ (0, L), t ∈ (0, T ),

with initial condition S. I. Kabanikhin · D. V. Klychinskiy · I. M. Kulikov · N. S. Novikov · M. A. Shishlenin (B) Institute of Computational Mathematics and Mathematical Geophysics, Prospect Akademika Lavrentjeva, 6, Novosibirsk, Russia e-mail: [email protected] S. I. Kabanikhin e-mail: [email protected] D. V. Klychinskiy e-mail: [email protected] I. M. Kulikov e-mail: [email protected] N. S. Novikov e-mail: [email protected] S. I. Kabanikhin · M. A. Shishlenin Sobolev Institute of Mathematics, pr. Academician Koptyug, 4, Novosibirsk, Russia S. I. Kabanikhin · D. V. Klychinskiy · N. S. Novikov · M. A. Shishlenin Novosibirsk State University, Pirogova St., 2, Novosibirsk, Russia © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_28

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w(x, 0) = w0 (x),

x ∈ (0, L).

Here t is time variable, x is space variable, w describes the concentration of the solid particles, f is flow function [1, 2]. Inverse problem is to determine the function f by using the additional information, which could represent the time evolution of two interfaces, that divides settled suspension from supernatant fluid and ascending sludge, or concentration at fixed points with respect to time or space. In Sect. 2 we describe the exact statement of direct problem for the hyperbolic system of the two dimensional acoustic waves propagation and numerical algorithms for solving the direct problem. In Sect. 3 we describe the gradient based numerical algorithm for solving inverse problem. We solve the inverse problem using the gradient-based approach [3]. We consider the mathematical model of acoustic tomography in the form of the system of hyperbolic equations of the first order because in this form written conservation laws of continuum mechanics. It allows us to control the preservation of the basic invariants during the solution of direct and inverse problems. This is important for solving unstable problems, as the conservation laws of the main invariants are the only criterion of the well-posedness of the solution. Using the problem formulation in the hyperbolic system we control for the numerical solution and the guarantee the numerical solution is close to the physical solution of the process.

2 Direct Problem The propagation of the acoustic waves in the 2D medium is described by the following system: 1 ∂p ∂v 1 ∂p ∂u + = 0, + = 0, ∂t ρ ∂x ∂t ρ ∂y   ∂p ∂v ∂u + ρc2 + = 0. ∂t ∂x ∂y

(1) (2)

Here u = u(x, y, t), v = v(x, y, t) are components of velocity vector, with respect to x and y respectively, p = p(x, y, t) is the acoustic pressure. The parameters of the system describe the properties of the medium: ρ(x, y) denotes the density of the medium, and c(x, y) is the speed of the wave propagation. The system (1)–(2) can be rewritten in the integral form that represents the conservation laws for the problem: 

 ρud xdy + pdydt = 0, ρvd xdy + pd xdt = 0,  pd xdy + ρc2 (udydt + vd xdt) = 0.

(3) (4)

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We consider the hyperbolic system of the two dimensional acoustic wave propagation in the form of conservation laws. These problems are related to ultrasound tomography [4, 5], aimed at the detection of inclusions in the soft human tissue. We use the hyperbolic first-order system to describe the propagation of the acoustic waves. It allows us to propose more realistic model from the physical point of view. We consider the numerical algorithm for solving direct problems, based on the Godunov scheme [6, 7]. The numerical methods, based on the Godunov approach, allows to find the balance between mathematical modeling of physical process and the effective numerical realization. The main element of this method is the solving of Riemann problem, which consists of an initial value problem composed of a conservation equation together with piecewise constant data having a single discontinuity. This problem can be solve explicitly or by using some approximation [7]. Let us introduce the mesh in R2 , provided by the lines x = x j and y = yk , and denote by h x , h y parameters of the mesh. We will also denote by pair of indices ( j − 1/2, k − 1/2) the grid cell, corresponded to the points (x j−1 , yk−1 ), (x j , yk−1 ), (x j , yk ) ,(x j−1 , yk ). We suppose, that values of the functions u(x, y, t), v(x, y, t), p(x, y, t), which describes the state of the medium, are constant in each cell, and denoted this values as u j−1/2,k−1/2 , v j−1/2,k−1/2 , p j−1/2,k−1/2 . Now let us consider the ratio (3) for the cell j − 1/2, k − 1/2: ρ(u j−1/2,k−1/2 − u j−1/2,k−1/2 )h x h y +  t0 +τ  yk + [ p(x j , y, t) − p(x j−1 , y, t)]dydt = 0, (5) t0

yk −1

The upper index indicates the values of u, v, p for the moment t0 + τ (τ is the time step), and the lower index corresponds to the values of functions at the moment t0 . We approximate the functions p(x j , y, t), p(x j−1 , y, t), corresponded to values at the boundary of the cell, by some constant values, which we denote as P j,k−1/2 and P j−1,k−1/2 respectively. Thus, we can obtain: ρ(u j−1/2,k−1/2 − u j−1/2,k−1/2 )h x h y + τ h y (P j,k−1/2 − P j−1,k−1/2 ) = 0.

(6)

We use the conservation laws (3), (4) to obtain two more ratios: ρ(v j−1/2,k−1/2 − v j−1/2,k−1/2 )h x h y + τ h x (P j−1/2,k − P j−1/2,k−1 ) = 0, (p

j−1/2,k−1/2

− p j−1/2,k−1/2 )h x h y +

+ τρc h y (U j,k−1/2 − U j−1,k−1/2 ) + τρc2 h x (V j−1/2,k − V j−1/2,k−1 ) = 0. 2

(7) (8)

We use the solution of the problem of “decay of discontinuity” to obtain values P, U , V at the boundaries of the cell:

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p j+1/2,k−1/2 − p j−1/2,k−1/2 u j−1/2,k−1/2 + u j+1/2,k−1/2 − (9) 2 2ρc p j−1/2,k−1/2 + p j+1/2,k−1/2 u j+1/2,k−1/2 − u j−1/2,k−1/2 = − ρc 2 2 (10)

u = U j,k−1/2 = p = P j,k−1/2

The same ratios can be obtain at the boundary y = yk p j−1/2,k+1/2 − p j−1/2,k−1/2 v j−1/2,k−1/2 + u j−1/2,k+1/2 − (11) 2 2ρc v j−1/2,k+1/2 − u j−1/2,k−1/2 p j−1/2,k−1/2 + p j−1/2,k+1/2 − ρc = 2 2 (12)

v = V j−1/2,k = p = P j−1/2,k

Therefore, according to (6), (7), (8), we can write the final ratios for computing the state of the medium for “upper” layer t = t0 + τ : τ (P j,k−1/2 − P j−1,k−1/2 ), ρh x τ v j−1/2,k−1/2 = v j−1/2,k−1/2 − (P j−1/2,k − P j−1/2,k−1 ), ρh y τ p j−1/2,k−1/2 = p j−1/2,k−1/2 − ρc2 (U j,k−1/2 − U j−1,k−1/2 )− hx τ − ρc2 (V j−1/2,k − V j−1/2,k−1 ). hy

u j−1/2,k−1/2 = u j−1/2,k−1/2 −

Here values U , P, V are defined by (9)–(12). The stability of the scheme can be achieved by using the CFL condition [6, 7]:  τ

C0 C0 + hx hy

 < 1.

The constant C0 is the upper-bound estimate for the speed of wave propagation in the computational domain.

3 Inverse Problem of Recovering the Density Let us consider the direct problem for the acoustic equations in the domain (x, y) ∈  = {[0, L] × [0, L]}, t ∈ (0, T ): 1 ∂p ∂u + = 0, ∂t ρ ∂x

∂v 1 ∂p + = 0, ∂t ρ ∂y

(13)

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  ∂u ∂v ∂p + ρc2 + = θk (x, y)I (t), ∂t ∂x ∂y u, v, p|(x,y)∈∂ = 0. u, v, p|t=0 = 0

(14) (15) (16)

Function θk describes the location of the sources k = 1, . . . , K , which has the form of Ricker wavelet:    2   1 −πν t− 1 (17) I (t) = 1 − 2 π ν0 t − e 0 ν0 ν0 The data of inverse problem is the pressure registered in the receivers: f k (t) = p(xk , yk , t), k = 1, . . . , K .

(18)

The inverse problems for hyperbolic systems were investigated in [8, 9]. The method of the solution of the direct problem for the acoustic tomograph which is based on the the two dimensional hyperbolic system of acoustic equations is developed in [10]. We consider the inverse problem for recovering the density ρ(x, y) from (13)– (18). We introduce the cost functional: J (ρ) =

K 

k=1

T

2 p(xk , yk , t) − f k (t) dt → min . ρ

0

(19)

and apply the gradient method to minimize (19) ρ (n+1) (x, y) = ρ (n) (x, y) − αn J  (ρ (n+1) )(x, y). Using apriori information we can prove the convergence [11, 12]. Here [10] J  (ρ) =



T

− 1t u − 2t v +

0

 1

3 (u x + v y ) dt ρ

(20)

and functions i , i = 1, 2, 3 solve the following adjoint problem [13–16]: 1 1

3x = 0;

2t + 3 y = 0; ρ ρ K

 + 2 y ) =2 δ(x − xk , y − yk ) p(x, y, t) − f k (t) ;

1t + 1

3t + ( 1x ρc2

(21) (22)

k=1

i (x, y, T ) = 0;

(23)

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i |(x,y)∈∂ = 0.

(24)

We apply S. K. Godunov scheme for solving the adjoint problem (21)–(22). Acknowledgements This work is supported by Russian Science Foundation (grant No. 19-1100154).

References 1. Kang, H., Tanuma, K.: Inverse problems for scalar conservation laws. Inverse Probl. 21, 1047– 1059 (2005) 2. Berres, S., Bürger, R., Coronel, A., Sepúlveda, M.: Numerical identification of parameters for a strongly degenerate convection-diffusion problem modelling centrifugation of flocculated suspensions. Appl. Numer. Math. 52, 311–337 (2005) 3. Holden, H., Priuli, F.S., Risebro, N.H.: On an inverse problem for scalar conservation laws. Inverse Probl. 30 (2014). https://doi.org/10.1088/0266-5611/30/3/035015 4. Duric, N., Littrup, P., Li, C., Roy, O., Schmidt, S., Janer, R., Cheng, X., Goll, J., Rama, O., Bey-Knight, L., Greenway, W.: Breast ultrasound tomography: bridging the gap to clinical practice. In: Proceedings of the SPIE Medical Imaging: Ultrasonic Imaging, Tomography, and Therapy (2012). https://doi.org/10.1117/12.9109 5. Jirik, R., Peterlik, I., Ruiter, N., Fousek, J., Dapp, R., Zapf, M., Jan, J.: Sound-speed image reconstruction in sparseaperture 3-D ultrasound transmission tomography. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 59(2), 254–264 (2012) 6. Godunov, S.K.: Differential method for numerical computation of noncontinuous solutions of hydrodynamics equations. Matematicheskiy Sbornik 47(3), 271–306 (1959) 7. Godunov, S.K., Kulikov, I.M.: Computation of discontinuous solutions of fluid dynamics equations with entropy nondecrease guarantee. Comput. Math. Math. Phys. 54(6), 1008–1021 (2014) 8. Romanov, V.G., Kabanikhin, S.I.: Inverse Problems for Maxwell’s Equations. VSP, Utrecht (1994) 9. Romanov, V.G.: Asymptotic expansion of the solution to the system of elasticity equations with a concentrated pulse force. J. Appl. Ind. Mathematics. 3(4), 482–495 (2009) 10. Kulikov, I.M., Novikov, N.S., Shishlenin, M.A.: Mathematical modelling of ultrasound wave propagation in two dimensional medium: direct and inverse problem. Sib. Electron. Math. Rep. 12, C.219–C.228 (2015) 11. Kabanikhin, S.I., Scherzer, O., Shishlenin, M.A.: Iteration methods for solving a two dimensional inverse problem for a hyperbolic equation. J. Inverse Ill Posed Probl. 11(1), 87–109 (2011) 12. Kabanikhin, S.I., Shishlenin, M.A.: Direct and iterative methods of solving inverse and ill-posed problems. Sib. Electron. Math. Rep. 5, 595–608 (2008) 13. Kabanikhin, S.I., Shishlenin, M.A.: Quasi-solution in inverse coefficient problems. J. Inverse Ill Posed Probl. 16(7), 705–713 (2008) 14. Kabanikhin, S.I., Nurseitov, D.B., Shishlenin, M.A., Sholpanbaev, B.B.: Inverse problems for the ground penetrating radar. J. Inverse Ill Posed Probl. 21(6), 885–892 (2013) 15. Kabanikhin, S.I., Shishlenin, M.A.: Acoustic sounding methods of linearization and conversion of the wave field. Sib. Electron. Math. Rep. 7, C.199–C.206 (2010) 16. Kabanikhin, S.I.: Definitions and examples of inverse and Ill-posed problems. J. Inverse Ill Posed Probl. 16(4), 317–357 (2008)

On a Heat Wave for the Nonlinear Heat Equation: An Existence Theorem and Exact Solution A. L. Kazakov, P. A. Kuznetsov and A. A. Lempert

Abstract The article deals with the nonlinear second order parabolic equation, which is known as heat equation with a source or the generalized porous medium equation. We construct solutions of a special type that describe disturbances propagating over a zero background with a finite velocity. Such solutions are called heat waves. Previously, we considered such problems in cases of one spatial variable, or without a source. In the present work, the obtained results are expanded to the case of two spatial variables. It leads to a transition to the polar coordinate system. The theorem on the existence and uniqueness of a solution, having the form of a heat wave, is proved for one boundary-value problem with special type degeneration in the class of analytical functions. The solution has the form of a multiple power series whose coefficients are determined when solving three-diagonal systems of linear algebraic equations with an infinitely increasing dimension. The convergence of the series is proved by the majorant method. We present an example where the conditions of the theorem are not fulfilled and show that the solution has the form of an immobile heat wave.

1 Introduction In this article, we deal with the nonlinear heat equation with a source or the generalized porous medium equation [1] Tt = div (K (T )∇T ) + Q(T ),

(1)

A. L. Kazakov · P. A. Kuznetsov (B) · A. A. Lempert Matrosov Institute for System Dynamics and Control Theory of SB RAS, 134 Lermontova ul., Irkutsk, Russia e-mail: [email protected] A. L. Kazakov e-mail: [email protected] A. A. Lempert e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_29

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where T (t, x) is medium temperature at point x ∈ Rν+1 , ν = 0, 1, 2 at time t; K (T ) = δT σ is a coefficient of heat conductivity, δ, σ > 0 are constants; Q(T ) is a source/sink function. Equation (1) has numerous applications. For example, it is used in the description of high-temperature plasma [2], processes of heat conductivity [3], filtration of polytropic gas in porous media [1, 4], migration of biological populations, etc. With the help of substitutions u = T σ , t1 = δt (the subscript is further omitted), Eq. (1) can be reduced to the standard form 1 (∇u)2 + F(u). σ

u t = uu + 1

(2)

1

Function F(u) has form F(u) = σ u 1− σ Q(u σ )/δ. Note that in Eq. (2) when u = 0 the first term uu vanishes, i.e. the equation cannot be resolved with respect to the highest derivative [5]. A large number of publications devoted to a comprehensive study of Eq. (2) can be mentioned. This can be explained both by numerous applications and non-trivial mathematical properties. It is known, for example, that the velocity of disturbances propagation described by Eq. (2) can be finite [3]. Here we consider such a type of solutions. A review of research in this domain is beyond the scope of this paper. We mention the publications devoted to the construction of exact [6] and approximate solutions [4], as well as the works of the authors, devoted to the proof of the analogue Cauchy–Kovalevskaya theorem [7] in the one-dimensional case [8] and the construction of solutions by the boundary elements method [9]. Note that Eq. (2) is usually considered when the source is absent, although there are papers, where F ≡ 0 (see [10]). We also mention [11], which deal with similar problems in a more general formulation.

2 Formulation We consider Eq. (2) in the case of two spatial variables. In polar coordinates ρ > 0, ϕ ∈ (−π, π ], we obtain the equation  ut = u

   1 1 1 1 2 2 u ρ + u ρρ + 2 u ϕϕ + u ρ + 2 u ϕ + F(u). ρ ρ σ ρ

(3)

The boundary condition for Eq. (3) has the form u(t, ρ, ϕ)|ρ=R(t,ϕ) = f (t, ϕ).

(4)

Suppose that the curve ρ = R(t, ϕ) is closed and bounds the star domain in R2 . The smooth functions R(t, ϕ) > 0 and f (t, ϕ) satisfy the inequalities Rt |t=0 ≥0, f t |t=0 ≥ 0 and

On a Heat Wave for the Nonlinear Heat Equation …

225

(Rt |t=0 )2 + ( f t |t=0 )2 > 0.

(5)

Let f (0, ϕ) = 0 and F(0) = 0. Previously, we considered such statements in the case when F ≡ 0 [12, 13].

3 Main Theorem Speaking an analytical function, we mean the function that coincides in a certain neighborhood with its Taylor decomposition. Theorem 1 Let R(t, ϕ), f (t, ϕ) be analytical functions in some neighborhood of t = 0 and F(u) be an analytical function in a neighborhood of u = 0. Then, in the neighborhood of t = 0, ρ = R(0, ϕ), problem (3), (4) has a nonzero analytical solution, which is unique when choosing the direction of the heat wave movement. Proof consists of two stages. At first, we construct the solution in the form of power (special [14]) series. At second, we prove it’s convergence by the majorant method [7]. For convenience, we introduce the new independent variable r = ρ − R(t, ϕ), then problem (3), (4) takes the form   1 2 u t − Rt u r = uu rr + u r g + G, σ

(6)

u(t, r, ϕ)|r =0 = f (t, ϕ),

(7)

where  G=u

u ϕϕ − 2Rϕ u ϕr − Rϕϕ u r ur + r+R (r + R)2  g(t, r, ϕ) = 1 +

 +

Rϕ r+R

u 2ϕ − 2Rϕ u r u ϕ σ (r + R)2

+ F(u),

2 .

We construct a solution in the form of Taylor series u(t, r, ϕ) =

∞  n,m=0

u n,m (ϕ)

tn rm . n! m!

(8)

Its coefficients are determined by successively solving three-diagonal systems of linear algebraic equations (SLAE) with an infinitely increasing dimension. By the condition of the theorem, R and f are analytical functions. Therefore, one can write decompositions

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R(t, ϕ) =

∞ 

∞

Rn (ϕ)

n=0

 tn tn , f (t, ϕ) = f n (ϕ) . n! n! n=0

(9)

From boundary condition (7) it follows that u n,0 = f n , and u 0,0 = f 0 = 0. The rest of the coefficients are determined by induction on the sum of the indices. Let the sum of the indices be 1. It is known, that u 1,0 = f 1 . Coefficient u 0,1 we find from Eq. (6) with t = r = 0: u 0,1 =

−R1 ±



R12 + 4g0,0 f 1 /σ

2g0,0 /σ

, g0,0 (ϕ) = g(0, 0, ϕ).

(10)

The choice of the sign in front of the root determines the direction of the heat wave. The remaining coefficients are found using the differential operator  ∂ n [.]  Dn−k,k [.] = n−k k  , n, k ∈ N, k ≤ n. ∂ t∂ r t=r =0 Applying it to Eq. (6), we obtain the equality u n−k+1,k + ak u n−k,k+1 + bn−k u n−k−1,k+2 = L n−k,k ,

(11)

where ak = −R1 − g0,0 u 0,1 (k + 2/σ ), bn−k = −(n − k) f 1 . All coefficients with the sum of indices n + 1 locate on the left side of Eq. (11). Functions L n−k,k , in particular, depend on Dn−k,k [F(u)]. Note that F(u) is a complex function, this leads to certain problems in its n-fold differentiation. With an increase in n, the number of terms increases: D0,0 [F(u)] = 0, D0,1 [F(u)] = F (0)u 0,1 , D0,2 [F(u)] = F (0)2 u 0,1 + F (0)u 0,2 , . . .. This fact is most significant in the case when it is necessary to obtain explicit formulas for the coefficients (for example, to verify numerical experiments [9]). Putting in Eq. (11) k = 0, 1, . . . , n, we obtain a SLAE with a non-degenerate (see [13]) square tridiagonal [15] matrix in the left-hand side. Therefore, the system is uniquely solvable and the coefficients of series (8) are uniquely determined. Let us turn to constructing a majorant problem. The existence of a heat wave front follows from the formulation of the problem (see [13]). In our case, the heat wave front is a closed analytic curve ρ = a(t, ϕ) satisfying the equalities u(t, r, ϕ)|r =a(t,ϕ) = 0, a(0, ϕ) = R(0, ϕ). Let b(t, ϕ) = a(t, ϕ) − R(t, ϕ) be a distance from curve ρ = R(t, ϕ) to the heat front. We make several consecutive substitutions in problem (6), (7). First, we introduce new independent variables τ = t, s = r − b(t, ϕ), χ = ϕ. Second, change the roles of unknown function u and independent variable s. Next, we introduce a new independent variable w = u − f (τ, χ ) and express τ as a function of u, w, χ using the implicit function theorem. Finally, we introduce a new unknown function S(u, w, χ ) using a partial expansion of s in a Taylor series by u:

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s(u, w, χ ) = s0 (w, χ ) + us1 (w, χ ) + u 2 S(u, w, χ ). Here s0 ≡ 0 and s1 is known. As a result, problem (6), (7) reduces to a single equation, in which condition (7) is already implicitly included



2 1 + σ1 S + 4 + σ1 u Su + u 2 Suu + +β0 (w, χ )S|w=0 + β1 (w, χ )(Su |w=0 )u + β2 (w, χ )(Suu |w=0 )u 2 = = h 0 + uh 1 + u 2 h 2 + u 3 h 3 .

(12)

Functions βi , i = 0, 1, 2, and h j , j = 0, 1, 2, 3 satisfy the conditions of Lemma 2 from [13]. Consequently, Eq. (12) is uniquely solvable in the class of analytical functions. 

4 Immobile Heat Wave Condition (5) plays an important role in the proof of the theorem. Let us show by example, what can happen when it is not valid. Consider Eq. (2) in the case of plane symmetry with power source function F(u) u t = uu ρρ +

1 2 u + αu β , α ∈ R, β ∈ N. σ ρ

(13)

Condition (4) is simplified to the form u(t, ρ)|ρ=R = 0, R ∈ R+ .

(14)

It is easy to see that the standard procedure for constructing a solution in the form of series (8) is possible only if we introduce an additional boundary condition u|t=0 = q(ρ) ≡ 0, q(0) = q (0) = 0. The question of the convergence of the series is still open. However, problem (13), (14) has some particular exact analytical solutions. Proposition 1 Problem (13), (14) has the following nonzero exact solutions of the immobile heat wave type [3]: 1. For α = 0: u(t, ρ) =

λ(ρ−R)2 . (2+4/σ )(c1 −λt)

2. For α = 0, α ∈ R, β = 1: u(t, ρ) =

λ(ρ−R)2 . (2+4/σ )(c2 exp(−αt)−λ/α)

Here λ, c1 , c2 = 0 are constants. Proof The proposition is proved by the Fourier method. After substitution u(t, ρ) = ψ(t)ϕ(ρ), we obtain a system of two ordinary differential equations; its integrating gives the desired solutions. 

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Thus it is shown that nonfulfilment with the conditions of the theorem can lead to the fact that the heat wavefront will be immobile.

References 1. Vazquez, J.L.: The Porous Medium Equation: Mathematical Theory. Clarendon Press, Oxford (2007) 2. Kovalev, V.A., Kurkina, E.S., Kuretova, E.D.: Thermal self-focusing during solar flares. Plasma Phys. Rep. 43(5), 583–587 (2017) 3. Samarskii, A.A., Galaktionov, V.A., Kurdyumov, S.P., Mikhailov, A.P.: Blow-up in quasilinear parabolic equations. Walter de Gruyte, Berlin (1995) 4. Filimonov, MYu., Korzunin, L.G., Sidorov, A.F.: Approximate methods for solving nonlinear initial boundary-value problems based on special construction of series. Rus. J. Numer. Anal. Math. Modelling. 8(2), 101–125 (1993) 5. Demidenko, G.V., Upsenskii, S.V.: Partial Differential Equations and Systems Not Solvable with Respect to the Highest-Order Derivative. Marcel Dekker, New York (2003) 6. Hayek, M.: An exact solution for a nonlinear diffusion equation in a radially symmetric inhomogeneous medium. Comput. Math. Appl. 68(12), 1751–1757 (2014) 7. Courant, R., Hilbert, D.: Methods of Mathematical Physics. Vol. II: Partial Differential Equations. Interscience, New York (2008) 8. Kazakov, A.L., Lempert, A.A.: Existence and uniqueness of the solution of the boundaryvalue problem for a parabolic equation of unsteady filtration. J. Appl. Mech. Tech. Phys. 54(2), 251–258 (2013) 9. Kazakov, A.L., Spevak, L.F.: An analytical and numerical study of a nonlinear parabolic equation with degeneration for the cases of circular and spherical symmetry. Appl. Math. Model. 40(2), 1333–1343 (2016) 10. Kudryashov, N.A., Sinelshchikov, D.I.: Analytical solutions for nonlinear convection-diffusion equations with nonlinear sources. Autom. Control. Comput. Sci. 51(7), 621–626 (2017) 11. Antontsev, S.N., Shmarev, S.I.: Evolution PDEs with Nonstandard Growth Conditions. Blowup, Existence, Uniqueness, Localization. Atlantis Press, Paris (2015) 12. Kazakov, A.L., Kuznetsov, P.A.: On one boundary value problem for a nonlinear heat equation in the case of two space variables. J. Appl. Ind. Math. 8(2), 227–235 (2014) 13. Kazakov, A.L., Kuznetsov, P.A.: On the analytic solutions of a special boundary value problem for a nonlinear heat equation in polar coordinates. J. Appl. Ind. Math. 12(2), 255–263 (2018) 14. Filimonov, MYu.: Application of method of special series for solution of nonlinear partial differential equations. AIP Conf. Proc. 40, 218–223 (2014) 15. Godunov, S.K., Malyshev, A.N.: On a special basis of approximate eigenvectors with local supports for an isolated narrow cluster of eigenvalues of a symmetric tridiagonal matrix. Comput. Math. Math. Phys. 48(7), 1089–1099 (2008)

Algorithm for Solving the Inverse Problem of 2-D Diffraction of a Linearly Polarized Elastic Transverse Wave for an N-Layer Medium with Composite Hierarchical Inclusions A. Yu. Khachay and O. A. Hachay Abstract A new approach to wave fields interpretation has been developed to determine the contours or surfaces of composite local hierarchical objects. An iterative process has been developed for solving the theoretical inverse problem for the case of determining the configurations of 2D hierarchical inclusions of the lth, mth, and sth ranks located one above the other in different layers of the N-layer medium and with various physic mechanical properties for acoustic monitoring. When interpreting the monitoring results, it is necessary to use the data of such observation systems that can be configured to study the hierarchical structure of the environment. Such systems include acoustic (in a dynamic version) and electromagnetic monitoring systems. The hierarchical structure of the geological environment is clearly seen in the analysis of rock samples taken in ore mines.

1 Introduction A major result of the research was the conclusion about the fundamental role of the block-hierarchical structure of rocks and massifs to explain the existence of a wide range of nonlinear geomechanical effects and the emergence of complex self-organizing geosystems. The hierarchical structure is characteristic of many systems, especially for the Earth’s lithosphere, where more than 30 hierarchical levels were distinguished from geophysical studies from tectonic plates thousands of kilometers long to individual millimeter-sized mineral grains [8]. Thus, the crust is a system of blocks and, like any synergetic discrete ensemble it has the properties of hierarchy and self-similarity [2]. It is known that the geological environment is an open dynamic system, experiencing natural and artificial influences at different scale A. Yu. Khachay UrFU, INS&M, Yekaterinburg, Russia e-mail: [email protected] O. A. Hachay (B) Institute of Geophysics UB RAS, Yekaterinburg, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_30

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levels, changing its state, resulting in a complex multi-rank hierarchical evolution [1]. When conducting mining operations in highly stressed rock massifs, man-made seismicity is manifested, the forecast and prevention issues of which receive a lot of attention in all countries with a developed mining industry. From the point of view of the paradigm of physical mesomechanics, which includes a synergistic approach to changing the state of rock masses of different material composition, this problem can be solved using monitoring methods tuned to the study of hierarchical structural media [7]. Changes in the environment that lead to short-term precursors of dynamic phenomena are explained in the framework of the concept of self-organized criticality, for which the key points are heterogeneity and non-linearity [7]. At present, theoretical results in modeling and interpretation of wave fields propagating in layered block media with hierarchical inclusions are in demand. In [5, 6], within the framework of this model, the problem of constructing an algorithm for solving the inverse problem using the equation of the theoretical inverse problem for the 2D Helmholtz equation was considered; explicit equations of the theoretical inverse problem are written for the cases of electromagnetic field scattering (E and H polarization) and linearly polarized elastic wave scattering in a layered conducting and elastic medium with a hierarchical conducting or elastic inclusion, which are the basis for determining the contours of misaligned inclusions of the lth rank of the hierarchical structure. In the present paper, an iterative algorithm for solving the inverse problem is constructed using the explicit equation of the theoretical inverse problem for a composite hierarchical inclusion according to acoustic monitoring data.

2 Main Result In the paper [6], the inverse problem of diffraction of a linearly polarized elastic transverse wave by a two-dimensional hierarchical elastic heterogeneity located in the ν layer of an N-layer medium was solved. Here this problem will be considered in the framework of a complicated model: a plastic hierarchical heterogeneity of the lth rank will be located in the ν-1 layer, the maximum value of the lth rank is L, the initial value of the lth rank is ll = 1; the elastic hierarchical heterogeneity of the mth rank is located in the layer ν; the maximum value of the mth rank is M, the initial value of the mth rank is mm = 1; and the anomalously stressed hierarchical heterogeneity of the sth rank is located in the layer ν + 1, the maximum value of s rank is equal to S, the initial value of rank s is ss = 1. We consider an algorithm for reconstructing 2D surfaces of hierarchical heterogeneities for the case when L L, and m ≤ M, we calculate u x(ν) (M0 ) using formula: u x(ν) (M0 ) = u 1x(ν+1) (M0 )+

2 2 ξa(ν+1)s (k2a(ν+1)s −k2(ν+1) )  S(ν+1)s u x(ν+1) (M)G SS (M, M0 )d S+ 2π ξ (M0 )  (ξν+1 −ξa(ν+1)s ) ∂G SS / S(ν+1)s . ∂ D(ν+1)s u x(ν+1) (M) ∂n d L; M0 ∈ 2π ξ(M0 )

(12)

mm = mm + 1 and we move to (4). If the rank is: m > M, then we calculate u x(ν+1) (M0 ), using formula: u x(ν+1) (M0 ) = u 1x(ν+1) (M0 )+ 2 2 ξa(ν+1)s (k2a(ν+1)s −k2(ν+1) )  2π ξ (M0 ) (ξν+1 −ξa(ν+1)s ) 2π ξ(M0 )

S(ν+1)s



∂ D(ν+1)s

u x(ν+1) (M)G SS (M, M0 )d S+

u x(ν+1) (M) ∂G∂nSS d L; M0 ∈ / S(ν+1)s .

(13)

ss = ss + 1 and we move to (8). If the rank is: s > S, then we calculate u x (M0 ), using formula (14) in all layers: ν = 1, . . . , N ; u x(ν+1) (M0 ) = u 1x(ν+1) (M0 )+ 2 2 ξa(ν+1)s (k2a(ν+1)s −k2(ν+1) )  2π ξ (M0 ) (ξν+1 −ξa(ν+1)s ) 2π ξ(M0 )



S(ν+1)s

∂ D(ν+1)s

u x(ν+1) (M)G SS (M, M0 )d S+

u x(ν+1) (M) ∂G∂nSS d L; M0 ∈ / S(ν+1)s .

(14)

The result is the defining of all nested contours of hierarchical heterogeneities according to the ranks of their hierarchy and given physical and mechanical properties.

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3 Conclusion When solving the inverse problem, monitoring systems configured to study the hierarchical structure of the environment should be used as the initial monitoring data. On the other hand, the more complex the medium, the each wave field brings its own information about its internal structure, therefore, the interpretation of the seismic and electromagnetic fields must be carried out separately, without mixing these databases. From the theory developed, it follows that with an increase in the degree of hierarchy of the medium, the degree of spatial non-linearity of the distribution of the components of the seismic and electromagnetic fields increases, which indicates the impossibility of using the linearization of the problem when creating interpretation methods.

References 1. Bulashevich, Yu.P., Khachai, Yu.V.: Convective stability of the earth’s interior with internal sources of heat. Izv. USSR Academy of Sciences. Physics of the Earth. 12, 13–19 (1975) 2. Kocharyan G.G., Spivak A.A.: The dynamics of the deformation of block arrays. IKC Akademkniga, Moscow (2003) 3. Khachai, A.Yu.: Algorithms for mathematical modeling of alternating electromagnetic and seismic fields in the source approximation. Ph.D. thesis. Ural State University named after A.M. Gorky, Yekaterinburg (2007) 4. Khachai, O.A., Khachai, A.Yu.: Modeling of electromagnetic and seismic fields in hierarchically heterogeneous media Bulletin of SUSU, series. Comput. Math. Comput. Sci. 2(2), 48–55 (2013) 5. Khachai, O.A., Khachai, AYu., Khachai, OYu.: Determination of the surface of a hierarchical plastic inclusion in a layered-block medium according to acoustic monitoring data. Geoinformatics 3, 30–36 (2017) 6. Khachai, O.A., Khachai, O.Yu., Khachai, A.Yu.: On the inverse problem of active electromagnetic and acoustic monitoring of a hierarchical geological environment. Geophys. Explor. 18(4), 71–84. (2017). https://doi.org/10.21455/gr2017/4-6 7. Panin V.E.: Physical mesomechanics and computer-aided design of materials.V1. Nauka, Novosibirsk (1995) 8. Prangishvili, I.V., Pashchenko, F.F., Busigin, B.P.: Systemic laws and patterns in electrodynamics, nature and society. Nauka, Moscow (2001)

Relaxation-Projection Schemes, the Ultimate Approximate Riemann Solvers Christian Klingenberg

Abstract A relaxation approach pioneered by Frédérique Coquel leads to very stable approximate Riemann solvers with positivity preserving properties. We outline this approach and give examples of its usefulness in practical applications.

1 Introduction A classical method to numerically solve hyperbolic conservation laws is the finite volume method, which involves as its essential ingredient the numerical solution of a Riemann problem. Phil Roe noticed in 1981 [14] that an approximation of the Riemann solver suffices. This led to a quest for finding particularly useful approximate Riemann solvers. Such an approximate Riemann solver was developed in Paris by Frédérique Coquel and co-workers around the turn of the century, see, e.g., [2, 5]. It is inspired by an idea of Shi Jin and Zhou-ping Xin [11], where the solutions to a system of conservation laws are approximated by a straightforward relaxation system. The ensuing French idea was twofold: • find a particularly clever relaxation system that approximates a given system of conservation laws and • translate this into a numerical scheme by first solving the left-hand side of the relaxation system (a linear transport and thus numerically easy to do) and then projecting the thus found solution to the equilibrium variables (again easy). We shall show how this leads to approximate Riemann solvers with good properties, like stability and entropy consistency, which implies positivity of density and temperature for the shallow water (see [12]), Euler (see [1, 6, 15, 16]), and the ideal magnetohydrodynamics equations (see [3, 4, 17]). C. Klingenberg (B) Department of Mathematics, Würzburg University, Emil Fischer Str. 40, 97074 Würzburg, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_31

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2 Approximate Riemann Solvers In the classical Godunov numerical algorithm to solve one-dimensional systems of hyperbolic conservation laws u t + f (u)x = 0, (see, e.g., [13]), the exact solution to the Riemann problem was originally solved numerically by Godunov in the 1950s via an iterative method. Phil Roe noticed in 1981 [14] that for this algorithm to be successful it is enough to approximate the Riemann solution by an approximate Riemann solver consisting of jumps separating constant states. Roe’s approach can be made quite accurate but suffers from not necessarily satisfying a discrete entropy inequality. Thus soon after approximate Riemann solvers were introduced of so-called HLL type [9] that were entropy stable but not quite accurate. A way forward was then shown by Coquel and co-workers (see, e.g., [5]) from the late 1990s on, who developed approximate Riemann solvers that were inspired by a relaxation approach (see [11]). To illustrate the underlying principle, consider a scalar conservation law u t + f (u)x = 0 which gets approximated à la Jin-Xin [11] by u t + v  = 0 1 vt + a 2 u x = ( f (u  ) − v  ). 

(1) (2)

This gets solved numerically by a particular splitting approach. First, solve the homogeneous, linearly degenerate equation

vt

u t + v  = 0 + a 2 u x = 0,

and in the second so-called projection step, solve u t = 0 1 vt = ( f (u  ) − v  ),  by setting  to zero. Notice that this gives rise to an approximate Riemann solver for u t + f (u)x = 0, which turns out to be equivalent to the Lax–Friedrichs solver. Also, note that this splitting procedure (transport—projection) is nothing but an analytic tool to determine the speed of the jumps and the intermediate constant states of the approximate Riemann solver for the original equation u t + f (u)x = 0. In the numerical algorithm, never is there an  present.

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In order to prove lim →0 u  = u (where u  is the solution to (1), (2), and u is the solution to u t + f (u)x = 0), one needs to impose −a < f  (u) < a, the so-called stability condition (see [11]). This condition ensures the entropy dissipative property of the associated approximate Riemann solver. Thus one can show that the relaxationprojection solver satisfies the discrete entropy inequality, which is at the core of their extreme usefulness also in more general cases. Historically, the first relaxation-projection solver of this type for the compressible Euler equations is what Coquel called the Siliciu solver. It is described in [2], Sect. 2.4.4. Note that this way we obtain an approximate Riemann solver of HLLC type (three waves with constant states in between) that gives rise to a finite volume solver which satisfies the discrete entropy inequality. From this, one deduces that this finite volume scheme gives positive density and internal energy. Thus, one has found an approximate Riemann solver which can be made both quite accurate and stable. As an aside, it is interesting to note that in [7] it was found that even though the Siliciu solver from the previous paragraph is positivity preserving for density and internal energy, it will not maintain all invariant domains of the Euler equations. This means, given a set of initial data in phase space for the Euler equations, consider the solution set in phase space. Does the finite volume scheme based on the approximate Riemann solver obtained by the Siliciu relaxation always stay in this solution set? It is shown in [7] that even though this property is true for the set positive density and internal energy, in general this will not be the case.

3 Applications We give a few examples of approximate Riemann solvers of relaxation-projection type, which have proven themselves quite useful in applications because of their accuracy and good stability property. Note that for each case there is no systematic procedure for finding the relaxation system analogous to (1), (2). This is a bit of an art, in particular proving the crucial stability estimates (the estimates corresponding to −a < f  (u) < a in the previous section) may turn out to be a lengthy calculation. Nonetheless, as the next examples show, it is worth the effort because they lead to approximate Riemann solvers that are very useful in practice. We begin with the Euler equations of compressible gas dynamics and seek a relaxation-projection approximate Riemann solver that is able to work for all Mach numbers, in particular in the limit of the Mach number going to zero, where the incompressible Euler equations are reached. This was achieved in [1, 15] by splitting the pressure into two parts and then finding a relaxation system that relaxes these two different parts separately. Next, we consider the Euler equations with gravity and seek a finite volume solver that maintains hydrostatic equilibria, which are stationary solutions with the velocity set to zero. These are called well-balanced numerical methods. In [6, 16],

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the relaxation system ensures that the discrete form of the hydrostatic equilibrium is always maintained. For the equations of ideal magnetohydrodynamics, a relaxation-projection solver has been found in [3, 4, 17]. The relaxation system takes its inspiration from the Siliciu solver for the Euler equations. In particular, it has been possible to extend the positivity preserving property to second order in multiple space dimensions on a Cartesian mesh. The solenoidal property of the magnetic field is taken into account by writing the equations in symmetrizable form via a Powell term, and all of this is achieved while still maintaining the positivity preserving property.

4 Conclusion The stable property of these solvers has been put to test in astrophysical codes. To name only one, the magnetohydrodynamic relaxation solver [3, 4, 17] was introduced into the FLASH code, giving rise to simulations (see, e.g., [10]) where flows with Mach number 100 could be simulated in a stable way. That these solvers may be useful also in a discontinuous Galerkin context can bee seen in [8]. These relaxation-projection solvers deserve to be better known, which we hope to have shown in this overview.

References 1. Berthon, C., Klingenberg, C., Zenk, M.: An all Mach number relaxation upwind scheme. SMAI J. Comput. Math. (2020). http://tinyurl.com/yxlhnqec 2. Bouchut, F.: Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for Sources. Frontiers in Mathematics Series. Birkhäuser, Basel (2004) 3. Bouchut, F., Klingenberg, C., Waagan, K.: A multiwave approximate Riemann solver for ideal MHD based on relaxation. I. Theoretical framework. Numer. Math. 108, 7–42 (2007) 4. Bouchut, F., Klingenberg, C., Waagan, K.: A multi-wave approximate Riemann solver for ideal MHD based on relaxation II: numerical implementation with 3 and 5 waves. Numerische Mathematik 115(4), 647–679 (2010) 5. Coquel, F., Perthame, B.: Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics. SIAM J. Numer. Anal. 35, 223–2249 (1998) 6. Desveaux, V., Zenk, M., Berthon, B., Klingenberg, C.: Well-balanced schemes to capture nonexplicit steady states on the Euler equation with a gravity. Int. J. Numer. Methods Fluids 81(2), 104–127 (2016) 7. Guermond, J.-L., Klingenberg, C., Popov, B., Tomas, I.: The Suliciu approximate Riemann solver is not invariant domain preserving. J. Hyperbolic Differ. Equ. 16(1), 59–72 (2019) 8. Guillet, T., Pakmor, R., Springel, V., Chandrashekar, P., Klingenberg, C.: High-order magnetohydrodynamics for astrophysics with an adaptive mesh refinement discontinuous Galerkin scheme. Mon. Not. R. Astron. Soc. 485(3), 4209–4246 (2019) 9. Harten, A., Lax, P.D., van Leer, B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25(1), 35–61 (1983)

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10. Hill, A.S., Joung, M.R., Mac Low, M.M., Benjamin, R.A., Haffner, L.M., Klingenberg, C., Waagan, K.: Vertical structure of a supernova-driven turbulent, magnetized interstellar medium. Astrophys. J. 750, 104 (19pp) (2012) 11. Jin, S., Xin, Z.: The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commun. Pure Appl. Math. 48(3), 235–276 (1995) 12. Klingenberg, C., Kurganov, A., Zenk, M.: Moving-water equilibria preserving HLL-type schemes for the shallow water equations (Submitted). http://tinyurl.com/yxuoppef (2019) 13. LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems, vol. 31. Cambridge University Press, Cambridge (2002) 14. Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357–372 (1981) 15. Thomann, A., Zenk, M., Puppo, G., Klingenberg, C.: An all speed second order IMEX relaxation scheme for the Euler equations. Commun. Comput. Phys. (2020). http://preview.tinyurl. com/y3anfrup 16. Thomann, A., Zenk, M., Klingenberg, C.: A second order well-balanced finite volume scheme for Euler equations with gravity for arbitrary hydrostatic equilibria. Int. J. Numer. Methods Fluids 89, 465–482 (2019) 17. Waagan, K., Federrath, C., Klingenberg, C.: A robust numerical scheme for highly compressible magnetohydrodynamics: Nonlinear stability, implementation and tests. J. Comput. Phys. 230(9), 3331–3351 (2011)

New Efficient Preconditioner for Helmholtz Equation Dmitriy Klyuchinskiy, Victor Kostin and Evgeny Landa

Abstract The capabilities of modern computers and mathematical software are entirely sufficient for the numerical solution of 2D PDEs. Two-dimensional problems naturally arise when the Fourier method is applied to a three-dimensional boundary value problem for PDEs with coefficients independent of one variable. Such specially constructed solvers can be used as preconditioners for iterative solvers of boundary value problems for PDEs with general case coefficients. In this paper, we develop the outlined above idea for the numerical solution of the Helmholtz equation. Keywords Helmholtz equation · Iterative methods · Preconditioning

1 Introduction Numerical simulation of acoustic wavefields serves as an engine for acoustic frequency-domain full-waveform inversion (FD FWI) (see, e.g., [9, 15] and references cited therein). Such simulation is performed several times for several low frequencies at each iteration of the process and consumes most of the computational time. In the frequency domain, the acoustic wave propagation in R 3 is governed by the Helmholtz equation supplied with some condition at infinity [14]. For the applicability of numerical methods, the Helmholtz equation with appropriate boundary conditions is discretized to a system of linear algebraic equations. The solution of those equations may be challenging due to the enormous sizes of the coefficient D. Klyuchinskiy Novosibirsk State University, Novosibirsk 630090, Russia V. Kostin (B) Trofimuk Institute of Petroleum Geology and Geophysics, Novosibirsk 630090, Russia e-mail: [email protected] E. Landa Tel-Aviv University, Tel Aviv, Israel © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_32

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matrices for real-life problems. For FD FWI applications, solution methods based on the triangular factorization of the coefficient matrix may be of great interest, though special efforts are needed to reduce memory requirements [8, 16]. Krylov-type iterative solvers with efficient preconditioners are preferable when only a few frequencies per inversion are used [5, 11]. An idea we use in this work is similar to ones from [1, 2]. As in those works, we construct the right preconditioner as a shifted Laplace operator [6, 10] but in this paper, a background velocity has more freedom—here it depends on two variables. The preconditioner in this paper is closer to the operator being inverted, and iterations should converge faster. Application of the Fast Fourier Transform (FFT) along the direction of the velocity constancy results in a separable system of linear equations. Due to the twodimensionality of boundary value problems numerical complexity of solution of respective linear systems tremendously decreased versus complexity of solving the initial problem. An additional advantage is the ideal parallelism of problems for different wavenumbers.

2 Helmholtz Equation in a 3D Inhomogeneous Medium Let D = {(x1 , x2 , x3 ) : X 1l < x1 < X 1r , X 2l < x2 < X 2r , X 3l < x3 < X 3r }

(1)

be some cuboid1 filled with an acoustic medium2 having sound velocity c(x1 , x2 , x3 ). Domain D is used for numerical simulation in frequency domain of acoustic waves governed by the Helmholtz equation Lu = u(x) +

ω2 u(x) = f (x). c2 (x)

(2)

Here x = (x1 , x2 , x3 ),  = ∂∂x 2 + ∂∂x 2 + ∂∂x 2 is the Laplace operator, ω is the angular 1 2 3 frequency, and f (x) is the source function. Usually, D is assumed to be embedded in the whole R 3 filled with some specially constructed acoustic medium which is homogeneous outside of some big ball. The sound velocity between domain D and the homogeneous medium varies smoothly. Sommerfeld’s Radiation Condition [14] is necessary to exclude nonphysical waves coming from the infinity. Constructions with unbounded domains, as in the previous paragraph, are not suitable for numerical computations. For practical purposes, the computational domain is a bounded cuboid 2

2

2

Dcomp = {(x1 , x2 , x3 ) : Xˆ 1l < x1 < Xˆ 1r , Xˆ 2l < x2 < Xˆ 2r , Xˆ 3l < x3 < Xˆ 3r } 1D

is assumed to contain the origin (0, 0, 0). density is constant and excluded from further considerations.

2 The

(3)

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that contains D as a subdomain. Restriction to D of the solution of some boundary value problem for a specially constructed PDE in Dcomp is considered as a result of frequency-domain acoustic wavefield simulation. The PDE coincides with the Helmholtz equation (2) in D and provides attenuation of solutions within Dcomp \ D to reduce amplitudes of waves reflected from the outer boundaries of Dcomp . The Perfectly Matched Layers (PMLs) (see, e.g., [3, 13]) proved to be a good technique to achieve desired behavior. The boundary value problem for u with PMLs in Dcomp reads as follows:   3 ∂ ∂u ω2 μ μ (x ) (x ) i i ∂ xi + c2 (x) u(x) = f (x), i=1 i i ∂ xi (4) u|∂ Dcomp = 0. Here, f (x) is the source function from (2) extended by zero to Dcomp , and complexvalued functions μi (xi ) are defined by explicit formulas μi (xi ) =

iω di (xi ) + iω

via damping functions of one variable di (xi ). The damping function di (xi ) is continuous, equal to zero within interval [X il , X ir ] and positive outside of [X il , X ir ]. Behavior of this function on the interval of positiveness determines attenuation properties of respective PML, and in practical computations di (xi ) are chosen experimentally. In terms of new variables  xi i x˜i = xi − di (s)ds, (i = 1, 2, 3) ω 0 PDE in (4) reads as follows: 

∂2 ∂2 ∂2 + 2+ 2 2 ∂ x˜1 ∂ x˜2 ∂ x˜3

 u(x) ˜ +

ω2 c2 (x) ˜

u(x) ˜ = f (x). ˜

(5)

Within D all x˜i are real and coincide with xi , all functions μi are equal to units, and PDE in (4) coincides with (2). If point (x1 , x2 , x3 ) lies in Dcomp \ D, some of respective x˜i , (i = 1, 2, 3) become complex-valued. Negative imaginary part of x˜i assures attenuation of solutions of Eq. (5) along O xi in Dcomp \ D. Finite difference approximation of boundary value problem (4) results in a system of linear equations with the coefficient matrix, which is complex sparse and symmetric. Gaussian elimination method to solve the system of linear equations is based on the triangular factorization of the coefficient matrix and has definite advantages for FD FWI applications. The fast growth of memory requirements with the increase of the matrix size is the main disadvantage of the method. Data compression based on the low-rank approximation is used in the factorization algorithm and helps to diminish memory requirements (see [8, 16]).

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3 Solving the Helmholtz Equation by Iterative Methods The main advantage of Krylov subspace iteration methods is much lower memory consumptions versus the direct LU-based methods. The price for this may be slow (or even lack of) convergence. Preconditioning the system of linear equation is targeted to improve the clustering of the matrix spectrum, which implies improving the convergence. To begin with, we sketch an idea we use to build a preconditioner for the Helmholtz equation. Roughly speaking, we precondition the Helmholtz operator by some other easily invertible Helmholtz operator (cf. [1, 2]). For brevity, an operator in the Hilbert spaces defined by a boundary value problem for the Helmholtz equation in some domain in R 3 is called the Helmholtz operator. The velocity function for the preconditioner depends on less than three spatial variables (one variable in the cited papers and two in the current work). Inversion of such the preconditioner can be done using the Fourier method. Now we proceed to details of the sketched idea. Let the Helmholtz equation (2) be given in domain D which is embedded in Dcomp . We extend the PDE from D to Dcomp but do this a bit differently than in the previous section. The complement of D in Dcomp is split into subdomains of two different types—PMLs and sponges (see Fig. 1). Interfaces between the PMLs and D are vertical faces of D parallel to the coordinate planes. Two sponges are separated from D by horizontal plane interfaces. The PDE in PMLs reads as follows: 2  i=1

μi (xi )

∂ ∂ xi

  ∂ 2u ∂u ω2 μi (xi ) + 2 + 2 u(x) = f (x), ∂ xi c (x) ∂ x3

(6)

where the velocity function is continuously extended across vertical interfaces from D to PML zones of Dcomp . The source function is extended by zero. Functions μi (xi ) are units within D, and Eq. (6) can be considered as the PDE in the union of D and PMLs.

Fig. 1 Schematic structure of Dcomp : domain D is white-colored, PMLs and sponges are gray of different nuances

3

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Now, consider the following PDE: 2 

∂ μi (xi ) ∂ xi i=1



∂v μi (xi ) ∂ xi

 +

∂ 2v ω2 v(x) = g(x), + (1 − iβ) 2 2 ∂ x3 c0 (x1 , x2 )

(7)

in Dcomp , where β > 0 is some constant parameter and  c0 (x1 , x2 ) =

X 3(r ) X 3(l)

d x3 c2 (x1 , x2 , x3 )

−1/2 .

The differential operator of Eq. (7) is similar to the shifted Laplacian (cf. [10]). Boundary conditions that we pose for Eq. (7) relate to the method we use to evaluate v. Independence of the velocity function c0 of variable x3 suggests using Fourier expansion of the unknown function. In its turn, this implies periodicity boundary conditions imposed for the unknown function. The solution should satisfy the homogeneous Dirichlet conditions on the remaining four vertical faces of Dcomp . The boundary value problem for Eq. (7) defines a differential operator L 0 . The domain of operator L 0 is a subspace of Sobolev functional space H 1 (Dcomp ) of functions that satisfy the described boundary conditions on ∂ Dcomp . To solve the boundary value problem for (7), we use the Fourier expansion g(x1 , x2 , x3 ) =



gk3 (x1 , x2 )eik3 x3

(8)

k3

of the source function. Let vk3 (x1 , x2 ) be solutions to the boundary value problems     ∂vk3 ∂ 2 2 1−iβ μ + ω μ (x ) (x ) − k i i ∂ xi 3 vk3 (x) = gk3 (x 1 , x 2 ), i=1 i i ∂ xi c2 (x1 ,x2 )  vk3 ∂ Dcomp = 0. (2)

2

(9)

PDE in (9) is considered in horizontal cross section (2) Dcomp = {(x1 , x2 ) : Xˆ 1l < x1 < Xˆ 1r , Xˆ 2l < x2 < Xˆ 2r }

of Dcomp . Solution v(x1 , x2 , x3 ) to boundary value problem (7) can be computed by summing up the Fourier series v(x1 , x2 , x3 ) =



vk3 (x1 , x2 )eik3 x3 .

(10)

k3

Formulas (8)–(10) outline an algorithm of inverting operator L 0 on any source function g. Now we are ready to define operator L. For this purpose, we use PDE in the form of Eq. (6) with the same boundary conditions as for operator L 0 , and the domains of

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operators L and L 0 are the same. Above we explained how to extend the velocity function from D to PMLs. The extension of the velocity function to the sponges should be continuous and subject to the condition that expression c2 (x1 , x2 , x3 ) −

c02 (x1 , x2 ) 1 − iβ

vanishes in some vicinities of horizontal faces of Dcomp . Notice that according to this definition c(x1 , x2 , x3 ) is complex-valued in the sponge zones, and positive imaginary part of the velocity assures decay of the solution within the sponge zones (see Fig. 3). The boundary value problem we described above takes the form of the equation Lu = f,

(11)

where u is a vector from the domain of operator L, and the right-hand side f stands for the source function in the Helmholtz equation. To solve (11), we apply the GMRES iterative algorithm [12] to the preconditioned equation (L · L −1 0 )v = f . The iterations stop when the residual becomes small enough, suggesting that the respective v is an approximate solution of the preconditioned equation. To find u, we should solve equation L 0 u = v.

4 Algorithm Implementation and Numerical Experiments To solve numerically Eq. (11), it should be discretized. Results described below were obtained using finite difference approximation of the boundary value problems for PDEs (6) and (7). This way we come up to a system of linear equations AU = F,

(12)

which is a discrete version of Eq. (11). Here, vector U represents values of the unknown function at grid points, F stands for the source function values at the grid points, and A is a sparse matrix. Now follow the steps to invert preconditioner L 0 described in the previous section. The Fast Fourier Transform (FFT) is used to compute approximations of integrals that represent the Fourier coefficients in (8). Using finite difference approximation, 2D boundary value problems (9) take the form of linear equations systems A(k3 ) V (k3 ) = G (k3 ) .

(13)

Note that the sizes of matrices A(k3 ) are much less than the size of matrix in (12) and are acceptable for use of direct methods to solve Eq. (13). We factor the coefficient matrices A(k3 ) once and use the factors in GMRES iterations. Equations in (13) are

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independent of each other, and this can be used for parallelization. To sum up the Fourier series (10), we use inverse FFT. We use Intel®MKL [7] that delivers highly optimized functionality needed for our purposes: FFT, the direct sparse solver PARDISO, BLAS, and LAPACK.

Homogeneous Medium We benchmarked the solver against the analytic solution in a homogeneous medium. In a cubic domain of linear size 3200 m filled with a homogeneous medium with velocity 1280 m/s a finite difference grid of equidistant step was used. The point source located at the cube center excites harmonic waves at frequency 4 Hz. The grid step of 32 m provides 10 ppw (points-per-wavelength). To illustrate the results, in Figs. 2, 3 we provide profiles of the analytic and numerical solutions along x1 -axis and x3 -axis. The intervals in Fig. 2 include PMLs of 640 m widths on both sides. Respectively, the intervals include sponges of 1600 m width. In the PMLs and the sponge zones, the numerical solution rapidly decays in outward directions.

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Re u(x,y,992)

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Fig. 4 Two cross sections of the solution in an inhomogeneous medium

Inhomogeneous Medium We tested the solver behavior in inhomogeneous media. Having no formula for the exact solution to use in comparisons, we provide only illustration of the solution computed by our solver. All parameters that define the test are the same as in the test for the homogeneous medium except the velocity is a linear function of coordinates. (n) (n−1)  of two sequenAfter n = 36 GMRES iterations, the relative difference u u−u (n−1)  −4 tial approximations to the solution became equal to 8.64 · 10 and computations stopped. A couple of cross sections of u (36) are depicted in Fig. 4.

Conclusions We described a new preconditioner for the Helmholtz equation. Its efficiency comes from the use of FFT, high-performance solvers of systems of linear equations with sparse coefficients of matrices, and internal parallelism of itself. Testing the Helmholtz solver with our preconditioner on homogeneous and inhomogeneous media demonstrated promising results. In future, we plan to extend the idea to preconditioners for the frequency-domain elastodynamic equations.

Author Contribution Evgeny Landa formulated the statement suited for FWI in the frequency-time domain; Victor Kostin proposed numerical method and featured algorithm for numerical solution; Dmitriy Klyuchinskiy developed the code and performed numerical experiments.

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Acknowledgements The main results of this paper were obtained by Victor Kostin during his stay at the Tel-Aviv University in the period 03.11–04.10 of 2018. Victor Kostin thanks the Russian Science Foundation for the financial support of the trip to Tel-Aviv University under the project 17-17-01128.

References 1. Belonosov, M., Dmitriev, M., Kostin, V., Neklyudov, D., Tcheverda, V.: An iterative solver for the 3D Helmholtz equation. J. Comput. Phys. 345, 330–344 (2017). https://doi.org/10.1016/j. jcp.2017.05.026 2. Belonosov, M., Kostin, V., Neklyudov, D., Tcheverda, V.: 3D numerical simulation of elastic waves with a frequency-domain iterative solver. Geophysics 83(6), T333–T344 (2018). https:// doi.org/10.1190/geo2017-0710.1 3. Berenger, J.-P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114(2), 185–200 (1994) 4. BLAS (Basic Linear Algebra Subprograms). http://www.netlib.org/blas/ 5. Erlangga, Y.A., Herrmann, F.J.: Full-waveform inversion with Gauss–Newton–Krylov method. In: 79th SEG Annual Meeting Expanded Abstracts, pp. 2357–2361. SEG (2009) 6. Erlangga, Y.A., Nabben, R.: On a multilevel Krylov method for the Helmholtz equation preconditioned by shifted Laplacian. Electron. Trans. Numer. Anal. 31, 403–424 (2008) 7. Intel® Math Kernel Library (Intel® MKL). https://software.intel.com/en-us/intel-mkl 8. Kostin, V., Solovyev, S., Bakulin, A., Dmitriev, M.: Direct frequency-domain 3D acoustic solver with intermediate data compression benchmarked against time-domain modeling for FWI applications. Geophysics 84(4), T207–T219 (2019). https://doi.org/10.1190/geo20180465.1 9. Mulder, W.A., Plessix, R.E.: How to choose a subset of frequencies in frequency-domain finitedifference migration. Geophys. J. Int. 158(3), 801–812 (2004). https://doi.org/10.1111/j.1365246X.2004.02336.x 10. Oosterlee, C.W., Vuik, C., Mulder, W.A., Plessix, R.-E.: Shifted-Laplacian preconditioners for heterogeneous Helmholtz problems. In: Advanced Computational Methods in Science and Engineering, pp. 21–46. Springer (2010). https://doi.org/10.1007/978-3-642-03344-5_2 11. Plessix, R.E.: Three-dimensional frequency-domain full-waveform inversion with an iterative solver. Geophysics 74, WCC149—WCC157 (2009). https://doi.org/10.1190/1.3211198 12. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003). https://doi.org/10.1137/1.9780898718003.bm 13. Singer, I., Turkel, E.: A perfectly matched layer for the Helmholtz equation in a semi-infinite strip. J. Comput. Phys. 201, 439–465 (2004). https://doi.org/10.1016/j.jcp.2004.06.010 14. Vainberg, B.: Principles of radiation, vanishing attenuation and limit amplitude in the general theory of partial differential equations. Usp. Mat. Nauk 21, 115–194 (1966). (in Russian) 15. Virieux, J., Operto, S., Ben-Hadj-Ali, H., Brosssier, R., Etienne, V., Sourber, F., Guraud, L., Haidar, A.: Seismic wave modeling for seismic imaging. Lead. Edge 28(5), 538–544 (2009). https://doi.org/10.1190/1.3124928 16. Wang, S., Li, X., Xia, J., Situ, Y., De Hoop, M.: Efficient scalable algorithms for solving dense linear systems with hierarchically semiseparable structures. SIAM J. Sci. Comput. 35(6), C519–C544 (2013). https://doi.org/10.1137/110848062

Edge-Based Reconstruction Schemes for Unstructured Meshes and Their Extensions Tatiana K. Kozubskaya

Abstract The paper presents an overview of the edge-based reconstruction schemes for solving Euler equations on unstructured meshes. Their idea, basic formulation and various extensions are discussed.

1 Introduction An attractive property of the Edge-Based Reconstruction (EBR) schemes for solving the Euler-type equations on unstructured meshes is their higher accuracy as compared with the traditional finite-volume second-order methods at lower costs as compared with the very high-order algorithms as discontinuous Galerkin, polynomial k-exact, spectral volume, etc. This property makes these schemes an efficient method for solving engineering problems of complex geometrical configurations in aviation industry applications. The higher accuracy of EBR schemes is provided, thanks to the quasi-1D reconstruction of variables on the extended edge-oriented stencils such that in the case of transnationally invariant (TI) (i.e. uniform gridlike) meshes these schemes reduce to the high-order (up to the 5th–6th) finite difference methods. The lower costs result from the quasi-1D nature of these finite-volume schemes. The practice shows that the price of extending the reconstruction stencil is about 15–20 % extra costs in comparison with the Godunov-type schemes with a two-point approximation of fluxes. Initially formulated as a vertex-centred method, the EBR schemes were generalized to the cell-centred framework. They were extended to hybrid unstructured meshes and equipped with TVD and WENO-based shock-capturing techniques. The hybrid version of EBR schemes was developed for scale-resolving simulations of turbulent flows. The schemes were implemented for interface regions of sliding meshes and are being adapted to prismatic boundary layers within the strand-mesh technology. T. K. Kozubskaya (B) Keldysh Institute of Applied Mathematics, Moscow, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_33

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The paper recalls the idea of EBR schemes and its original and simplified vertexcentred implementations, and overviews the extensions of these schemes.

2 Original and Simplified Vertex-Centred Formulations Consider the vertex-centred EBR schemes for solving the Euler equation system ∂Q + ∇ · F (Q) = 0, ∂t

F = (ρu, ρuu + pI, (E + p) u)T

(1)

written with respect to conservative variables Q = (ρ, ρu, E)T . Here, ρ is the density, u is the velocity vector, p is the pressure, I is the identity matrix, and F is the flux vector. The general formulation of edge-based vertex-centred schemes can be given as     1  dQ (2) =− hi j ni j , dt i vi j∈N (i) 1

where vi is the volume of the dual cell built around vertex i, N1 (i) is the set of first-level neighbours   of node  i, hi j is the numerical flux which is calculated as n ds, ∂Ci j is the boundary of cell interface between hi j = Fi j · ni j /ni j , ni j = ∂Ci j

nodes i and j, and ni j is the oriented square of cell surface Ci j . As applied to the EBR schemes, the key point of formulation (2) is that both the numerical flux hi j and the oriented square ni j are evaluated at the edge i j midpoint only. A specific scheme of the EBR family is defined by the method of calculating the flux hi j on an extended edge-based quasi-1D stencil. We calculate hi j as a solution of approximate Godunovtype Riemann solver or flux-splitting methods basing on the left and right states L/R L/R Fi j , F = Fx n x + F y n y + Fz n z or/and Qi j which, in their turn, are determined with the use of quasi-1D reconstructions. The reconstructions are built so that they transform to the corresponding high-order finite-difference approximations when applied to TI-meshes. We denote an EBR scheme as EBRn scheme if its highest theoretical order reachable on TI-meshes is equal to n. Figure 1 schematically shows the extended quasi-1D edge-based stencil we use for the numerical flux evaluation on a triangular unstructured mesh. The reconstructions for calculating the left and right states are built from the divided differences basing on the points with local numeration {ri } , i = 1, 6. In the simplified formulation proposed in [2], the values in these points are determined with the help of linear interpolation on the correspondent edges which are intersected by the edge i j direction. In 3D, the reconstructions of left and right states are constructed in a similar way with replacing the intersected edges with the intersected faces. In particular, in the EBR5 scheme, L/R the reconstruction operators Ri j acting on mesh function ψ can be written in terms of divided differences ψi+1/2 = (ψi+1 − ψi )/ri+1/2 , ri+1/2 = |ri+1 − ri | as

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Fig. 1 Quasi-1D stencils of EBR schemes for a triangular mesh

  r7/2 1 11 4 1 − ψ3/2 + ψ5/2 + ψ7/2 − ψ9/2 ({ψ}) = ψi + 2 15 30 5 10   r 1 11 4 1 7/2 R − ψ11/2 + ψ9/2 + ψ7/2 − ψ5/2 . Ri j ({ψ}) = ψ j − 2 15 30 5 10 (3) In the original version of EBR schemes, the authors avoid calculating the values in the points of intersections by replacing the divided differences ψ3/2 , ψ5/2 , ψ9/2 , ψ11/2 with the P1 Galerkin gradients defined on the corresponding firstand second-level supporting triangles which lay the edge-based direction upstream and downstream the cell interface [1, 3, 19]. All the stencils and metrical coefficients which provide the algorithm in both formulations are determined at the preprocessing stage, and the total costs of EBR schemes appear not significantly higher than in the case of second-order finite-volume methods. RiLj

3 Cell-Centred Formulation We implement the quasi-1D idea also for cell-centred finite-volume schemes [12] for convex elements. The difference with respect to the EBR schemes is that not one but two quasi-1D stencils are needed for calculating the numerical fluxes at the element faces. The two stencil supporting directions connect the face mass centre with the barycenters of neighbouring elements shared by this face. For tetrahedral meshes, the resulting Barycenters-Based Reconstruction (BBR) schemes are more expensive than the vertex-centred EBR method; however, the schemes become comparable in costs for hybrid hex-dominant meshes and unstructured meshes consisting of polyhedral elements. In comparison with the vertex-centred formulations, the BBR schemes are more naturally adapted to the walls and sliding interfaces of mesh blocks.

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4 EBR Schemes for Hybrid Unstructured Meshes To provide the 1-exactness (i.e. exactness on linear functions) of EBR schemes on hybrid unstructured meshes, we introduce a new type of control volumes, called as semitransparent cells [9, 13]. For tetrahedral meshes, this property is satisfied by using barycentric dual volumes. In the case of hybrid unstructured mesh, the 1-exact method is built basing on local decompositions of prisms, pyramids and hexahedrons in simplexes.

5 Shock Capturing EBR Schemes To treat possible discontinuities and solutions with high gradients, we equip the EBR schemes with the shock-capturing techniques well elaborated for 1D formulations. Following this way, we developed the quasi-1D EBR-WENO scheme [11]. To implement shock capturing, we use the Riemann solvers (in most cases, the Roe solver) or flux-splitting methods written in characteristic variables. The WENO-type reconstruction is built in the concordance with the classical finite-difference WENO scheme. For this purpose, we represent the involved lower order approximations and corresponding smoothness monitors in terms of divided differences defined along the edge-oriented direction. The resulting WENO-reconstruction operators are applied to the characteristic variables to calculate the left and right states. The TVD-type EBR schemes and corresponding sensors are built in a similar way.

6 Hybrid EBR Scheme for LES-Based Simulations As is known, to provide correct LES-based turbulent-flow simulations, the scale resolving approaches require a carefully calibrated balance between the numerical dissipation and instability. The method should be both accurate and stable enough, to allow for a correct representation of arising small physical instabilities and damping “parasitic” numerical oscillations. To meet the above requirements, in most cases, we use the hybrid scheme [20] that combines upwind and central-difference parts for the numerical flux evaluation. In contrast to the original scheme, our implementation is based on the high-order edge-based reconstructions. The presence of shocks in the regions of developed turbulence strongly complicates the process of building an appropriate hybrid scheme. In [16], we design the three-term hybrid approximation blending the central difference, upwind and shock-capturing reconstructions:   R L ({ψ}) + RiRj ({ψ}) 3 RW,L ({ψ}) + σ − σ 3 R L ({ψ}) + (1 − σ ) i j RiH,L = σ ({ψ}) ij j ij 2

(4)

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where reconstructions RiLj ({ψ}) and RiRj ({ψ}) are defined in (3), the method of deriving the WENO-reconstruction RiW,L ({ψ}) is explained in Sect. 5, and σ ∈ [0, 1] j is the weight coefficient of upwind component of the hybrid scheme proposed in [20]. According to (4), the central-difference reconstruction and the corresponding numerical flux dominate when σ varies near zero, the upwind reconstruction works when σ is of moderate values and the WENO reconstruction gets involved when σ approaches to 1. The development of sensor for switching to the complete shock-capturing EBR-WENO scheme still presents a problem. In practice, we use different problem-adjusted techniques. In most case, we control the value of | pi+1 − pi |/min { pi , pi+1 } where pi , pi+1 are the pressure values in two neighbouring points.

7 Other Extensions of EBR Scheme The recent implementations of the EBR schemes mainly concern the extensions to sliding interfaces of unstructured mesh blocks [10], immersed boundary method both for fixed and moving objects [4, 8] and dynamic adaptation of moving-mesh type [6, 7]. The EBR schemes are actively used for simulating complex turbulent flows and predicting aerodynamic and acoustic characteristics of various aviation-industryoriented configurations [5, 14–18].

8 Conclusion The paper presents the EBR schemes and their extensions. We pay attention to these schemes as an efficient compromise between costs and accuracy for solving applied unsteady aerodynamics and aeroacoustics problems on unstructured meshes. The EBR schemes are implemented as the basic numerical scheme the in-house code NOISEtte. Acknowledgements The work is supported by the Russian Science Foundation (Project 16-1110350).

References 1. Abalakin, I., Bakhvalov, P., Kozubskaya, T.: Edge-based reconstruction schemes for prediction of near field flow region in complex aeroacoustic problems. Int. J. Aeroacoustics 13(3–4), 207– 234 (2013) 2. Abalakin, I., Bakhvalov, P., Kozubskaya, T.: Edge-based reconstruction schemes for unstructured tetrahedral meshes. Int. J. Numer. Methods Fluids 81(6), 331–356 (2016)

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3. Abalakin, I., Kozubskaya, T., Dervieux, A.: High Accuracy Finite Volume Method for Solving Nonlinear Aeroacoustics Problems on Unstructured Meshes. Chin. J. Aeronaut. 19(2), 97–104 (2006) 4. Abalakin, I., Zhdanova, N., Kozubskaya, T.: Immersed boundary penalty method for compressible flows over moving obstacles. In: Owen, R., de Borst, R., Reese, J., Pearce C. (eds.) Proceedings of ECCM 6/ECFD 7, 11–15 June 2018, Glasgow, UK, pp. 3449–3458. International Center for Numerical Methods in Engineering (CIMNE), Barcelona, Spain (2018) 5. Abalakin, I.V., Anikin, V.A., Bakhvalov, P.A., Bobkov, V.G., Kozubskaya, T.K.: Numerical investigation of the aerodynamic and acoustical properties of a shrouded rotor. Fluid Dyn. 51(3), 419–433 (2016) 6. Abalakin, I.V., Bahvalov, P.A., Doronina, O.A., Zhdanova, N.S., Kozubskaya, T.K.: Simulating aerodynanics of a moving body specified by immersed boundaries on dynamically adaptive unstructured meshes. Math. Models Comput. Simul. 11(1), 35–45 (2019) 7. Abalakin, I.V., Kozubskaya, T.K., Soukov, S.A., Zhdanova, N.S.: Numerical simulation of flows over moving bodies of complex shapes using immersed boundary method on unstructured meshes. In: Garanzha, V., Kamensky, L., Si, H.(eds.) Numerical Geometry, Grid Generation and Scientific Computing. Lecture Notes in Computational Science and Engineering, vol. 131, pp. 171–180. Springer Int. Publishing (2019) 8. Abalakin, I.V., Zhdanova, N.S., Kozubskaya, T.K.: Immersed boundary method for numerical simulation of inviscid compressible flows. Comput. Math. Math. Phys. 58(9), 1411–1419 (2018) 9. Bakhvalov, P., Kozubskaya, T.: On efficient vertex-centered schemes on hybrid unstructured meshes, AIAA paper 2016-2966. In: 22nd AIAA/CEAS Aeroacoustics Conference, Lyon, France (2016) 10. Bakhvalov, P.A., Bobkov, V.G., Kozubskaya, T.K.: Application of schemes with a quasionedimensional reconstruction of variables for calculations on nonstructured sliding grids. Math. Models Comput. Simul. 9(2), 155–168 (2017) 11. Bakhvalov, P.A., Kozubskaya, T.: EBR-WENO scheme for solving gas dynamics problems with discontinuities on unstructured meshes. Comput. Fluids 157, 312–324 (2017) 12. Bakhvalov, P.A., Kozubskaya, T.K.: Cell-centered quasi-one-dimensional reconstruction scheme on 3D hybrid meshes. Math. Models Comput. Simul. 8(6), 625–637 (2016) 13. Bakhvalov, P.A., Kozubskaya, T.K.: Construction of edge-based 1-exact schemes for solving the Euler equations on hybrid unstructured meshes. Comput. Math. Math. Phys. 57(4), 680–697 (2017) 14. Dan’kov, B.N., Duben’, A.P., Kozubskaya, T.K.: Numerical modeling of the self-oscillation onset near a three-dimensional backward-facing step in a transonic flow. Fluid Dyn. 51(4), 534–543 (2016) 15. Dankov, B.N., Duben, A.P., Kozubskaya, T.K.: Numerical simulation of the transonic turbulent flow around a wedge-shaped body with a backward-facing step. Math. Models Comput. Simul. 8(3), 274–284 (2016) 16. Duben, A., Kozubskaya, T.: Jet noise simulations using quasi-1D schemes on unstructured meshes, AIAA 2017-3856 17. Duben, A., Kozubskaya, T.: On scale-resolving simulation of turbulent flows using higheraccuracy quasi-1d schemes on unstructured meshes. In: Hoarau, Y., Peng, S.H., Schwamborn, D., Revell, A. (eds.) Progress in Hybrid RANS-LES Modelling, Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 137, pp. 169–178. Springer International Publishing, Cham (2018) 18. Duben, A., Zhdanova, N., Kozubskaya, T.: Numerical investigation of the deflector effect on the aerodynamic and acoustic characteristics of turbulent cavity flow. Fluid Dyn. 52(4), 561–571 (2017)

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19. Koobus, B., Alauzet, F., Dervieux, A.: Numerical algorithms for unstructured meshes. In: F. Magoulès (ed.) Computational Fluid Dynamics, Numerical Analysis and Scientific Computing, pp. 131–204. CRC Press, Boca Raton (2011) 20. Travin, A., Shur, M., Strelets, M., Spalart, P.R.: Physical and numerical upgrades in the detached-eddy simulation of complex turbulent flows. In: Friedrich, R., Rodi, W. (eds.) Advances in LES of Complex Flows, Fluid Mechanics and its Applications, vol. 65, pp. 239– 254. Springer, Dordrecht (2002)

The Collision of Giant Molecular Cloud with Galaxy: Hydrodynamics, Star Formation, Chemistry Igor Kulikov, Igor Chernykh, Dmitry Karavaev, Viktor Protasov, Vladimir Prigarin, Ivan Ulyanichev, Eduard Vorobyov and Alexander Tutukov

Abstract This paper will present the new results of mathematical modeling of the collision of giant molecular cloud and galaxy. The numerical model includes selfgravity hydrodynamics equation for gas component of galaxy and collisionless Boltzmann equation for stellar component. Numerical model includes important sub-grid physics: star formation, supernova feedback, stellar wind, cooling and heating function, and nonequilibrium chemistry to ion helium hydride. The numerical simulation of destroy of galaxy in high-density ICM will be demonstrated. Keywords Computational astrophysics · Galaxy simulation · Numerical methods

I. Kulikov (B) · I. Chernykh · D. Karavaev ICMMG SB RAS, Novosibirsk, Russia e-mail: [email protected] I. Chernykh e-mail: [email protected] D. Karavaev e-mail: [email protected] V. Protasov · V. Prigarin · I. Ulyanichev NSTU, Novosibirsk, Russia e-mail: [email protected] V. Prigarin e-mail: [email protected] I. Ulyanichev e-mail: [email protected] E. Vorobyov University of Vienna, Vienna, Austria e-mail: [email protected] A. Tutukov Institute of Astronomy RAS, Moscow, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_34

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1 Introduction The subject of modern astrophysics is the study of physical processes in the universe, their influence on the self-organization, and evolution of astronomical objects, as well as on their further dynamics and interaction. The description of astronomical objects is based on hydrodynamic processes. It is hydrodynamics that determines character of astrophysical flows, which leads to the evolution of astrophysical objects. Mathematical modeling is the main and often the only way for theoretical investigation of astrophysical flows due to the impossibility of carrying out total experiments. The numerical model of interacting galaxies is based on gravitational gas dynamics equations to describe the gas component and equations for the first moment of the collisionless Boltzmann equation with full tensor of velocities dispersion to describe the star component [1].

2 Hydrodynamical Model of Galaxies To describe the gas components, we will use the system of single-speed component gravitational hydrodynamics equations, which is written in Euler coordinates: ∂ρ + ∇ · (ρu) = S − D, ∂t

ρi ∂ρi ρi + ∇ · (ρi u) = −si + S − D , ∂t ρ ρ

∂ρu + ∇ · (ρuu) = −∇ p − ρ∇() + vS − uD, ∂t ∂ρ S + ∇ · (ρ Su) = (γ − 1) ρ 1−γ ( − ) , ∂t S D ∂ρ E + ∇ · (ρ Eu) = −∇ · ( pv) − (ρ∇(), u) −  +  + ρ γ − ρ γ , ∂t ρ ρ ρE =

1 2 ρu + ρε, 2

p = (γ − 1)ρε = Sρ γ .

To describe the collisionless components, we will use the system of equations for the first moments of the Boltzmann collisionless equation, which is also written in Eulerian coordinates: ∂n + ∇ · (nv) = D − S, ∂t

∂nv + ∇ · (nvv) = −∇ − n∇() + uD − vS, ∂t

∂nWi j D S + ∇ · (nWi j v) = −∇ · (vi j + v j i ) − (n∇(), v) + ρ γ − ρ γ , ∂t ρ ρ

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Wi j = vi × v j + i j . The Poisson equation can be written as

 = 4π G(ρ + n), where p—gas pressure, ρi —density of ith species, si —speed of formation of ith species, ρ—density of gas mixture, n—density collisionless component, u—speed gas component, v—speed collisionless component, ρ E—density of total mechanical gas energy, ρWi j —density of total mechanical collisionless components energy, —gravitational potential, ε—density of internal energy of gas, S—entropy, γ — adiabatic index, i j —a tensor of dispersion of speeds collisionless components, S— the speed of formation of supernova stars, D—star formation speed, —function of Compton cooling, and —function of heating from explosion of supernova stars. The details of initial profile can be found in [2].

3 Sub-grid Physics 3.1 Star Formation and Supernova Feedback The supernovae feedback we should use for SNII, SnIa, and other stars by stellar wind S = S S N I I + S S N I a + S∗ . To describe supernova feedback, we should use the initial mass function [3]  φ(M∗ ) =

AM∗−1.3 , M∗ ≤ 0.5M AM∗−2.3 , M∗ > 0.5M

stellar lifetimes function [4]  τ (M∗ ) = where f (M∗ ) =

1.2M∗−1.85 + 0.003Gyr, M∗ ≥ 7.45M 10 f (M∗ ) , M∗ < 7.45M

0.334 −

√ 1.79 − 0.2232 × (7.764 − log (M∗ )) 0.1116

and nucleosynthesis [5]. For each type of stars, their share is determined R I I,I a,∗ (see details in work [1]) and the rate of star formation is determined by the simple formula: S = ρ × (R S N I I + R S N I a + R∗ )

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We note that the supernova explosion also contributes to the heating of the gas  = 1051 × (R S N I I + R S N I a ) × M × V −1 The rate of star formation is taken from work [6].

3.2 Cooling and Heating Functions The contribution to the cooling and heating functions is made by chemistry, which was described in the previous subsection, as well as supernovae feedback, in the following we will consider other sources of cooling/heating. We will consider cooling functions in two temperature modes: 1. The low-temperature cooling. At low temperatures, the ionization of the elements H, O, C, N, Si, and Fe occurs due to collision. The collision frequency and the corresponding cooling function can be found in the work [7]. 2. The high-temperature cooling. At high temperatures, the emission process for the elements H, He, C, N, O, Ne, Mg, Si, and Fe occurs. The cooling function can be found in paper [8]. To describe the heating function, we will consider the following two processes: 1. Cosmic ray heating. The process of ionization of hydrogen and helium atoms. The heating function can be found in operation [9]. 2. Photoelectric heating from small dust grains [10].

3.3 Supermassive Black Holes In recent years, many important results have been obtained in studying the physics of supernovae, the remnants of their explosion, and their feedback to more massive structures. So it was justified that dwarf galaxies cannot be the birthplace of supermassive black holes, which in turn is associated with the restriction of the mass of stars in such galaxies. In this regard, given that our main interest in the future is dwarf galaxies, we will not introduce a model of supermassive black holes in the mathematical model in the near future and dwell on their inclusion in a collisionless model [11].

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4 The Chemistry The collision of a giant gas cloud like the Smith cloud [12] with a Milky Way type galaxy is considered. The collision is considered in the model of the motion of a dense gas cloud through a rarefied medium. The characteristics of the gas components of the galaxy are as follows: n A = 10−1 cm−3 —density of interstellar gas, TA = 104 K— interstellar gas temperature, n H : n He = 9 : 1—chemical composition of the gas component. Cloud Characteristics: n C = 1 cm−3 —density of cloud, TC = 10 K—cloud temperature, v = 3 × 105 m/sec—cloud speed, and n C = n H —chemical composition of the cloud. The chemical composition of the gas mixture of interstellar gas and clouds are as follows: H, He, e, H+ , He+ , HeH+ , H+ 2 . The physics model: • • • • • • • •

Multicomponent one-speed gravitational hydrodynamics [13]. The process of star formation [14]. Effect of supernova explosions and stellar wind [3–5]. Cooling functions at low temperature [7]. Cooling functions at high temperature [8]. Heating by cosmic rays [9]. Photoelectric heating [10]. Chemical kinetics to helium hydride ion [15]. The chemical reaction network:

Ionization of hydrogen by cosmic rays H + c.r. → H+ + e [16]. Collisional ionization of hydrogen H + e → H+ + 2e [17]. Collision ionization of helium H + e → He+ + 2e [18]. Dissociative recombination of atomic hydrogen H+ + e → H + γ [19]. Dissociative recombination of molecular hydrogen H+ 2 + e → 2H + γ [20]. Radiative association of the helium ion and hydrogen He+ + H → HeH+ + hν [21]. 7. Dissociative recombination of the helium hydride ion HeH+ + e → He + H [22]. 8. Collisional recombination of the helium hydride ion HeH+ + H → He + H+ 2 [23].

1. 2. 3. 4. 5. 6.

Adiabatic index [24] (Table 1): γ =

5n H + 5n He + 5n e + 5n H+ + 5n He+ + 7n HeH+ 7n H+2 3n H + 3n He + 3n e + 3n H+ + 3n He+ + 5n HeH+ 5n H+2

Acknowledgements The authors were supported by the Russian Foundation for Basic Research (project no. 18-01-00166) and ICMMG SB RAS project 0315-2019-0009.

266 Table 1 The rate of chemical reactions

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Speed reaction cm3 /sec

1

10−17

2

e−32.7+13.5×ln T −5.7×ln

3 4

e−44.1+23.9×ln T −10.8×ln T 3.92 × 10−13 × T −0.6353 2 e−28.6−0.7×ln T −0.02×ln T 10−8 1.32 × 10−6 × T −0.76 1.4 × 10−16 3.0 × 10−10 × (T /104 K)−0.47 1.2 × 10−9 × (T /104 K)−0.11

5 6 7 8

Activation (K) –

2

T 2

– 900 – 5500 – 617 5000 1000 200

References 1. Vorobyov, E., Recchi, S., Hensler, G.: Stellar hydrodynamical modeling of dwarf galaxies: simulation methodology, tests, and first results. Astron. Astrophys. 579 (2015). Article Number A9 2. Springel, V., Di Matteo, T., Hernquist, L.: Modelling feedback from stars and black holes in galaxy mergers. Mon. Not. R. Astron. Soc. 361, 776–794 (2005) 3. Kroupa, P.: On the variation of the initial mass function. Mon. Not. R. Astron. Soc. 322(2), 231–246 (2001) 4. Padovani, P., Matteucci, F.: Stellar mass loss in elliptical galaxies and the fueling of active galactic nuclei. Astrophys. J. 416, 26–35 (1993) 5. Van den Hoek, L.B., Groenewegen, M.A.T.: New theoretical yields of intermediate mass stars. Astron. Astrophys. Suppl. Ser. 123(2), 305–328 (1997) 6. Kulikov, I., Chernykh, I., Tutukov, A.: A new hydrodynamic model for numerical simulation of interacting galaxies on intel xeon phi supercomputers. J. Phys. Conf. Ser. 719 (2016). Article Number 012006 7. Dalgarno, A., McCray, R.A.: Heating and ionization of HI regions. Annu. Rev. Astron. Astrophys. 10, 375–426 (1972) 8. Boehringer, H., Hensler, G.: Metallicity-dependence of radiative cooling in optically thin, hot plasmas. Astron. Astrophys. 215(1), 147–149 (1989) 9. Van Dishoeck, E.F., Black, J.H.: Comprehensive models of diffuse interstellar clouds - Physical conditions and molecular abundances. Astrophys. J. Suppl. Ser. 62, 109–145 (1986) 10. Bakes, E.L.O., Tielens, A.G.G.M.: The photoelectric heating mechanism for very small graphitic grains and polycyclic aromatic hydrocarbons. Astrophys. J. 427(2), 822–838 (1994) 11. Mezcua, M.: Dwarf galaxies might not be the birth sites of supermassive black holes. Nat. Astron. 3, 6–7 (2019) 12. Smith, G.P.: A peculiar feature at lII = 40◦ .5, bII = −15◦ .0. Bull. Astron. Inst. Neth. 17, 203–208 (1963) 13. Kulikov, I.M., Chernykh, I.G., Tutukov, A.V.: A new parallel intel xeon phi hydrodynamics code for massively parallel supercomputers. Lobachevskii J. Math. 39(9), 1207–1216 (2018) 14. Springel, V., Hernquist, L.: Cosmological smoothed particle hydrodynamics simulations: a hybrid multiphase model for star formation. Mon. Not. R. Astron. Soc. 339(2), 289–311 (2003) 15. Gusten, R., et al.: Astrophysical detection of the helium hydride ion HeH+. Nature 568, 357– 359 (2019)

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16. Lepp, S.: The Cosmic-ray ionization rate. In: Astrochemistry of Cosmic Phenomena: Proceedings of the 150th Symposium of the International Astronomical Union, held at Campos do Jordao, Sao Paulo, Brazil, 5–9 August 1991, P. 471–475 (1992) 17. Abel, T., Anninos, P., Zhang, Y., Norman, M.: Modeling primordial gas in numerical cosmology. New Astron. 2(3), 181–207 (1997) 18. Janev, R.K., Langer, W.D., Evans, K.J., Post, D.E.J.: Elementary Processes in HydrogenHelium Plasmas: Cross sections and Reaction Rate Coefficients. Springer Series on Atomic, Optical, and Plasma Physics, vol. 4, 326 p (1987) 19. Ferland, G.J., Peterson, B.M., Horne, K., Welsh, W.F., Nahar, S.N.: Anisotropic line emission and the geometry of the broad-line region in active galactic nuclei. Astrophys. J. 387, 95–108 (1992) 20. Schneider, I.F., Dulieu, O., Giusti-Suzor, A., Roueff, E.: Dissociate recombination of H2(+) molecular ions in hydrogen plasmas between 20 K and 4000 K. Astrophys. J. 424, 983–987 (1994) 21. Zygelman, B., Dalgarno, A.: The radiative association of He+ and H. Astrophys. J. 365, 239– 240 (1990) 22. Stromholm, C., et al.: Dissociative recombination and dissociative excitation of 4HeH+: absolute cross sections and mechanisms. Phys. Rev. A 54, 3086–3094 (1996) 23. Bovino, S., Tacconi M., Gianturco, F.A., Galli, D.: Ion chemistry in the early universe. Revisiting the role of HeH+ with new quantum calculations. Astron. Astrophys. 529 (2011). Article Number A140 24. Glover, S., Mac Low, M.: Simulating the formation of molecular clouds. I. Slow formation by gravitational collapse from static initial conditions. Astrophys. J. Suppl. Ser. 169(2), 239–268 (2007)

Model Dirac and Dirac–Hestenes Equations as Covariantly Equipped Systems of Equations N. G. Marchuk

Abstract We define a new class of partial differential equations of first order (complex covariantly equipped systems of equations), which are invariant with respect to (pseudo)orthogonal changes of cartesian coordinates of (pseudo)Euclidian space. It is shown that for pseudo-Euclidian spaces of signature (1, n−1) covariantly equipped systems of equation can be written in the form of Friedrichs symmetric hyperbolic systems of equations of first order. We prove that Dirac and Dirac–Hestenes model equations belong to the class of covariantly equipped systems of equations.

1 Complex Covariantly Equipped Systems of Equations of First Order (Pseudo)Euclidian space Rr,s . Let n be a natural number, r, s be nonnegative integer numbers, and Rr,s be the n = r + s dimensional (pseudo)Euclidian space with cartesian coordinates x μ , μ = 1, . . . , n and with a metric tensor given by the diagonal matrix η = ημν  = diag(1, . . . , 1, −1, . . . , −1) with r pieces of 1 and s pieces of −1 on the diagonal. Genforms. In [1], we define a set (an algebra) of genforms [h] (Rr,s ) considered as a special geometrized representation of Clifford algebra. We use notations from [1]. Let e1 , . . . , en be generators of Clifford algebra C(r, s) and e be the identity element. μ μ And let ya be a tetrad in Rr,s . We may take a genvector h μ := ya ea ∈ C1 (r, s)T1 . Consider the set of genforms [h] r,s r,s [h] (Rr,s ) = [h] even (R ) ⊕ odd (R ) =

n 

r,s [h] k (R ),

k=0

N. G. Marchuk (B) Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina St., Moscow 119991, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_35

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where [h] [h] r,s r,s r,s [h] even (R ) = 0 (R ) ⊕ 2 (R ) ⊕ . . . , [h] [h] r,s r,s r,s [h] odd (R ) = 1 (R ) ⊕ 3 (R ) ⊕ . . . .

We see that a genform U = U (x) ∈ [h] (Rr,s ) of the form U = ue +

n n    1 u μ1 ...μk h μ1 ...μk u μ1 ...μk h μ1 ∧ . . . ∧ h μk = ue + k! k=1 k=1 μ 0 such that (H1 ξ, ξ ) > γ (ξ, ξ ) for all nonzero complex N -dimensional vectors ξ and for all x ∈ . ¯ ⊂ Rn is bounded by the surface ∂. For • (b) For x 1 > 0 the closure of domain  this surface, a vector of exterior normal τ = (τ1 , . . . , τn ) at any point x ∈ ∂ is n Hi τi is positive defined. such that the matrix i=1 • (c) The functions with values in matrices Hi = Hi (x),

Q = Q(x),

j = j (x), x ∈ 

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are smooth (infinitely differentiable) functions of x ∈ . The boundary ∂ of ¯ for x 1 > 0 is smooth. The vector function ψ = ψ(x) ´ is a smooth the domain  function of x´ ∈ S. The boundary of the domain S¯ is smooth. According to the theory of Friedrichs symmetric hyperbolic systems of first order, if conditions (a), (b), (c) are satisfied, then there exists a classical (continuously differentiable) solution of the Cauchy problem (6), (7). Also, there is an a priory estimate that gives us correctness of the Cauchy problem (see [2–4]). Now we want to write down a covariantly equipped system of equations in pseudoEuclidian space R1,n−1 in the form of Friedichs symmetric hyperbolic system of first order. The operation of Hermitian conjugation of genforms in R1,n−1 was considered in details in the book [5] (Sect. 3.9). For an arbitrary genform V ∈ C ⊗ [h] (R1,n−1 ) the Hermitian conjugated genform is defined by the formula ˜¯ V † := β Vβ, where β = e1 is the generator of Clifford algebra C(1, n − 1) such that (e1 )2 = e. By V˜ we denote the reverse operation and by V¯ we denote the complex conjugation operation. For a genvector h μ we have (βh μ )† = βh μ , μ = 1, . . . , n.

(8)

We may multiply both parts of the system of Eq. (2) from left by the genform β and, as a result, we get the system of equations βh μ ∂μ φ + β Q(φ) = β f,

(9)

which is symmetric according to the identity (8). To prove the hyperbolicity in Friedrichs sense of the system of Eq. (9), we must prove that the genform βh 1 is positive defined. For Minkowski space R1,3 , this proposition was proved in [5] (Sect. 5.7, Theorem 5.2). It is easy to check that the proof of Theorem 5.2 in [5] is valid for an arbitrary even n ≥ 2. Finally, to get a connection with the theory of Friedrichs symmetric hyperbolic systems of first order, we can use a matrix representation of Clifford algebra C(1, n − 1) that consistent with the operation of Hermitian conjugation of genforms (see [5], Sect. 11.3). So, we see that a complex covariantly equipped system of equations can be written in the form of Friedrichs symmetric hyperbolic system of equations of first order.

3 Modified Dirac–Hestenes Equation as a Covariantly Equipped System of Equations In [6–10] one consider a modification of the Dirac equation of an electron, which uses differential forms and a wave function has 16 complex components. Hestenes

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[11] has found a new form of the Dirac equation with the aid of even elements of Clifford algebra C(1, 3). A modification of the Dirac–Hestenes equation using even and odd genforms was considered in [5] (Sect. 9.1). This modification in Mikowski space R1,3 looks as follows: h μ (∂μ  + aμ  K ) + m Kβ = 0,

(10)

[h] 1,3 1,3 23 1 where  ∈ [h] even (R ) or  ∈ odd (R ), K = −e , β = e . Note that a genform K satisfy conditions 1,3 K ∈ [h] 2 (R ),

K 2 = −e, [β, K ] = 0.

So, as a K we may take ±e23 , ±e24 , ±e34 or a linear combination of these elements with real coefficients (and with the condition K 2 = −e). Using the operator ð = 1,3 h μ ∂μ and a genform A = aμ h μ ∈ [h] 1 (R ), the Eq. (10) can be written in the form of covariantly equipped system of equations ð + A K + m Kβ = 0.

(11)

According to results of previous section, the system of Eq. (11) can be written as Friedrichs symmetric hyperbolic system of first order. This leads to the correctness of Cauchy problem for (11) in a proper domain of Minkowski space.

4 How to Reduce a Cauchy Problem for a Model Dirac Equations with Nonabelian Gauge Symmetry to a Cauchy Problem for Covariantly Equipped System of Equations Hermitian idempotents and related structures. An element (scalar) t ∈ C ⊗ [h] (Rr,s ) that satisfy conditions t 2 = t, t † = t, ∂μ t = 0, (μ = 1, . . . , n),

(12)

is called a Hermitian idempotent (projector).1

1 It was proved in [5] (Sect. 3.12) that for Minkowski space R1,3

in the complexified Clifford algebra

C ⊗ C(1, 3) there are five types of Hermitian idempotents t(0) = 0, t(1) = t(3) =

1 1 (e + e1 + ie23 + ie123 ), t(2) = (e + e1 ), 4 2

1 (3e + e1 + ie23 − ie123 ), t(4) = e. 4

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Let us introduce the following sets of genforms from C ⊗ [h] (Rr,s ): I (t) = {U ∈ C ⊗ [h] (Rr,s ) : U = U t}, K (t) = {U ∈ I (t) : U = tU }, L(t) = {U ∈ K (t) : U † = −U }, G(t) = {U ∈ C ⊗ [h] (Rr,s ) : U † U = e, U − e ∈ K (t)}. These sets of genforms I (t), K (t), L(t), G(t) can be interpreted as follows: I (t) is a left ideal; K (t) is two-sided ideal; L(t) is a real Lie algebra of a unitary Lie group; and G(t) is a unitary Lie group. A model Dirac equation. A model Dirac equation ([5], Chap. 5) in a (pseudo-) Euclidian space Rr,s with a fixed Hermitian idempotent t ∈ C ⊗ [h] (Rr,s ) can be written in the form (13) h μ (∂μ ψ + ψ Aμ ) + imψ = 0, where h μ is a genvector, ψ = ψ(x) ∈ I (t), Aμ = Aμ (x) are components of a covector with values in L(t), i is the imaginary unit, m is a real constant (a mass of a particle). Note that the genform ψ = ψ(x) ∈ I (t), which is unknown in Eq. (13), is supposed to be a scalar in Rr,s . A procedure of covariant equipment of a Cauchy problem for the model Dirac equation. According to the definition (in the first section) of a covariantly equipped system of equations, we see that system of Eq. (13) is a covariantly equipped system of equations iff t = e. What to do for Hermitian idempotents t = e? In what follows, we restrict ourselves by pseudo-Euclidian spaces R1,n−1 with even n ≥ 2. Consider a Cauchy problem for system of Eq. (13) h μ (∂μ ψ + ψ Aμ ) + imψ = 0, x 1 > 0 ψ = ψ0 , x 1 = 0,

(14) (15)

where ψ0 = ψ0 (x 2 , . . . , x n ) = ψ0 t ∈ I (t) are a known finite smooth genform of initial data of the Cauchy problem. With Cauchy problem (14), (15) one can associate the following Cauchy problem for a covariantly equipped system of equations: h μ (∂μ  +  Aμ ) + im = 0, x 1 > 0

(16)

 = ψ0 , x = 0,

(17)

1

where  = (x) ∈ C ⊗ [h] (R1,n−1 ), x ∈ R1,n−1 . Theorem 1 • (I) If a genform ψ ∈ I (t) is a solution of Cauchy problem (14), (15), then the genform  ≡ ψ ∈ C ⊗ [h] (R1,n−1 ) satisfies Cauchy problem (16), (17). • (II) If a genform  ∈ C ⊗ [h] (R1,n−1 ) is a solution of Cauchy problem (16), (17), then the genform  belongs to the set I (t) and the genform ψ =  satisfies Cauchy problem (14), (15).

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Proof The proposition (I) of theorem is evident. Let us prove the proposition (II). For a Hermitian idempotent t ∈ C ⊗ [h] (R1,n−1 ), one can define the dual Hermitian idempotent t := e − t, where e is the identity element of Clifford algebra C(1, n − 1) and simultaneously e is the identity element of the set of genforms C ⊗ [h] (R1,n−1 ). Let a genform  ∈ C ⊗ [h] (R1,n−1 ) be a solution of Cauchy problem (16), (17). One can represent the genform  as the sum  = e = (t + t ) = t + t = φ + φ , where φ ∈ I (t), φ ∈ I (t ). Let us substitute this decomposition of  into Cauchy problem (16), (17). We see that the genforms φ, φ satisfy two independent Cauchy problems. Namely, the genform φ ∈ I (t) satisfies Cauchy problem (14), (15) and the genform φ ∈ I (t ) satisfies the Cauchy problem h μ ∂μ φ + imφ = 0, x 1 > 0

φ = 0, x = 0. 1

(18) (19)

Let us multiply the left-hand part of system of Eq. (18) from left by β = e1 . Then we get the system of equations that can be written in the form of Friedrichs symmetric hyperbolic system of equations of first order. A Cauchy problem for a Friedrichs symmetric hyperbolic system of equations is well posed in Hadamard sense. Hence, a solution of Cauchy problem (18), (19) exists and is unique. So φ ≡ 0 is the unique solution of (18), (19). This completes the proof of the theorem.

References 1. Marchuk, N.G.: A generalization of Yang-Mills equations. Proc. Steklov Inst. Math. 306 (2019) 2. Godunov, S.K.: Uravneniya matematicheskoy fiziki (Equations of mathematical physics). Nauka, Moscow (1979) 3. Friedrichs K.O.: Symmetric hyperbolic linear differential equations. Commun. Pure Appl. Math. 7(2), 345–391 (1954) 4. Mizohata, S.: The Theory of Partial Differential Equations. Cambridge University Press, Cambridge (1979) 5. Marchuk, N.: Field theory equations. Amazon, CreateSpace (2012) 6. Iwanenko D., Landau L.: Zur theorie des magnetischen electrons. I, Zeitschrift fur Physik, Bd. 48, 340–348 (1928) 7. Kahler E.: Rendiconti di Mat. (Roma) ser. V, 21, 425 (1962) 8. Atiyah, M.: Vector Fields on Manifolds, Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen. Heft 200 (1970) 9. Benn, I.M., Tucker, R.W.: An Introduction to Spinors and Geometry with Applications to Physics. Adam Hilger, Bristol (1987)

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10. Obukhov, Y.N, Solodukhin, S.N.: Reduction of the Dirac equation and its connection with the Ivanenko-Landau-Kahler equation. Theoret. Math. Phys. 94(2), 198–210 (1993) 11. Hestenes, D.: Space-Time Algebra. Gordon and Breach, New York (1966)

Developments of the Godunov Method: Explicit–Implicit Scheme, Aeroacoustics Igor Menshov

Abstract The present paper addresses the numerical method for solving the gas dynamics equations that has been proposed by S. K. Godunov almost 60 years ago. The method reads as a first-order explicit finite-volume scheme with the numerical flux approximation based on the exact solution to the Riemann problem. We consider the extension of the Godunov method to an unconditionally stable second-order accurate implicit–explicit time marching scheme, introduce the variational Riemann problem (VRP), and discuss possible applications of this problem to solving the hybrid implicit–explicit Godunov scheme and also problems of aeroacoustics.

1 Introduction The numerical method known in literature as the Godunov method is by essence a finite-volume discretization of the gas dynamics equations where the numerical flux at the cell face is approximated on the exact solution to the Riemann problem. Originally it was developed for the 1D model [1, 2] and then extended to the multidimensional case [3, 4]. Basic ideas of this method were also implemented in developing a discrete model for stationary supersonic gas dynamics [5, 6]. Here the analog of the break-up of initial discontinuity (Riemann problem) is the problem on the interaction between two uniform supersonic flows, which is also self-similar and possesses the exact analytical solution under arbitrary initial data. One distinctive feature of the Godunov approach is that it mainly lies on the theoretical basis of compressible fluid mechanics rather than formal mathematical operations; it is mainly based on physical principles of approximation rather than on mathematical ones. That is why Godunov’s ideas were given wide distribution not only in the USSR but also in other countries, so that we have now a well-established terminology of Godunov’s method in literature. In its original formulation, the Godunov method is explicit and has the first order of approximation in space and time. Its generalization to a higher order of accuracy is I. Menshov (B) Keldysh Institute for Applied Mathematics RAS, Miusskaya sq., 4, Moscow 125047, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_36

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rather transparent. One needs just to replace piece-wise constant solution approximation by higher order approximations (e.g., piece-wise linear, piece-wise parabolic). For the first time, this was proposed by Kolgan in [7] and then developed by van Leer [8], Colella and Woodward [9], Rodionov [10], Harten and Osher [11], and others. A weak point of this extension is that the Riemann problem is no longer selfsimilar and does not admit simple analytical solutions. Many studies that addressed the Riemann problem with non-constant initial data were carried out by the team of Russian scientists headed by Teshukov [12]. Non-self-similar Riemann problem is referred now in literature as Generalized Riemann problem (GRP) after the work by Ben-Artzi and Falcovitz [13]. In this paper, the GRP has been introduced and investigated for 1D equations in Lagrangian variables, and analytical solutions have been derived for time derivatives of flow parameters along the contact discontinuity at the initial time moment t = 0 for the case of linearly distributed initial data. Menshov in [14] studied the GRP solution in Eulerian variables in the (x, t)plane in the small vicinity of the point of initial discontinuity. The exact analytical solution was obtained in the first approximation with respect to the parameter θ = √ x 2 + t 2 when θ → 0. If we consider some piece-wise linear initial data given by q = q ± + |x|S ± , where “+” and “−” stands for x ≥ 0 and x ≤ 0, respectively, then the GRP solution is written in the following form: Q R (λ, θ ) = Q 0 (λ, q ± ) + Q 1 (λ, q ± , S ± )θ + O(θ 2 )

(1)

where λ = x/t, Q 0 (λ, q ± ) denotes the standard self-similar solution, Q 1 represents the solution with the accuracy O(θ 2 ). The latter depends on the self-similar solution Q 0 and can be found in the explicit analytical form [14]. This solution was then implemented to achieve the second order of accuracy in the Godunov method for the 1D Euler equations [15]. In this method, the discrete solution in each computational cell i is given by two state vectors—qin and Sin that represent cell averaged values and slopes so that the numerical solution within each cell is given by a linear distribution qin (x) = qin + Sin (x − xi ). Once we have solution to the GRP problem (1) for arbitrary initial data, one can suggest a simple direct second-order accurate sequel of the Godunov method. It reads as follows [15]: t [Fi+1/2 − Fi−1/2 ] x 1 [Q i+1/2 − Q i−1/2 ] Sin+1 = x   n+1 n+1 ← minmod qi , x Sin+1 , qi+1 qin+1 = qin +

Sin+1

(2) (3) (4)

where the r.-h.s. is calculated on the GRP solution as Fi+1/2 = F[Q R (0, 0.5t)] Q i+1/2 = Q R (0, t)

(5) (6)

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n n with initial data given by q +(−) = qi+1(i) + Si+1(i) (xi+1/2 − xi+1(i)) and S +(−) = n +(−)Si+1(i) . The numerical scheme (2)–(6) is the second-order explicit Godunov scheme that is stable under the standard CFL condition. To reduce this time-step limitation, next we consider how to transform this explicit scheme to a hybrid implicit–explicit unconditionally stable analog.

2 Hybrid Implicit–Explicit Godunov Scheme We devise this hybrid scheme under the following requirements: (1) local and continuous switching between implicit and explicit constituents, (2) second order of accuracy in space and time in the pure explicit mode, and (3) maximum norm diminishing (MND) property for any value of the time step. Basic principles of constructing such a hybrid scheme from an explicit counterpart were discussed in [16]. In accordance with these principles, once we have a second-order explicit scheme, the hybrid scheme in question can be seen as this explicit scheme launched to the upper time level not from the lower level but from an intermediate one. The intermediate time level is defined by a scalar ω, 0 ≤ ω ≤ 1, properly defined for each computational cell. With this parameter, one can introduce an intermediate state vector qiω = qin + (1 − ωi )n qi . If the explicit scheme is cast as n qi = t L 2 (t, q n ) ,

(7)

where L 2 (·) represents the discrete second-order accuracy transfer operator, then the hybrid scheme reads as (8) n qi = t L 2 (ωi t, q ωi ) , This hybrid scheme transforms to its explicit counterpart when the hybridization parameter equals one for all the computational cells. Therefore, the question is to adopt this parameter as much closer to unity provided that the numerical solution remains to be MND. Recasting (8) as qin+1 = qiω + ωi t L 2 (ωi t, q ωi ) ,

(9)

one can see that the hybrid scheme considered exactly coincides with the explicit scheme (7) where time integration is fulfilled in each cell from the intermediate time level. Therefore, we can meet the MND property, if the following condition will be valid: ωi t ≤ x/λ(qiω ), where λ denotes the maximal characteristic velocity. This makes it possible to assign the intermediate parameter for each cell as follows:  ωi = min 1,

x tλ(q ω )

 (10)

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The penalty for this hybridization is the system of nonlinear equations (8) and (10) that have to be simultaneously solved for each time step. This can be done with the dual-time approach by searching the unknown vector q n+1 as a steady state to the following problem: (1 − ω)t

∂q n+1 = −R(q n+1 ) , R(q n+1 ) = n qi − t L 2 (ωi t, q ωi ) ∂τ

(11)

The steady state of (11) is obtained by implementing standard one-step implicit time marching scheme in τ followed by Newton iterations. The resulting linear system is solved with the matrix-free approximate factorization LU-SGS method of [17]. The linear system comes from the linearization of the residual R(q n+1 ). To do this, one needs to linearize the function of the Godunov’s numerical flux, i.e., the solution of the Riemann problem Q R (q − , q + ) with respect to initial data. Such linearization can be carried out analytically in terms of the solution to the variational Riemann problem to be considered in the next section.

3

Variational Riemann Problem and Aeroacoustics

The variational Riemann problem (VRP) is to determine the first variation to the standard self-similar Riemann problem solution arising when the initial data undergo small variations. From the physical point of view, the VRP can be thought as describing the interaction of two abutting parcels of small disturbances on the background of a given base flow. Once the VRP solution is given, one can employ it to update the acoustic filed for a small time interval in the same way as it is done in the Godunov method for fluid dynamics. In other words, the VRP can be used in computational aeroacoustics in the similar way as the Riemann problem is used in Godunov-type methods for computational fluid dynamics. When solving the implicit–explicit scheme discussed in Sect. 2 and linearizing the Godunov numerical flux, we have to consider the variation of the exact RP solution caused by small variations of initial data values. This variation is expressed in a linear form with respect to initial data variations in terms of variation matrices associated with the left- and right-hand side of the initial discontinuity. Thus, we also come to the VRP that is stated as follows. Let us consider the Riemann problem with constant initial data q ± and its selfsimilar solution Q 0 (λ, q ± ) which is assumed to be given. The problem in hand is to define how this solution is changed when the initial data undergoes small variations: q ± → q ± + δq ± . The first variation of the solution Q 0 (λ, q ± ) is determined in terms of left and right variation matrices: (12) δ Q 0 = M − δq − + M + δq + where the variational matrices M ± are given by

Developments of the Godunov Method: Explicit–Implicit Scheme, Aeroacoustics

M ± = M ± (λ, Q 0 ) =

∂ Q0 ∂q ±

281

(13)

Thus, the VRP is to define these variational matrices for arbitrary initial data. Variational matrices are piece-wise analytical matrix-functions. Their domains of analyticity coincide with those of the self-similar solution Q 0 and are defined by the wave pattern arisen in the baseline RP. The matrices are constant with λ in subdomains of uniform flow in the RP, i.e., in contact and unperturbed zones. In the region of the rarefaction wave they are variable, M ± = μ±RW (λ). The matrices in the contact zone μc and in the rarefaction wave μ RW are found analytically [18].The solution to the VRP can be expressed in a compact analytical form by introducing the notation of proper or conjugate value. A parameter takes a proper value if λ corresponds to the parameter proper side, and a conjugate value otherwise. For example, if λ lies on the left of the contact discontinuity, then left attributed parameters take proper values, and right attributed - conjugate values. With this notation, the solution of the VRP can be written as δ Q 0 = Mδq + M ∗ δq ∗ , where with no superscript denotes the proper value and the superscript ∗ denotes the conjugate value, and M = I, M ∗ = 0 in the non-perturbed region M = μ RW , M ∗ = 0 in the rarefaction wave M = μc ,

M ∗ = μ∗c in the contact zone

(14) (15) (16)

Matrices μ RW , μc , and μ∗c are given in [18]. Now, having the solution of VRP in hand, we can extend the Godunov method to aeroacoustics equations that model the evolution of the acoustic field on the background of a variable base flowfield [19]. These equations are commonly derived by linearizing the system of compressible Euler equations with respect to the base flow that is assumed to be available from other numerical or analytical studies. The resulting linearized Euler equations (LEE) are not conservative. Moreover, coefficients in general case can be piece-wise smooth functions with discontinuities (shock waves, contacts). This makes solving the LEE model a challenging problem. To overcome these difficulties, one can suggest an alternative approach. Instead of linearizing the Euler equations and trying to solve numerically the resulting system, we consider first discretization of the Euler equations with the Godunov method. In the resulting discrete equations, we then split the state vector into base flow and acoustical constituents, qi = q¯i + qˆi and linearize the residual with respect to the base vector q¯i . The acoustical discrete field is derived in this way in terms of the first variation of the Godunov’s numerical flux which is determined on the solution to the VRP, δ F = A( Q¯ 0 ) Qˆ 0 , where A = ∂ F/∂ Q is the Jacobian matrix of the local one-dimensional inviscid flux, and Q¯ 0 = Q 0 (0, q¯ ± ) is the solution to the base Riemann problem stated for the cell interface. The vector Qˆ 0 is the VRP solution, which represents the variation of the base Riemann problem solution due to disturbances qˆ ± —the acoustical field in hand. It is readily written through variational matrixes as Qˆ 0 = M − (0, q¯ ± )qˆ − + M + (0, q¯ ± )qˆ + .

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Acknowledgements The research was supported in part by RFBR (project No 18-01-00921).

References 1. Godunov, S.K.: A difference scheme for numerical computation of discontinuous solution of hydrodynamic equations. Math. Sb. 47, 271 (1959) [in Russian]; Translation, U.S. Joint Publ. Res. Service, JPRS 7226 (1969) 2. Alalikin, G.B., Godunov, S.K., Kireeva, I.L., Pliner, L.A.: Solution of One-Dimensional Gas Dynamics Problems Using Moving Grids. Nauka, Moscow (1970). [In Russian] 3. Godunov, S.K., Zabrodin, A.V., Prokopov, G.P.: A difference scheme for two-dimensional unsteady aerodynamics and an example of calculations with a separated shock wave. J. Comput. Math. Math. Phys. 2(6), 1020 (1961). [In Russian] 4. Godunov, S.K., Zabrodin, A.V., Ivanov, M.I., Kraiko, A.N., Prokopov, G.P.: Numerical Solution of Multidimensional Gas Dynamics Problems. USSR: MIR, Moscow (1979). (in Russian) 5. Ivanov, M.Y., Kraiko, A.N.: Shock capturing method for two-dimensional and threedimensional supersonic flows (in Russian). J. Comput. Math. Math. Phys. 12(2), 441–463 (1972) 6. Taylor, T.D., Mason, B.S.: Application of the unsteady numerical method of Godunov to computation of supersonic flows past bell shaped bodies. J. Comput. Phys. 5(3), 443 (1970) 7. Kolgan, V.P.: Implementationn of the principle of minimal slopes to constructing finitedifference schemes for calculation of discontinuous solutions of gas dynamics equations (in Russian). Studies TSAGI 3(6), 68–77 (1995) 8. van Leer, B.: Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme. J. Comput. Phys. 14, 164–169 (1974) 9. Colella, P., Woodward, P.R.: The piecewise-parabolic method (PPM) for gas-dynamics simulations. J. Comput. Phys. 54, 174–201 (1984) 10. Rodionov, A.V.: Methods for increasing the accuracy in Godunov’s scheme. USSR Comput. Math. Math. Phys. 27(6), 321–354 (1987) 11. Harten, A., Osher, S.: Uniformly high-order accurate nonoscillatory schemes. SIAM J. Num. Anal. 24(2), 279–309 (1987) 12. Teshukov, V.M.: Break-up of initial discontinuity on a curvilinear surface (in Russian). J. Comput. Math. Math. Phys. 2, 126–133 (1980) 13. Ben-Artzi, M., Falcovitz, Z.: A second-order Godunov-type scheme for compressible fluid dyamics. J. Comput. Phys. 55(1), 1–32 (1984) 14. Menshov, I.S.: Generalized problem on the break-up of initial discontinuity. Appl. Math. Mech. (in Russian) 1, 86–94 (1991) 15. Menshov, I.S.: Increasing the order of approximation of Godunov’s scheme using solutions of the generalized Riemann problems. USSR Comput. Math. Math. Phys. 30(5), 54–65 (1990) 16. Menshov, I.S., Nakamura, Y.: Hybrid explicit-implicit, unconditionally stable scheme for unsteady compressible flows. AIAA J. 42(3), 551–559 (2004) 17. Menshov, I.S., Nakamura, Y.: Implementation of the LU-SGS method for an arbitrary finite volume discretization. In: Proceedings of the 9th Conference on CFD, Tokyo, 1995, pp. 123– 124 (1995) 18. Menshov, I.S., Nakamura, Y.: On implicit Godunov’s method with exactly linearized numerical flux. Comput. Fluids 29, 595–616 (2000) 19. Menshov, I.S., Nakamura, Y.: Implementation of the variational Riemann problem solution for calculating propagation of sound waves in nonuniform flow fields. J. Comput. Phys. 182, 118–142 (2002)

A Note on Construction of Continuum Mechanics and Thermodynamics Michal Pavelka, Ilya Peshkov and Martin Sýkora

Abstract The standard way towards continuum thermodynamics based on conservation laws is discussed. Modelling of complex materials, where detailed state variables are necessary, requires insight beyond the conservation laws to determine the evolution equations for the detailed variables in a unique way. Several such approaches are discussed, including the natural configurations, SHTC equations and GENERIC.

1 Introduction When teaching continuum mechanics, e.g. [1, 2], one typically starts with a reference configuration or material configuration and a mapping to the actual configuration (inertial reference frame). The mapping expresses kinematics of mass points and, consequently, of material volumes. To obtain dynamics of material volumes, the laws of conservation of mass, linear momentum and energy are invoked. By means of the Reynolds transport theorem, one obtains the integral versions of equation of continuity, balance of momentum and balance of energy of the volume situated in the actual configuration (Eulerian inertial frame). The balance of momentum contains M. Pavelka (B) · M. Sýkora Faculty of Mathematics and Physics, Mathematical Institute, Charles University, Sokolovská 83, 18675 Prague, Czech Republic e-mail: [email protected] M. Sýkora e-mail: [email protected] I. Peshkov Department of Civil, Environmental and Mechanical Engineering, University of Trento, Via Mesiano 77, 38123 Trento, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_37

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a priori unknown surface terms, which using the Cauchy theorem are expressed by a second-order tensor field. Balance of energy contains unknown heat flux. The balances are then localized by shrinking the material volume to infinitesimal size, which yields the partial differential equations (balance laws with unknown Cauchy stress and heat flux) in the Eulerian inertial frame. A General balance equation can be seen as (1) ∂t  = −∂i Ji , where Ji is the flux of quantity . Because there are some unspecified fields present in the balance equations (Cauchy stress and heat flux, and possibly others), thermodynamics is used to find suitable constitutive relations for them. Firstly, the set of state variables is declared, on which the entropy density (or free energy density) depends. Derivatives of the entropy potential (or free energy) with respect to the state variables (energy density, density, velocity, and extra state variables) are denoted as temperature, pressure, etc., using the assumption of local thermodynamic equilibrium for interpretation. This is called a Gibbs relation. Time derivative of entropy density is then calculated by means of chain rule and, using the balances of mass, momentum and energy, it is rewritten into a form of divergence of entropy flux, while the remaining terms interpreted as the entropy production. The remaining terms contain the so far unspecified quantities and suitable constitutive relations (for instance, linear force–flux relations) are sought so that entropy production be non-negative. This ensures validity of the second law of thermodynamics. When seeking the constitutive relations, further axioms are often invoked, like objectivity or material frame indifference, Galilean invariance, Curie principle and Onsager–Casimir reciprocal relations [3]. Because the entropy is allowed to depend on some extra state variables (not only mass, momentum and energy densities), their time derivatives are present in the formula for entropy production, and their evolution equations are thus part of the closure relations. This way a closed set of evolution equations for the state variables is obtained. Let us refer to this last step (seeking suitable constitutive relations) as the modelling step because actual modelling and creativity is needed in that step. In the case of simple fluids (Navier–Stokes–Fourier system), there is not much freedom in the modelling, since there are no extra state variables. However, in non-equilibrium thermodynamics beyond the Classical Irreversible Thermodynamics (formulated in [1, 4]), some extra state variables (fluxes of other variables, internal variables, etc.) are typically present, and their evolution equations are to be found in the modelling step. If an extra variable is a scalar quantity, it is usually clear that the substantial derivative (including convection) expresses evolution of the variable. On the other hand, in case of tensorial variables, one has the freedom to choose any objective time derivative (upper or lower convective, Jaumann, etc.) with no theoretical lead which to choose. The conservation laws with the additional requirements above are typically not enough to fully specify the complex kinematics of tensorial state variables, and additional insight is needed. Let us make a non-exhaustive review of several possibilities.

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2 Some Sources of Additional Insight into Modelling Further insight can be gained from kinetic theory, e.g. the Grad hierarchy [5, 6] for the Boltzmann equation or the BBGKY hierarchy, results of which can be implemented into non-equilibrium thermodynamics [4, 7, 8]. In the case of dense gases, a new double hierarchy was proposed in [9]. One can also exploit the Liu procedure for generating the closure relations [10, 11]. Additional insight can be obtained by introducing other configurations like the natural configuration [12, 13] or relaxed metric [14, 15]. In the case of natural configurations, evolution is split into plastic processes from the reference configuration to the natural configuration (irreversible) and elastic processes from the natural to the actual configuration (reversible). Dynamics of the natural configuration (pure elasticity) then provides the additional insight, namely, the evolution for deformation tensor or its variant. One can also exploit the space–time formulations of kinematics, see, e.g. [15, 16], which provides additional leads for choosing time derivatives. Another source of additional insight is provided by the requirement that the evolution equations be symmetric hyperbolic, quasilinear first-order, dissipative and obey certain gauge invariance (w.r.t. rotations, etc.) [17–25]. This line of research is culminating with the Symmetric Hyperbolic Thermodynamically Compatible equations (SHTC) [26–28], a simplified version of which is ∂t ρ = −∂i (ρ E m i )

(2a)

∂t m i = −∂ j (m i E m j ) − ρ∂i E ρ − m j ∂i E m j − s∂i E s − A L l ∂i E A Ll +∂i (A L l E A Ll ) − ∂l (A L i E A Ll ) ∂t s = −∂i (s E m i ) +

E AI i Es

MliL I

E AI i Es

,

∂t A L l = −∂l (A L i E m i ) + (∂l A L i − ∂i A L l )E m i − MliL I

(2b) (2c)

E AI i Es

.

(2d)

These equations express evolution of density, total momentum density, entropy density (per volume) and distortion A, which can be interpreted as inverse deformation gradient (at least in the absence of irreversible terms, see below). The symmetric positive definite tensor M is typically considered isotropic, in which case it can be interpreted as inverse relaxation time and typically depends on the state variables (ρ, m, s, A). These evolution equations represent a set of first-order quasilinear symmetric hyperbolic equations that conserve energy, produce entropy and are Galilean invariant. Note that irreversible terms (those with the time-reversal behaviour opposite to the corresponding left-hand side [29]) are present only in the equation for the distortion.

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Fig. 1 Vertical vibrations in an elastoplastic solid with parameters specified in [31]. The figure shows y-velocity profiles times t = 0.00005 (green), t = 0.00005 + P (red) and t = 0.00005 + 2P (blue), where P = 1/2141 is the period of the linearized elastic problem. Dissipation gradually diminishes the velocity

Hyperbolicity of the SHTC equations is advantageous in numerical simulations. In Fig. 1, a simple one-dimensional finite string at three subsequent time instants is shown. The numerical results were obtained using publicly available code [30] in [31]. Further insight into the modelling step can be found in geometry. For instance, one can take advantage of variational principles [32–34], which yields evolution equations for the chosen state variables once a Lagrangian is specified. Finally, another possibility is to formulate the evolution equations as Hamiltonian mechanics. For instance, on the material manifold (sometimes also called reference configuration) each continuum particle is equipped with a label X. The state variables are then the field of positions in an inertial frame (actual configuration) of the particles and the field of their momenta, x(X) and M(X). The canonical Poisson bracket expressing kinematics of these fields is {F, G}(L) =



 dX

 δG δG δF δF − , δx i (X) δ Mi (X) δx i (X) δ Mi (X)

(3)

where F and G are two arbitrary functionals and subscripts stand for functional derivatives. The Hamilton canonical equations implied by this bracket are ∂t x i (X) =

δE , δ Mi (X)

∂t Mi (X) = −

δE . δx i (X)

(4)

Once the energy is known, a closed set of reversible evolution equations is obtained. A transformation of the canonical Poisson bracket to the Eulerian fields (ρ, m, s, A) then leads to a non-canonical Poisson bracket for the fields, which leads to the reversible part of SHTC evolution Eq. (2), see [35, 36]. This Lagrange-to-Euler transformation, c.f. [37], can be used to derive boundary conditions for free surfaces [38]. The reversible part of SHTC equations is Hamiltonian.

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The irreversible part of the SHTC equations, which provides dissipative algebraic terms, can be seen as a particular instance of gradient dynamics as shown in [8, 36]. SHTC equations are thus fully compatible with the framework of General Equation for Non-Equilibrium Reversible–Irreversible Coupling (GENERIC) [8, 39–41]. Yet, further insight can be gained by firstly formulating the motion as an infinitedimensional Lie group as in [42–44]. For instance, the SHTC equations then possess the structure of a semidirect product, which has further mathematical implications about hyperbolicity, gauge invariance, etc., see [35]. Finally, one can also formulate evolution equations for fluid dynamics and Grad’s hierarchy in differential geometry without Lie groups [45], which leads to sort of GENERIC without Poisson brackets.

3 Conclusion In this note, several approaches to continuum thermodynamics have been briefly recalled. It was emphasized that only balance laws are not sufficient for modelling complex materials, where extra state variables (not only density, momentum and energy density) are necessary. Indeed, their kinematics is often too complex to be derived based just on the conservation laws, entropy principle and objectivity. Several approaches that go beyond the conservation laws were then discussed. The framework of Symmetric Hyperbolic Thermodynamically Compatible equations has been recalled and illustrated on a simple numerical example. Finally, geometric techniques based on Poisson geometry and Lie groups were mentioned, and the Hamiltonian origin of SHTC equations was exposed. Acknowledgements I.P. acknowledges a financial support by Agence Nationale de la Recherche (FR) (Grant No. ANR-11-LABX-0040-CIMI) within the program ANR-11-IDEX-0002-02. M.P. and M.S. were supported by Czech Science Foundation, Project No. 17-15498Y, and by Charles University Research Program No. UNCE/SCI/023.

References 1. Meixner, J., Reik, H.G.: Thermodynamik der Irreversible Prozesse. In: Handbuch der Physik, vol. 3/II. Springer, Berlin (1959) 2. Truesdell, C.: Rational Thermodynamics, 2nd edn. Springer, New York (1984) 3. Lebon, G., Jou, D., Vázquez, J.C.: Understanding Non-Equilibrium Thermodynamics: Foundations, Applications, Frontiers. Springer London Limited, London (2008). SpringerLink: Springer e-Books 4. de Groot, S.R., Mazur, P.: Non-equilibrium Thermodynamics. Dover Publications, New York (1984) 5. Grad, H.: Principles of kinetic theory of gases. In: Encyclopedia of Physics, vol. 12. Springer, Berlin (1958) 6. Struchtrup, H.: Macroscopic Transport Equations for Rarefied Gas Flows. Springer, Berlin (2005)

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7. Jou, D., Casas-Vázquez, J., Lebon, G.: Extended Irreversible Thermodynamics, 4th edn. Springer, New York (2010) 8. Pavelka, M., Klika, V., Grmela. M.: Multiscale Thermo-Dynamics. de Gruyter, Berlin (2018) 9. Ruggeri, T., Sugiyama, M.: Rational Extended Thermodynamics Beyond the Monoatomic Gas. Springer, Heidelberg (2015) 10. Mongiovì, M.S., Restuccia, L.: Hydrodynamical model of anisotropic, polarized turbulent superfluids. i: constraints for the fluxes. Zeitschrift für angewandte Mathematik und Physik 69(1), 2 (2017) 11. Müller, I.: Thermodynamics. Interaction of Mechanics and Mathematics Series. Pitman, Borough (1985) 12. Málek, J., Pr˚uša, V.: Derivation of equations for continuum mechanics and thermodynamics of fluids. In: Giga, Y., Novotný, A. (eds.) Handbook of Mathematical Analysis in Mechanics of Viscus Fluids. Springer, Berlin (2016) (Submitted) 13. Rajagopal, K.R., Srinivasa, A.R.: On the thermomechanics of materials that have multiple natural configurations part i: viscoelasticity and classical plasticity. Zeitschrift für angewandte Mathematik und Physik ZAMP 55(5), 861–893 (2004) 14. Berezovski, A., Ván, P.: Internal Variables in Thermoelasticity. Solid Mechanics and Its Applications. Springer International Publishing, Berlin (2017) 15. Fülöp, T., Ván, P.: Kinematic quantities of finite elastic and plastic deformation. Math. Methods Appl. Sci. 35(15), 1825–1841 (2012) 16. Matolcsi, T., Ván, P.: Can material time derivative be objective? Phys. Lett. A 353(2), 109–112 (2006) 17. Friedrichs, K.O.: Conservation equations and the laws of motion in classical physics. Commun. Pure Appl. Math. 31(1), 123–131 (1978) 18. Godunov, S., Mikhailova, T., Romenskii, E.: Systems of thermodynamically coordinated laws of conservation invariant under rotations. Sib. Math. J. 37(4), 690–705 (1996) 19. Godunov, S.K.: An interesting class of quasi-linear systems. Soy. Math. 2, 947 (1961) 20. Godunov, S.K., Romenskii, E.I.: Elements of Continuum Mechanics and Conservation Laws. Kluwer Academic/Plenum Publishers, New York (2003) 21. Godunov, S.K.: Symmetric form of the magnetohydrodynamic equation. Chislennye Metody Mekh. Sploshnoi Sredy 3(1), 26–34 (1972) 22. Godunov, S.K., Romensky, E.: Thermodynamics, conservation laws and symmetric forms of differential equations in mechanics of continuous media. In: Computational Fluid Dynamics Review, pp. 19–31. Wiley, New York (1995) 23. Müller, I., Ruggeri, T.: Rational Extended Thermodynamics, 2nd edn. Springer, New York (1998) 24. Ruggeri, T., Strumia, A.: Main field and convex covariant density for quasi-linear hyperbolic systems. Ann. Inst. H. Poincaré 34, 65 (1981) 25. Zhu, Y., Hong, L., Yang, Z., Yong, W.A.: Conservation-dissipation formalism of irreversible thermodynamics. J. Non-Equilib. Thermodyn. 40(2), 67–74 (2015) 26. Dumbser, M., Peshkov, I., Romenski, E., Zanotti, O.: High order ader schemes for a unified first order hyperbolic formulation of continuum mechanics: viscous heat-conducting fluids and elastic solids. J. Comput. Phys. 314, 824–862 (2016) 27. Dumbser, M., Peshkov, I., Romenski, E., Zanotti, O.: High order ADER schemes for a unified first order hyperbolic formulation of Newtonian continuum mechanics coupled with electrodynamics. J. Comput. Phys. 348, 298–342 (2017) 28. Peshkov, I., Romenski, E.: A hyperbolic model for viscous Newtonian flows. Contin. Mech. Thermodyn. 28, 85–104 (2016) 29. Pavelka, M., Klika, V., Grmela, M.: Time reversal in nonequilibrium thermodynamics. Phys. Rev. E 90(062131) (2014) 30. Jackson, H., Nikiforakis, N.: A numerical scheme for non-newtonian fluids and plastic solids under the gpr model. J. Comput. Phys. 387, 410–429 (2019) 31. Sýkora, M.: Hamiltonian and thermodynamic theory of solids and fluids. Master’s thesis, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic (2019)

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A Method of Boundary Equations for Unsteady Hyperbolic Problems in 3D S. Petropavlovsky, S. Tsynkov and E. Turkel

Abstract We propose an efficient algorithm based on boundary operator equations for the numerical simulation of time-dependent waves in 3D. The algorithm employs the method of difference potentials combined with the (strong) Huygens’ principle (lacunae of the solution). It can handle nonconforming boundaries on regular structured grids with no loss of accuracy and offers sublinear computational complexity.

1 Introduction We present a boundary method for computing the scattering of unsteady waves about general three-dimensional shapes. Boundary methods are common for elliptic and time-harmonic problems, but their generalization to the time-dependent case is far less straightforward. A major limitation is the ever-expanding backward dependence of the solution on time. It manifests itself via the boundary convolutions that go all the way back from the current to the initial moment of time, which makes the computation too costly. The key contribution of the current work is a boundary timemarching algorithm that requires only a narrow time window, which slides over the (2+1)-dimensional boundary of a (3+1)-dimensional space–time computational domain. The reduced dimension of the proposed boundary formulation compared to that of a volumetric integration results in a sublinear complexity of the algorithm. Truncation of the backward temporal dependence of the solution is an implication of the (strong) Huygens’ principle. It is a rare occurrence, but it holds for several S. Petropavlovsky National Research University Higher School of Economics, Moscow 101000, Russia e-mail: [email protected] S. Petropavlovsky · S. Tsynkov (B) North Carolina State University, Raleigh, NC 27695, USA e-mail: [email protected] E. Turkel Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_38

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PDE models important for applications: the wave (d’Alembert) equation, acoustics equations (linearized Euler’s), Maxwell’s equations in vacuum, and elasticity equations. The Huygens principle as a vehicle for truncating the convolutions is time has been exploited in some numerical methods based on retarded potential boundary integral equations [1]. Our implementation is fundamentally different in that it relies on the lacunae in the solutions and does not require the sophisticated numerical quadratures over the intersection of the light cone with the space–time boundary. The boundary operators for the sliding window time-marching algorithm are computed by finite differences, including high order, on Cartesian grids. No sophisticated grid generation is required (which is typical, e.g., for finite elements). In doing so, the method of difference potentials (MDP) by Ryaben’kii [2] guarantees no deterioration of accuracy even for nonconforming boundaries. We demonstrate the performance of our algorithm by solving both interior and exterior initial boundary value problems (IBVPs) for the 3D wave equation.

2 Description of the Method Consider an exterior IBVP for the homogeneous 3D wave equation: c u = 0, on R3 \ × [0, T ], l  u = φ, on ∂ × [0, T ], u|t=0 = ∂u/∂t|t=0 = 0,

(1a) (1b) (1c)

where c ≡ c12 ∂t∂ 2 −  is the d’Alembert operator and c is the propagation speed which is assumed constant. The solution u is to be calculated outside a bounded domain  ⊂ R3 until the final time T . System (1) describes scattering about the shape , and the unknown quantity u represents the scattered field. The homogeneous initial condition (1c) implies that the incident wave u inc has just reached  by the initial time t = 0. The operator l  in (1b) determines the type of scattering. The simplest examples of the boundary conditions (BCs) on  ≡ ∂ are the Dirichlet inc ∂u = − ∂u∂n conditions written in terms of l  u ≡ u = −u inc and Neumann l  u ≡ ∂n the incident field. Our approach does not impose any limitations on the type of the initial or boundary conditions except that the entire problem (1) must be well-posed. Next, consider a larger auxiliary domain  ⊃  and denote by G(x, t) the fundamental solution of the 3D d’Alembert operator of (1a). We will be interested in ¯ (the horizontal bar obtaining the scattered field u on the complimentary domain  \ stands for the closure of ). Let us apply the Green’s formula to the four-dimensional bounded domain  \ × (0, t], t ≤ T (see, e.g., [3, Eq. (7.3.5)]): 2

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u(x, t) =

1 c2

  ¯  \

293

 ∂u ∂G ( y, 0)G(x − y, t) − u( y, 0) (x − y, t) d y ∂t ∂t

 

 ∂u    ∂G  + ( y, t )G(x − y, t − t ) − u( y, t ) (x − y, t − t ) dt  d S y ∂n ∂n t    ∂u    ∂G  + ( y, t )G(x − y, t − t ) − u( y, t ) (x − y, t − t ) dt  d S y . ∂n ∂n t

(2) The first integral in (2) is zero because of (1c). The last integral in (2) is taken over the artificial boundary t ≡ ∂ × (0, t] and is also equal to zero. Given that 1 δ(t − |x|/c) is the Green’s function of the 3D wave equation in free G(x, t) = 4π|x| space, this implies that all the waves at the outer boundary t are outgoing, and there are no waves coming through the boundary t back to the domain  . As a result, only the second integral on the right-hand side of (2) remains, where the integration is performed over t ≡ ∂ × (0, t]. Thus, formula (2) allows us to express the solution ¯ at a given time t only through the boundary u inside the computational domain  \ values on the surface of the scatterer ∂ at the previous moments of time. By analogy with (2), we introduce the Calderon potential with vector density ξ t = (ξ0 , ξ1 ) defined on t : P ˜ ξ t (x, t) =

   ∂G (x − y, t − t  ) dt  d S y , (3) ξ1 ( y, t  )G(x − y, t − t  ) − ξ0 ( y, t  ) ∂n

t

˜ = R3 \. ¯ Comparison of (2) and (3) suggests that ξ0 and ξ1 can be where x ∈  interpreted as the trace of an outgoing solution to the wave equation (1a) and its normal derivative on ∂, respectively. It turns out that this is the case if and only if the boundary equation with projection (BEP) holds (see [2]): P t ξ t = ξ t ,

(4)

where P t ≡ T r t P ˜ ξ t is the Calderon projection and for any function w defined    . To obtain a unique outgoing solution, Eq. (4) ˜ T r t w(x, t) def = w, ∂w on , ∂n t should be considered along with the BC (1b) reformulated in terms of ξ t = (ξ0 , ξ1 ): l t ξ t = φ.

(5)

Thus, Eqs. (4) and (5) form a problem equivalent to the IBVP (1) but set only on the lateral boundary ∂ rather than on the entire 3D domain R3 \. Once the density ξ t is found from (4), (5), the solution u on R3 \ is given by formula (3). The surface integral in (3) may be difficult to compute, especially for a complicated geometry of the scatterer. Therefore, we redefine the Calderon poten-

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tial. Take an arbitrary function w(x, t) compactly supported in space and such that T r t w = ξ t . Then, P ˜ ξ t (x, t) = v(x, t), where v(x, t) is a solution to    c w, x ∈ ,  wt=0 , x ∈ , ∂v  the Cauchy problem: c v = = , v t=0 = , 0, x∈ / , 0, x∈ / , ∂t t=0   ∂w  , x ∈ , ∂t t=0 This Cauchy problem is referred to as the auxiliary problem (AP). 0, x∈ / . While formulated for all x ∈ R3 and t  0, in practice it will need to be solved only over a finite short interval of time, which, in turn, requires only a bounded auxiliary domain  . For a given ξ t = (ξ0 , ξ1 ), the function w(x, t) in the vicinity of t can be built p by Taylor’s formula: w(x, t) = Pp=0 l!1 ∂∂nwp (x 0 , t)ρ p , x 0 ∈ ∂, where w(x 0 , t) = ξ0 (x 0 , t), ∂w (x 0 , t) = ξ1 (x 0 , t), and higher order derivatives are obtained by differ∂n entiating the d’Alembert equation (1a). The density ξ t is represented as an expansion N N with respect to a basis on t : ξ0 = i=1 c0,i ψi (x, t), ξ1 = i=1 c1,i ψi (x, t) so that Eqs. (4), (5) can be reformulated and solved for the coefficients c0,i , c1,i . The details of the discretization by means of the MDP can be found in [4]. However, as the time elapses, the boundary t extends, which makes Eqs. (4), (5) increasingly costly to solve. To overcome this fundamental difficulty, we partition the boundary T (where T is the final time) into K equal parts: T = 1 ∪ 2 ∪ ... ∪  K , where k = ∂ × ((k − 1)T0 , kT0 ]. Then, we recast system (4), (5) (for t = T ) in such a way that the corresponding partial densities can be determined consecutively. From the BEP (4), we derive the following equation for the partial density ξ  K : P T0 ξ  K +

K −1 k=1

T r  K P ˜ ξ k = ξ  K .

(6)

It is supplemented by the boundary condition derived from (5): l  K ξ  K = φ.

(7)

Assuming that all ξ k for k = 1, 2, . . . , K − 1 are known, Eqs. (6), (7) allow us to determine ξ  K . As T and K are arbitrary, we can continue consecutive updates K → K + 1. This is a time-marching algorithm along the boundary. Still, it inherits full backward dependence on time: ξ  K = ξ  K (ξ  K −1 , . . . , ξ 1 ). Let us, however, choose T0  1c diam . As P ˜ is convolution with G, there will be no contribution from any of ξ k , k = K − 2, . . . , 1. Therefore, from (6) we have P T0 ξ  K + T r  K P ˜ ξ  K −1 = ξ  K .

(8)

Thus, ξ  K depends only on the immediately preceding ξ  K −1 . Combining (8) with (7), we have a sliding window time marching with respect to K performed along a (2+1)-dimensional boundary t . The limited backward dependence on time in (8), (7) is an implication of the Huygens’ principle. Indeed, the Calderon operators take

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advantage of the lacuna of G or, equivalently, of the lacuna in the solution of the AP driven by a source compactly supported on  and operating over a finite time T0 . Expanding the densities ξ  K and ξ  K −1 with respect to the basis {ψi }, we recast (8): ) −1) −1) Q (0) c0(K ) + Q (1) c(K = −R(0) c(K − R(1) c(K , 1 0 1

(9)

) (K ) ) (K ) −1) (K −1) where c(K = {c0,i }, c(K = {c1,i } are the unknown coefficients and c(K , c1 0 1 0 are considered known. Equation (9) is to be solved together with the boundary condition (7) reformulated using the same basis {ψi }. The columns of the matrices Q (0,1) and R(0,1) in (9) correspond to individual basis functions ψi on T0 . Namely, each vector function (ψi , 0) or (0, ψi ) is extended by Taylor’s formula, and the resulting w is used to generate the source term for the AP. Then, the AP is solved on the time interval 2T0 and the solution on the special discrete sets of grid nodes near T0 yields the aforementioned columns. Solving multiple APs is the costliest part of the algorithm, because the number of basis functions ψi may be substantial. Yet different APs can be very efficiently solved in parallel, because different basis functions ψi are completely independent. As the AP is solved on a short time interval (we assume T0  T ), we approximate it numerically on a bounded computational domain  ⊃  chosen sufficiently large so that no wave reflected back from the outer boundary ∂ can reach the physical boundary ∂ before t = 2T0 . This approach eliminates the need for artificial boundary conditions on ∂ and is equivalent to a perfect reflectionless treatment of the outgoing waves. It is important that Eq. (9) does not depend on the boundary condition (7). The two need to be combined to obtain a unique solution. Therefore, changing the boundary condition is numerically inexpensive, and the proposed methodology appears very well suited for solving multiple similar problems. Moreover, applying the proposed methodology to interior problems requires only insignificant changes.

3 Numerical Demonstrations For our numerical simulations (of both interior and exterior problems), we choose a unit sphere as the computational domain . The auxiliary domain is a larger cube:  ⊃ . It is discretized using a uniform Cartesian mesh, and the boundary ∂ is nonconforming. The wave equation (1a) is approximated using two schemes—the standard explicit second-order accurate central difference scheme and a compact implicit fourth-order accurate scheme [5] that uses a full 3 × 3 × 3 × 3 stencil. First, we solve the interior problem with second-order accuracy. The test solution is a plane wave (λ = diam ), and we reconstruct it numerically for the boundary condition (1b) of a Dirichlet, Neumann, or Robin type. Table 1 shows the design rate of grid convergence. There is no deterioration of accuracy even though the boundary is nonconforming, and no indication of any difficulties related to cut cells [6].

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Table 1 Time-averaged ∞ error on three successively refined grids (by a factor of 2) and the respective convergence rates for the interior problem Grid 1 Grid 2 Rate Grid 3 Rate 8.03 × 10−5 2.64 × 10−5 1.74 2.89 × 10−4 7.50 × 10−5 1.96 4.86 × 10−4 1.29 × 10−4 1.94

Dirichlet BC Robin BC Neumann BC (∇)

6.87 × 10−6 1.96 1.87 × 10−5 1.99 3.48 × 10−5 1.92

Table 2 Average values of the ∞ error over 0  t  Tsim on three successive grids and the respective convergence rates for the exterior scattering problem solved using second-order scheme Grid Dirichlet Neumann Robin Average error Rate Average error Rate Average error Rate 1x 2x 4x

6.55 × 10−3 1.52 × 10−3 2.91 × 10−4

– 4.27 5.25

1.84 × 10−2 4.84 × 10−3 9.69 × 10−4

2.02 × 10−2 5.74 × 10−3 1.41 × 10−3

– 3.81 4.99

– 3.52 4.06

Table 3 Average values of the ∞ error over 0  t  Tsim on two successive grids and the respective convergence rates for the exterior scattering problem solved using fourth-order scheme Grid Dirichlet Neumann Robin Average error Rate Average error Rate Average error Rate 2x 4x

6.52 × 10−5 2.44 × 10−6

– 26

2.09 × 10−4 8.60 × 10−6

– 24

2.86 × 10−4 1.31 × 10−5

– 21

Table 4 CPU time needed to advance the solution to the scattering problem over the time T0 Grid Volumetric method+PML MDP+lacunae CPU time, sec Scaling, times CPU time, sec Scaling, times 1x 2x 4x

1.26 19.8 322

– 15.7 16.3

0.0474 0.421 3.56

– 8.87 8.46

The exterior problem that we solve is that of the scattering of an impinging plane wave about a unit sphere. The exact solution for the scattered field is known in the form of an expansion in spherical harmonics and Hankel functions [3, Vol. 2, Chap. 11, p. 1483]. Tables 2 and 3 present the results of computations with secondand fourth-order accuracy, respectively. The design rates of grid convergence are achieved for all three types of the boundary conditions. Again, the boundary ∂ is nonconforming. Table 4 compares the CPU time needed to compute the solution by a traditional volumetric scheme terminated with a PML against that for the proposed boundary approach. One can see that in the volumetric case the time scales as 24 , whereas for the proposed method it scales as 23 . This is a demonstration of sublinear complexity.

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Acknowledgements Work supported by US ARO, grants W911NF-16-1-0115 and W911NF-14C-0161, and US–Israel BSF, grant 2014048.

References 1. Sayas, F.J.: Retarded Potentials and Time Domain Boundary Integral Equations. A Road Map. Springer Series in Computational Mathematics, vol. 50. Springer, Cham (2016). https://doi.org/ 10.1007/978-3-319-26645-9 2. Ryaben’kii, V.S.: Method of Difference Potentials and Its Applications. Springer Series in Computational Mathematics, vol. 30. Springer, Berlin (2002) 3. Morse, P.M., Feshbach, H.: Methods of Theoretical Physics, vol. 2. International Series in Pure and Applied Physics. McGraw-Hill Book Co., Inc., New York (1953) 4. Petropavlovsky, S., Tsynkov, S., Turkel, E.: J. Comput. Phys. 365, 294 (2018). https://doi.org/ 10.1016/j.jcp.2018.03.039 5. Smith, F., Tsynkov, S., Turkel, E.: J. Sci. Comput, 1–29 (2019). https://doi.org/10.1007/s10915019-00970-x 6. Sticko, S., Kreiss, G.: Computer methods in applied mechanics and engineering 309, 364 (2016). https://doi.org/10.1016/j.cma.2016.06.001

Solution of Deformable Solid Mechanics Dynamical Problems with Use of Mathematical Modeling by Grid-Characteristic Method Igor B. Petrov

Abstract The most suitable method for hyperbolic equation systems, e.g., equations describing elastic waves, is the grid-characteristic method. This method allows us to simulate physically correctly wave processes in heterogeneous media and take into consideration boundary conditions and conditions on contact surfaces. Most fully the advantages of the method are for one-dimensional equations, especially in combination with a fixed difference grid, as in conventional grid-based methods. However, in the multidimensional case using the algorithms of splitting with respect to spatial variables, the author has managed to preserve its positive qualities. The use of method of Runge–Kutta type for hyperbolic equations makes it possible to effectively carry out the generalization of methods developed for linear equations, in nonlinear case. Several important problems of exploration seismology, seismic resistance, global seismic studies on Earth and Mars, medical applications, nondestructive testing of railway lines, and other areas of practical application were numerically solved. A significant advantage of the constructed method is the preservation of its stability and precision at the strains of the environment. This article presents the results of numerical solution based on the grid-characteristic method to such problems as modeling elastic–plastic deformation in traumatic brain injury, direct problem of seismology, and impact task.

1 Introduction The numerical solution to nonstationary dynamic problems of elastic and elastic– plastic deformations of isotropic solids in case of two or three spatial coordinates is the object of many works; a fairly detailed overview is given in [1, 2]. In solid mechanics, various mathematical rheological models have been developed, among which an important place is occupied by those which are described by systems of I. B. Petrov (B) Moscow Institute of Physics and Technology, Dolgoprudniy, Institutskiy Pereulok, 9, Moscow region, 141701 Dolgoprudny, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_39

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equations of hyperbolic type, and, above all, the models of linear and nonlinear theory of elasticity and plasticity. The most important concept for such equations is their characteristic properties. The use of the characteristic records of the original equations when constructing a particular numerical methods (i.e., approximation of not the original equations, but the equivalent to the conditions of compatibility along some characteristic directions) allows in the most natural way building computational algorithm on the borders of the region of integration, to a certain extent taking into account solutions dependence domain, etc. Most fully the advantages of the characteristic approaches occur for onedimensional equations, especially in combination with a fixed difference grid, as in conventional grid-based methods (grid-characteristic methods [3–5]; however, in the multidimensional case using the algorithms of spatial splitting, the variables are able to preserve their positive qualities. The use of the equations of Runge–Kutta type or integro-interpolation method [6, 7] makes it possible to effectively carry out the generalization of methods developed for linear equations, in nonlinear case, including by ensuring the implementation of the difference analogs of the conservation laws, which is important for shock capturing of discontinuous solutions.

2 Mathematical Model Using conventional kinematic equations for a symmetric tensor of velocities of deformations [11] in spatially fixed orthogonal curvilinear coordinate system x1 , x2 , and x3 emn =

1 2



1 ∂vn 1 ∂vm + Hn ∂ xn Hm ∂ xm

 +

1 Hn

 δmn

 3     vs ∂ Hm ∂ Hn ∂ Hm 1 + − vm , Hs ∂ xs 2Hm ∂ xn ∂ xm s=1

(1) m, n = 1, 2, 3, choosing the defining relations in the form ∂σi j  dσi j = + dt ∂t k=1 3



vk ∂σi j Hk ∂ xk

 (2)

we write a closed system of two-dimensional nonstationary equations in the form of  ∂u ∂u + Ai = f. ∂t ∂ξi i=1,2

(3)

The symbol u = {V1 , V2 , σ11 , σ12 , σ22 , σ33 , T } means the vector of desired variables, including the components v1 and v2 of the velocity vector V (on the axes x1 , x2 ,

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respectively) nonzero components σ i j of the symmetric stress tensor and the temperature T ; f(t, x1 , x2 , u) = ( f 1 , f 2 , f 11 , f 22 , f 33 , f T )—vector of right parts with the following components in the curvilinear orthogonal coordinate system x1 , x2 , and x3 .

3 Results of Numerical Modeling 3.1 Traumatic Brain Injuries This task was set by the specialists of the Institute named after Sklifosovsky and Ambulance Central clinical military hospital named after Burdenko. Working together with the neurosurgeons, we formulated three mechanical– mathematical models of human head. The simplest of these is a two-component model, in which bone tissue and brain are described by homogeneous isotropic materials with average mechanical properties; more complex models take into account the presence of ventricle and two membranes. The external load is set as the collision of skull–brain system with a completely fixed rigid barrier with a given initial velocity (1–3 m/s). It is known that a significant contribution to the formation of a wave pattern of disturbance propagation in an elastic medium makes heterogeneity. Therefore, a two-component model was further developed in the form of a three-component model, consisting of bones and brain and ventricle. The dural membranes have an inhibiting effect on the movement of brain inside skull. Therefore, subsequently, it was decided to introduce a model of the falx cerebri, with a vertical membrane separating the hemispheres in the parietal region. The distribution of the shear stress in the formulation of various conditions is shown in Fig. 1.

Fig. 1 The distribution of shear stresses a complete adhesion; b slide; c two-component model

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Model of traumatic brain sections: a—two-component model, b—model of ventricles, and c—model with ventricles and membrane (falx cerebri). Figure 1 shows the overall characteristics of the mechanical effects on the brain at a side impact, obtained using different models with free slip on the border of the skull–brain. The most dangerous are concentrations of maximum tensile (positive) and shear stress. The use of conditions for the complete adhesion on the border of the skull–brain leads to concentration of the shear stress along the contact boundary on the lateral surfaces, while the sliding contact completely removes them. Accounting for the presence of ventricles almost has no effect on the distribution of the regions of maximum compression and extension but significantly affects the distribution of shear loads. The presence of membrane is more essential to localize areas of compression–tension in side impacts.

3.2 Direct Problems of Exploration Seismology Figure 2 shows the isosurfaces of the modulus of the velocity for ten-layer geological environment at four points in time effects of seismic loading at the upper boundary. Let’s consider the problem of reflection of plane Ricker pulse, the direction of propagation of which is 14 degrees with vertical (this corresponds to the displacement of the point of the explosion along the surface to 0.5 km) from the reservoir five vertical filled cracks [8, 9]. The pattern of reflected waves at various distances between the cracks is examined. Figure 3 shows the corresponding velocity fields. A comparison of the results of calculations of the geological problems on two-dimensional triangular grid on a three-dimensional cubic grid is done. This comparison shows that the discrepancy of the results obtained is not more than 30%. Thus, the twodimensional calculation gives not only a qualitatively correct result but also a good quantitative assessment. Therefore, in the study of three-dimensional dynamic processes in geological environments, it is possible to use two-dimensional calculations.

Fig. 2 The isosurfaces of the modulus of the velocity for ten-layer geological environment at four points in time (see Figure a, b, c, d for 500 ms, 600 ms, 700 ms, 800 ms, respectively)

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Fig. 3 Velocity fields from fractures

This can significantly speed up computation time and reduce storage costs and data processing. It is seen that the used in the paper numerical scheme gives well the complex wave pattern resulting from the interaction of seismic waves with geological layered rock. On the calculated seismogram, the reflected from all contact boundaries waves are clearly visible. The obtained calculations show the possibility of using a full numerical simulation of complex wave processes in layered geological media to obtain data and study of the structure of rocks.

3.3 Impact Task Similar designs are used as protective ones (e.g., body armor). The lower (in the figures on the right) calculated layer corresponds to the protected from hitting human body [10]. As it can be seen from the calculations, after hitting the upper boundary of the five-layer construction, the effect of elastic–plastic compression waves leads to a collision of layers from top to bottom, after which the compression waves go to a protected environment. By simulating attacks on the multilayer barrier, we obtained wave pattern corresponding to the distribution of the secondary waves generated by the collisions of obstacles and directed against the impact. In addition, the layers in such structures face then go away from each other, depending on the sign of the normal stress (negative or positive) on the contact boundaries. These processes (collision and detachment) initiate the emergence of secondary waves of compression stretching in the design and the protected body (Fig. 4).

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Fig. 4 The results of calculations (velocity fields) for the impacts on several layer barriers

Acknowledgements This work was carried out with the financial support of the Russian Science Foundation, project no. 19-11-00023.

References 1. Kukudzhanov, V.N.: Chislennoe reshenie neodnomernykh zadach rasprostraneniya voln napryazheniy v tverdykh telakh (Numerical solution of multidimensional problems of propagation of stress waves in solid bodies). Soobshch. po prikl. matem., (6), 65p Moscow, VTS AN USSR (in Russian) (1976) 2. Novatskii, V.K.: Volnovye zadachi teorii plastichnosti (Wave problems of the theory of plasticity). M.: Izd. Mir 307p (in Russian) (1978) 3. Magomedov, K.M., Kholodov, A.S.: O postroenii raznostnykh skhem dlya uravneniy giperbolicheskogo tipa na osnove kharakteristicheskikh sootnosheniy (On construction of difference schemes for equations of hyperbolic type on the basis of the characteristic ratios). Zh. vychisl. matem. i matem. fiz. 9(2), 373–386 (in Russian) (1969) 4. Kholodov, A.S.: O postroenii raznostnykh skhem s polozhitel’noy approksimatsiey dlya uravneniy giperbolicheskogo tipa (On construction of difference schemes with positive approximation for equations of hyperbolic type). Zh. vychisl. matem. i matem. fiz. 18(6), 1476–1492 (in Russian) (1978) 5. Kholodov, A.S.: O postroenii raznostnykh skhem povyshennogo poryadka tochnosti dlya uravneniy giperbolicheskogo tipa (On construction of difference schemes of increased order of accuracy for equations of hyperbolic type). Zh. vychisl. matem. i matem. fiz. 20(6), 1601– 1620 (in Russian) (1980) 6. Rusanov, V.V.: Raznostnye skhemy tretego poryadka tochnosti dlya skvoznogo scheta razruvnykh resheniy (Difference schemes of third order of accuracy for calculation of discontinuous solutions). Dokl. AN USSR. 180(6), 1303–1305 (in Russian) (1968) 7. Tikhonov, A.I., Samarskiy, A.A.: Ob odnorodnykh raznostnykh skhemakh (On homogeneous difference schemes). Zh. vychisl. matem. i matem. fiz. 1(1), 5–63 (in Russian) (1961)

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8. Petrov, I.B., Muratov, M.V.: The application of grid-characteristic method in solution of fractured formations exploration seismology direct problems (review article). Matem. Mod. 21(4), 33–56 (2019) 9. Kvasov, I.E., Leviant, V.B., Petrov, I.B.: Chislennoe issledovanie volnovykh protsessov v poristoy srede s ispolzovaniem setochno-kharakteristicheskogo metoda (Numerical study of wave processes in a porous medium using grid-characteristic method). Zh. vychisl. matem. i matem. fiz. 56(9), 1645–1656 (in Russian) (2016) 10. Petrov, I., Vasyukov, A., Ermakov, A., Favorskaya, A.: Numerical modeling of non-destructive testing of composities. Sci. Process. Comput. Sci. 96, 930–938 (2016)

Set of Solutions of Ordinary Differential Equations in Stability Problems A. N. Rogalev

Abstract The article studies the characteristics of the sets of solutions and boundaries of the set of solutions of ordinary differential equations. These sets of solutions appear when there are known only uncertain parameters of the problem (only inequalities for this parameter are known). Estimates of solutions of ODEs are constructed based on perturbation of the initial data or parameters of the ODE system. Formulas are obtained and investigated that make it possible to evaluate the convergence and divergence of the trajectories of the solutions of ODEs when the input data changes.

1 Some Properties of Sets of Solutions of Ordinary Differential Equations We consider a system of ordinary differential equations (ODEs) that describes the state of a real object (system) and its evolution in time dz = g(t, z(t), u(t)), dt

(1)

where z(t) is the state vector of the object (system), and u(t) is a function that was determined by our choice, as well as perturbations of the system that we cannot influence. The dependence of the right-hand side of (1) on the perturbing (or control) action changes the properties of the sets of solutions; therefore, we first proceed to the next problem with the initial data dz = f (t, z(t)) , t ∈ I := [t0 , T ] , dt z(t0 ) ∈ Z 0 ,

(2) (3)

A. N. Rogalev (B) Institute of Computing Modelling, Akademgorodok, build.90, Krasnoyarsk, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_40

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where Z 0 —is some set, for example, set defined by the boundaries for each component of the solution vector, or by the formula. Let there be at least one solution to this problem y on the interval I belonging to the corresponding function space. It is required to define (to construct boundaries) the set of solutions Z (t) such that z(t, z 0 ) ∈ Z (t)

(4)

on the interval t ∈ I and for all solutions z(t, z 0 ) of the problem (2), (3). The set Z (t) is represented as any admissible set for which it is easy to determine the boundaries (ball, sphere, n—dimensional (rectangular) box). The inclusion (4) will be dense at some point in time t ∈ I at the inclusion of initial value (3) if ∀z ∈ Z (t) ∃z 0 ∈ Z 0 : z = z(t, z0 ), i.e. the solutions z(t, z0 ) completely fill the range of values of the multivalued function Z (t) for y0 in Y0 . We introduce the concepts of the internal and boundary points of the set of solutions. Let a system of ordinary differential equations be given dy = f (t, y(t)) dt

(5)

and a set of initial data densely filling the initial data region on the plane Y0 . For each problem (5), the conditions for the existence and uniqueness of solutions are satisfied, then a unique solution (a single integral curve) leaves the point of the domain (set) of solutions. The set of all solutions to this problem will contain the internal points of the domain (set) of solutions and the boundary points of the domain (set) of solutions. For set valued solutions (like set valued mappings), you can enter the distance [3] d( f, g) = sup {ρ ( f (s) , g (s)) : s ∈ X } , where ρ (·, ·) is the Euclidean metric, and X —an arbitrary set is the solution domain. Definition 1 A solution y in (t) is called an internal solution of the region of solution if there exists ε—a neighborhood of the solution that is included in the solution domain. Definition 2 The solution y outer (t) is called the boundary solution of the region of solution if any ε—neighborhood of this solution will contain elements (points) that are not a solution to the problem. If the right-hand side in the problem (5) is a linear function, then the mapping from the set of initial data to the value range of the solutions of the problem (5) will be the affine mapping [6]), which displays convex set to convex set, that is, the boundary of the set of initial data goes to the boundary of the set of solutions at the time t. Theorem 1 Let the right-hand side of the f system (2) be the continuous and linear function, and the set y0 of initial values form a polyhedron with the vertices y01 , y02 , . . . , y0m ∈ {Y0 }. Choosing y(0) = y0k , as initial conditions k = 1, . . . , m we

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  1 2 y) of solutions to find the solutions  y , y , . . . ,  y m of problem  (2). Then the set ( y2, . . . ,  y m as vertices problem (2), (3) has points  y1,  Proof of the theorem based on the linearity of right side. Any initial value y0∗ ∈ Y 0 will map to internal value of solution set at any t. Any initial value of bound solutions y0bound ∈ Y 0 will map to boundary solution in solution set at any t. Let the right-hand side in the problem (5) f (t, y) be a nonlinear function, f, y are elements of some Banach space E. The problem (5) is equivalent t to the integral equation y(t) − y0 − t0 f (τ, y(τ ))dτ = 0. We write this equation in the form F(y0 , y(τ )) = 0. Thus, F is an operator that maps the direct sum of the space E and the space C E1 [t0 , t1 ] of continuously differentiable functions with values in E to the space C E1 [t0 , t1 ]. If the function f (t, y) is continuous and has a derivative continuous in (t, y), then the expression  y(t) −

t

f (τ, y(τ ))dτ

(6)

t0

defines a differentiable map of the space C E1 [t0 , t1 ] into itself. Therefore, F(y0 , y(t)) is an operator differentiable with respect to y(t), and since y0 is included in F(y0 , y(t)) is additive, then F is a differentiable function on E × C E1 [t0 , t1 ]. The differential of this function with respect to y has the form Fy h = h(t) − t  t0 f y (τ, y(τ ))h(τ )dτ . The right-hand side of the equality (6) defines an operator that maps C E1 [t0 , t1 ] into itself. This operator is invertible. Indeed, for any function z(t) ∈ C E1 [t0 , t1 ] the equation Fy h(t) = z(t), 

or h(t) −

t0

t

f y (τ, y(τ ))h(τ )dτ = z(t),

equivalent to the differential equation dh(t) − f y (τ, y(t))h(t) = z  (t) dt

(7)

with the initial condition h(t0 ) = z(t0 ). The equation(7) is a linear equation with continuous coefficients; therefore, by well-known theorems [5] there is a unique solution to this equation, defined on the interval [t0 , t1 ] and satisfying the initial condition specified above, which means the invertibility of the operator Fy . The obtained result means that the implicit function theorem can be applied to the equation F(y0 , y(t)) = 0.

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By virtue of this theorem, the solution y = y(t) of this equation, which can be considered as a function of the variable initial value y0 : y = y(t, y0 ), is differentiable with respect to y0 . It follows that in the case of nonlinear ODE systems (5), each internal point of the solution domain goes to the internal point of the solution domain, which has a neighborhood composed again of the solutions of the original system. For ODE systems, the conditions for the existence and uniqueness of solutions are satisfied; therefore, for boundary points lying on the boundary of the solution domain, the only possibility remains, and the boundary points will be reflected in the boundary points of this set of solution along the trajectories defined by the equation (5). This means that for many problems it is useful to study the behavior of the points of the set of solution (integral curves) starting at the boundary of the set of initial data.

2 Estimates of the Solutions of ODE Systems with a Change (Disturbance) in the Initial Data For most ODE systems in practical problems, solutions must exist for a finite period of time, and it is necessary to study not only the fact of their stability or instability, but also quantitative estimates of these solutions, as well as the acceptability of these estimates in real conditions. Lyapunov stability is a qualitative characteristic of the behavior of systems on an infinite time interval and does not always characterize the behavior of a system on a finite interval of time with respect to predetermined regions in the state space. Arnold [2] stated that to describe all the solutions of ODE systems with close parameters (initial data) it is enough to calculate the derivative of the solution with respect to the parameter (according to the initial data). In the theory of ordinary differential equations and in their applications for many problems, it is necessary to evaluate the perturbations of solutions when the right-hand sides of ODE systems change, as well as the initial conditions. The simplest such estimates can be found in many works, but these estimates increase with time. This contradicts practice. In [1] for an ODE system (5) with a fairly smooth right-hand side, a shift operator along the path is introduced, denoted by Sτt y. The solution to this system, for t = τ , passes through the point y; thus Sττ y = y. Operator Sτt y = y exists on time interval T . This interval depends on the choice of y and τ . We now give the properties of the shift operator along the trajectory of the ODE system. The notion of a shift operator along the trajectories of the differential equation [4] is close to the notion of a general solution of an ODE (ODE system). Both of these definitions are not expressed by an explicit formula, but the properties of the shift operator can be derived directly from the properties of the right-hand side of the system. For the purpose of interpreting the properties of sets of ODE solutions, we can interpret the system (7) [6] as the law of motion of the image of point y in the phase space R n . Then the set of the system (7) will correspond in R n to an ensemble of points reflecting the mechanical state of the system. Let all the image points of the

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ODE system, denoted by Y0 at the initial time t = t0 , form a compact set. Any of these points y0 ∈ Y0 , shifting along the trajectories of the ODE system (7), will fall into the new state y(t) = T (t0 , t)y 0 ∈ Y t over the time from t0 to t, where T (t, t 0 ) is the shift operator along the trajectories of the system (7); Zt —is image of the set Z0 by the system (7); Zt = T(t0 , t)Z0 . Definition 3 The trajectory shift operator (Cauchy operator) of a system of ordinary differential equations dy = f (t, y), y ∈ Rn dt is called the operator S, depending on the parameters θ and τ , such that S(θ, τ ) : Rn → Rn , and corresponding to the value of every solution y(t) of the system at the point t = θ the value of the same solution at the point t = τ : S(θ, τ ) y(τ ) = y(θ ). For a nonlinear system that satisfies the conditions of the existence and uniqueness theorem for a solution to the Cauchy problem, similar equalities are also valid with due reservations regarding the domain of definition of the operators included in it. Under the assumptions made, Sτt y is differentiable with respect to t, τ, y. By definition,   ∂ t

Sτ y = F t, Sτt y . ∂t Next, we introduce the notation J (t, y) =

∂ ∂  t  S y , F(t, y), ϕ (τ, t, y) = ∂y ∂τ τ A (t, τ, y) =

∂  t  S y . ∂y τ

From here, it is possible to get that   ∂ A (t, τ, y) = J t, Sτt y A, r m A(τ, τ, y) = E, ∂t   = J t, Sτt y ϕ, ϕ(τ, τ, y) =  −F(τ, y). Based on the concepts introduced in y0 − Stt0 y0  ≤ δ(t), for all t ∈ [t0 , τ ), where | y0 − y0 | ≤ δ(t0 ), [1], the estimates  Stt0 d y N (t)—some estimate of the defect (t, y) = dt − F(t, y(t)) when substituting into | (t, the formula instead of the exact approximate solution, x)| ≤ N (t), |A(t1 , t, y)|  ∂ t  =  ∂ y St1 y  ≤ (t1 , t)—estimate of the derivative of the shift operator along the trat jectory. The deviation of the perturbed solution δ(t) = (t0 , t) δ (t0 ) + t0 (τ, t) N (τ )dτ. The notation can be simplified. Denote by y and z the solutions of the problem ∂ϕ ∂t

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dy = f (t, y), y(t0 ) = y0 dt

(8)

dz = f (t, z) + g(t, z), z(t0 ) = z0 , dt

(9)

respectively, and suppose that ∂∂ yf exists and is continuous. Then the solutions of the system (8) and the “perturbed” system (9) are related by the relation z(t) = y(t)+ t 1 + 0 ∂∂yy0 (t0 , t, y0 + s (z 0 − y0 )) · (z 0 − y0 ) ds + t0

∂y ∂ y0

(t0 , s, z(s)) · g (s, z(s)) ds.

The previous formula allows you to perform a certain correction of the estimates of the sets of solutions of ODE systems. In addition, knowledge of estimates of the values of derivatives with respect to the initial data is used in a formula that estimates changes in the Lebesgue measure of the set of solutions vol( Yt ) =

 t t0

t

 ∂ F1 ∂ F2 ∂ F3  + + dydτ + vol( Yt0 ). ∂ y1 ∂ y2 ∂ y3  y=y(y0 ,t0 ,t)

(10)

The estimates constructed in the article compute the range of all possible values of the solution sets, that is, they investigate practical stability. A formula is obtained that estimates the divergence or condensability of the trajectories, which is the basis for the study of methods for including solutions.

References 1. Alekseev, V.M. On an estimate of perturbations of solutions of ordinary differential equations. I. Bulletin of Moscow University. A series of mathematics and mechanics. No.2, 28-36 (1961) 2. Arnold, V.I.: Ordinary differential equations. Nauka, Moskva (1975) 3. Filippov, V.V.: Set of solutions of ordinary differential equations. MGU, Moskva (1993) 4. Krasnoselsky, M.A.: Operator of shift along trajectories. Nauka, Moskva (1966) 5. Kolmogorov, A.N., Fomin, S.V.: Elements of function theory and functional analysis. Nauka, Moskva (1976) 6. Rashevsky, P.K.: Riemannian geometry and tensor analysis. Nauka, Moskva (1967)

Two-Phase Computational Model for Small-Amplitude Wave Propagation in a Saturated Porous Medium Evgeniy Romenski, Galina Reshetova, Ilya Peshkov and Michael Dumbser

Abstract The new two-phase model for compressible fluid flows in nonlinear poroelastoplastic media is presented. The derivation of the model is based on the symmetric hyperbolic thermodynamically compatible systems theory, which is developed with the use of the first principles and fundamental laws of irreversible thermodynamics. The governing PDEs form the first-order hyperbolic system and can be used for studying a wide variety of processes in saturated porous medium, including smallamplitude wave propagation. The theory predicts the three types of waves: fast and slow pressure waves and a shear wave, as it is in Biot’s model. The material constants of the model are fully determined by the properties of the solid and fluid constituents, and unlike Biot’s model do not contain empirical parameters. The governing PDEs for small-amplitude wave propagation in a saturated porous medium are presented and studied by means of the efficient numerical method, which is based on the finite difference fourth-order staggered-grid discretization.

1 Introduction The modeling of fluid flows in porous media is of permanent interest in many geophysical and industrial applications. The starting point of research developments in this field was the series of pioneering work of Bio [1–3], in which a model of elastic E. Romenski (B) Sobolev Institute of Mathematics, Novosibirsk, Russia e-mail: [email protected] G. Reshetova Institute of Computational Mathematics and Mathematical Geophysics, Novosibirsk, Russia I. Peshkov Institut de Mathématiques de Toulouse, Université Toulouse III, Toulouse, France M. Dumbser Department of Civil, Environmental and Mechanical Engineering, University of Trento, Trento, Italy © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_41

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wave propagation in saturated porous media was proposed. Some modification and generalization of the model have been done in the past (see, for example [4, 11] and references therein) and at present, Biot’s approach is the commonly accepted and widely used in geophysical community. Nevertheless, many actual technological and scientific problems such as geothermal energy extraction, CO2 storage, and hydraulic fracturing require a development of new advanced models of fluid flow in poroelastic media admitting the use of complex equations of state of the fluid and nonlinear elastic and elastoplastic deformations of the solid matrix. In this paper, based on the theory of Symmetric Hyperbolic Thermodynamically Compatible (SHTC) systems [8, 9, 15], SHTC model of compressible multi-phase flows [14, 16–18], and a recently developed unified SHTC model of Newtonian continuum mechanics [5, 6], we propose a new finite strain poroelastoplasticity model. The interfacial friction between liquid and solid phases and shear stress relaxation are implemented in the model as relaxation-type source terms in accordance with the laws of thermodynamics. The linearized version of the model is applied to study small-amplitude waves in a saturated poroelastic medium.

2 Thermodynamically Compatible Master System for Two-Phase Saturated Porous Medium The following PDE master system for two-phase solid–fluid flows can be derived as a subsystem of the SHTC master system studied in [7–9, 15], e.g., see [14, 16–18]: ∂(ρvi vk + pδik + wi E wk − σik ) ∂ρvi + = 0, ∂t ∂ xk   ∂ Ai j ∂ Aik ∂ Aik ψik ∂ Aim vm =− + , + vj − ∂t ∂ xk ∂x j ∂ xk θ ∂ρ ∂ρvk ∂ρα1 vk ∂ρα1 ρϕ + + = 0, =− , ∂t ∂ xk ∂t ∂ xk θ1 (∂ρc1 vk + ρ E wk ) ∂ρc1 + = 0, ∂t ∂ xk  k  ∂w λk ∂(wl vl + E c1 ) ∂wl ∂wk =− , + + vl − ∂t ∂ xk ∂ xl ∂ xk θ2 k ∂ρsv ρ ρ 2 ρ ∂ρs + ψik ψik + ϕ + λk λk ≥ 0, = ∂t ∂ xk θT θ1 T θ2 T

(1a) (1b) (1c) (1d) (1e) (1f)

Here, the poroelastoplastic medium saturated by a fluid is considered as a solid–fluid mixture characterized by the volume fraction α1 of the liquid component, which also can be identified with the porosity usually denoted as φ. Then, α2 = 1 − α1 is the volume fraction of the solid component. The mixture density ρ is connected with the phase mass densities ρ1 and ρ2 via ρ = α1 ρ1 + α2 ρ2 . Parameter c1 represents the

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mass fraction of the liquid component, vi = c1 v1i + c2 v2i , i = 1, 2, 3 is the velocity of the mixture, and wi = v1i − v2i is the relative velocity between the two phases. Here, we consider a single entropy approximation of the two-phase medium, and introduce the entropy of the mixture s. Finally, as a deformation measure of an element of the mixture, we consider the distortion matrix A [5], while the tangential stress σik and temperature T are defined as σi j = −ρ Aki ∂∂AEk j , T = ∂ E/∂s. Also, θ , θ1 , and θ2 are relaxation parameters. In order to apply system (1) to a certain physical situation, the total energy of the system E has to be specified. Here, we use the following energy potential: E = E 1 (α1 , c1 , ρ, s) + E 2 (c1 , ρ, s, A) + E 3 (v) + E 4 (c1 , w),

(2)

where E 1 = c1 e1 (ρ1 , s) + c2 e2 (ρ2 , s), E 3 = 21 v2 , E 4 = c1 c2 21 w2 , and the energy of shear deformation E 2 is constructed in the following way. In order to provide zero trace of the stress tensor tr(σ ) = 0, we take E 2 depending on A via the normalized strain tensor g = a T a, where a = A/(det A)1/3 . It gives us g = G/(detG)1/3 , where  G = AT A. Then, the energy E 2 can be defined as in [13], E 2 = 18 cs2 tr(g 2 ) − 3 , where cs is the shear velocity of sound at the reference conditions. With the use of the above definition, we can compute the derivative: ∂∂ EA = ∂∂EA2 =   2 c2 −ρ −1 F T σ = 2s F T g 2 − tr(g3 ) I , where F = A−1 . Then, the shear stress is trace  2 c2 free tr(σ ) = 0 and reads as σ = −ρ 2s g 2 − tr(g3 ) I . Details of the computations of other derivatives of the energy potential can be found, for example, in [16, 18]. As soon as we define the total energy, the closing relations in the phase interaction source terms are defined via the thermodynamic forces ψik = E Aik , λk = E wk , ϕ = E α1 . The solution to system (1) satisfies the energy conservation law which reads   ∂ ρvk E + vi ( pδik + ρwi E wk − σik ) + ρ E c1 E wk ∂ρ E + = 0. ∂t ∂ xk

(3)

3 Small-Amplitude Waves in a Saturated Porous Medium Small-amplitude wave propagation can be described by the PDE system obtained as a linearization of (1) with the coefficients defined in the equilibrium state of the original system. Thus, we assume that the medium with a given volume fractions of constituents is at rest and under the reference conditions, which means that its parameters of state are v1k = v2k = 0, ρ1 = ρ10 , ρ2 = ρ20 , α1 = α10 , α2 = α20 = 1 − α10 , Ai j = δi j , and s = 0. We assume also that the phase pressure relaxation to the common pressure is instantaneous due to the small scale of pore space in the medium. Thus, the resulting linear system can be written in terms of the mixture velocities, relative velocities, pressure, and shear stress:

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 ∂wk ∂vk ∂P ∂P ∂vi ∂sik α0 α0  +K + + 1 0 2 ρ20 − ρ10 K = 0, ρ 0 − α20 = 0, ∂t ∂ xk ρ ∂ xk ∂t ∂ xi ∂ xk (4a)   i  0 0 k j k ∂sik ∂v 1 ∂v 2 ∂v 1 ∂P c c ∂w + − δik − 0 = − 1 2 wk , , =μ + 0 ∂t ∂ xk ∂ xi 3 ∂x j ∂t θ2 ρ1 ρ2 ∂ xk (4b)  −1 where sik = σik /α20 , K = α10 K 1−1 + α20 K 2−1 , μ = ρ20 cs2 , and τ = shear-stress relaxation time.

ρ0θ 2ρ20 cs2

is the

4 Numerical Examples 4.1 Dispersion Relation Here, we compare the dispersive properties of the solution to the SHTC system (4) and the one-dimensional first-order form of the governing equations of the so-called “low-frequency limit” of the Biot theory [4, 11] ∂p η ∂vs T ∂q + ρf + = − q, ∂t φ ∂t ∂x κ ∂q ∂σ ∂vs − (λu + 2μu ) − αM = 0, ∂t ∂x ∂x ρf

∂vs ∂σ ∂q + ρf − = 0, ∂t ∂t ∂x ∂p ∂vs ∂q + αM +M = 0, ∂t ∂x ∂x

ρ

(5a) (5b)

where φ is the porosity, vs and q = φ(vf − vs ) are the solid and fluid (relative to the solid) particle velocities, and σ and p are the bulk stress and fluid pressure, respectively. Additionally, ρf and ρs are the fluid and solid densities and ρ = (1 − φ)ρs + φρf is the total (mixture) density. Furthermore, the material parameters are defined as λu = K u − 23 μu = K m + α 2 M − 23 μu , M = B K u /α, α = 1 − K m /K u , K u = K m /(1 − α B), B = (1/K m − 1/K s )(1/K m − 1/K s + φ(1/K f − 1/K s ))−1 , where the subscript “u” denotes the quantities characterizing the undrained solid matrix, while the subscript “m” denotes the quantities characterizing the dry matrix. Thus, λu , K u , and μu are the undrained Lamé parameter, bulk modulus, and the shear modulus of the undrained matrix, while K m is the bulk modulus of the dry matrix. Additionally, K s and K f are the bulk moduli of the grains and fluid, while B is the Biot parameter and T is the tortuosity. Also, it is implied that the shear modulus of the undrained and dry matrix coincide, μu = μm . Dispersion relations for the nonhomogeneous first-order hyperbolic systems (4) and (5) can be obtained in the standard way, e.g., see [5]. We plot the phase velocities Vph (ω) = ω/Re(k) and attenuation factor per wavelength aλ (ω) = −2π Im(k)/ Re(k) of the fast and slow sound waves for the both models in Fig. 1. Here, ω is the real frequency and k is the complex wave number. One may note the difference

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Fig. 1 Phase velocities (first row) and attenuation factors per wavelength (second row) for the SHTC and Biot models. The material parameters are as follows. Grain: ρs = ρ2 = 2500 kg/m3 Cs = 4000 m/s for the Biot and 4332 m/s for the SHTC model, Csh = 3787 m/s, K s = K 2 = ρ2 Cs2 = 40, 2 for the Biot and 46.9 GPa 35.85 GPa for the SHTC model, κ = 2 · 10−14 m2 , θ = μs = μ = ρ2 Csh 2 9.5 · 10−7 s. Matrix: K m = 9.48 GPa, μm = 24.5 GPa, T = 3.75. Fluid: ρf = ρ1 = 1040 kg/m3 , Cf = 1500 m/s, K f = K 1 = ρf Cf2 = 2.34 GPa, η = 10−3 Pa· s,

in the high-frequency limit of the fast modes which is, in fact, due to the discrepancy in the characteristic speeds Cfast and Cslow (eigenvalues of the homogeneous systems) depicted in Fig. 1 by the dashed lines. Indeed, recall that the eigenvalues of the homogeneous hyperbolic system are the high-frequency limits (ω → ∞) of the sound speeds (eigenvalues of the nonhomogeneous system) [12]. The slow mode dispersion curves of the both models are almost indistinguishable.

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4.2 Two-Dimensional Wave Propagation To discretize the governing equations (4), the difference scheme for the velocity– stress formulation, proposed for elasticity equations [10, 19], is used. The choice of the numerical scheme on staggered grids looks very natural due to the fact that equations form the symmetric first-order system of evolution equations for the components of mixture and relative velocities, pressure, and shear stress. A forcing function f (t, x, y) = (1 − 2π 2 f 02 (t − t0 )2 )ex p[−π 2 f 02 (t − t0 )2 ] is introduced as the source term on the right-hand side of the pressure equation or the equations for the normal components of the deviatoric stress in system (4). It is defined as the product of Dirac’s delta in space and Ricker’s wavelet in time where f 0 is the source peak frequency and t0 is the wavelet delay. Figure 2 depicts the solution to the SHTC model (4) for different values of the source peak frequency f 0 . Parameters of the medium were taken the same as in Fig. 1. For each time–frequency, a snapshot is recorded at the time 5/ f 0 (including the shift wavelet delay 1/ f 0 ) for the square domains with the side of 5 m (for f 0 = 104 ), 0.5 m (for f 0 = 105 ), and 0.05 m (for f 0 = 106 ). The time and size are chosen in such a way that in the isotropic elastic case we can obtain three identical snapshots. As it is expected for the poroelastic case, we observe wavefield differences which are most clearly seen in seismograms: the lower the frequency, the stronger the dispersion and attenuation of the slow P-wave. -2.5

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Acknowledgements Results by E.R. and G.R. obtained in Sects.2–4 were supported by the Russian Science Foundation grant (project 19-77-20004), whereas results obtained in Sect. 5 were supported by RFBR grant 19-01-00347. I.P. greatly acknowledges the support by Agence nationale de la recherche (FR) (Grant No. ANR-11-LABX-0040-CIMI) within the program ANR-11-IDEX-000202. The work by M.D. was partially funded by the European Union’s Horizon 2020 Research and Innovation Programme under the project ExaHyPE. Furthermore, MD acknowledges funding from the Italian Ministry of Education, University and Research (MIUR) in the frame of the Departments of Excellence Initiative 2018–2022 attributed to DICAM of the University of Trento, as well as financial support from the University of Trento in the frame of the Strategic Initiative Modeling and Simulation.

References 1. Biot, M.: Theory of propagation of elastic waves in a fluid-saturated porous solid. II. higher frequency range. J. Acoust. Soc. Am. 28(2), 179–191 (1956) 2. Biot, M.: Theory of propagation of elastic waves in fluid-saturated porous solid. I. lowfrequency range. J. Acoust. Soc. Am. 28(2), 168–178 (1956) 3. Biot, M.: Mechanics of deformation and acoustic propagation in porous media. J. Appl. Phys. 33(4), 1482–1498 (1962) 4. Carcione, J.M., Morency, C., Santos, J.E.: Computational poroelasticity-A review. Geophysics 75(5), 75A229–75A243 (2010). https://doi.org/10.1190/1.3474602 5. Dumbser, M., Peshkov, I., Romenski, E., Zanotti, O.: High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: Viscous heat-conducting fluids and elastic solids. J. Comput. Phys. 314, 824–862 (2016). https://doi.org/10.1016/j.jcp.2016. 02.015 6. Dumbser, M., Peshkov, I., Romenski, E., Zanotti, O.: High order ADER schemes for a unified first order hyperbolic formulation of Newtonian continuum mechanics coupled with electrodynamics. J. Comput. Phys. 348, 298–342 (2017). https://doi.org/10.1016/j.jcp.2017.07.020 7. Godunov, S.K.: An interesting class of quasilinear systems. Dokl. Akad. Nauk SSSR 139(3), 521–523 (1961) 8. Godunov, S.K., Mikhaîlova, T.Y., Romenskiî, E.I.: Systems of thermodynamically coordinated laws of conservation invariant under rotations. Sib. Math. J. 37(4), 690–705 (1996). https:// doi.org/10.1007/BF02104662 9. Godunov, S.K., Romenski, E.I.: Thermodynamics, conservation laws, and symmetric forms of differential equations in mechanics of continuous media. In: Computational Fluid Dynamics Review 95, pp. 19–31 John Wiley, NY (1995) 10. Levander, A.R.: Fourth-order finite-difference P-SVseismograms. Geophysics 53(11), 1425– 1436 (1988). https://doi.org/10.1190/1.1442422 11. Masson, Y.J., Pride, S.R., Nihei, K.T.: Finite difference modeling of Biot’s poroelastic equations at seismic frequencies. J. Geophys. Res.: Solid Earth 111, B10305 (2006). https://doi.org/10. 1029/2006JB004366 12. Muracchini, A., Ruggeri, T., Seccia, L.: Dispersion relation in the high frequency limit and non linear wave stability for hyperbolic dissipative systems. Wave Motion 15(2), 143–158 (1992) 13. Ndanou, S., Favrie, N., Gavrilyuk, S.: Criterion of hyperbolicity in hyperelasticity in the case of the stored energy in separable form. J. Elast. 115(1), 1–25 (2014). https://doi.org/10.1007/ s10659-013-9440-7 14. Peshkov, I., Grmela, M., Romenski, E.: Irreversible mechanics and thermodynamics of twophase continua experiencing stress-induced solid-fluid transitions. Contin. Mech. Thermodyn. 27(6), 905–940 (2015). https://doi.org/10.1007/s00161-014-0386-1 15. Peshkov, I., Pavelka, M., Romenski, E., Grmela, M.: Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations. Contin. Mech. Thermodyn. 30(6), 1343–1378 (2018). https://doi.org/10.1007/s00161-018-0621-2

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16. Romenski, E., Drikakis, D., Toro, E.: Conservative Models and Numerical Methods for Compressible Two-Phase Flow. J. Sci. Comput. 42(1), 68–95 (2010) 17. Romenski, E., Resnyansky, A.D., Toro, E.F.: Conservative hyperbolic model for compressible two-phase flow with different phase pressures and temperatures. Q. Appl. Math. 65(2), 259–279 (2007) 18. Romenski, E., Belozerov, A.A., Peshkov, I.M.: Conservative formulation for compressible multiphase flows. Q. Appl. Math. 74(1), 113–136 (2016). https://doi.org/10.1090/qam/1409 19. Virieux, J.: P-SV wave propagation in heterogeneous media; velocity-stress finite-difference method. Geophysics 51(4), 889–901 (1986). https://doi.org/10.1190/1.1442147

Godunov Symmetric Systems and Rational Extended Thermodynamics Tommaso Ruggeri

Abstract The compatibility between balance laws and the entropy principle implies that any hyperbolic system becomes a Godunov symmetric system if we choose as variables the main field. The consequence is the well-posedness of Cauchy problem. Rational Extended Thermodynamics (RET) is a theory for nonequilibrium gases that belongs in this mathematical framework and is motivated at the mesoscopic scale by the kinetic theory. We present in this contribution some recent results and some open problems on RET for both monatomic and polyatomic gases.

1 Balance Laws, Entropy Principle, and Godunov Systems The physical laws in continuum theories are expressed by balance laws. By assuming the required regularity, and considering constitutive equations of local type, the system becomes a first-order quasi-linear system of N balance laws for the R N unknown vector u(x α ) ; (x α ≡ (t, x i ), α = 0, 1, 2, 3; i = 1, 2, 3; ∂α = ∂/∂ x α ) : ∂α Fα (u) = f(u).

(1)

The system (1) becomes a particular case of generic quasi-linear first-order systems: Aα (u)∂α u = f(u),

(2)

where the matrices Aα are Aα = ∂Fα /∂u. Def. (Friedrichs Symmetric Systems) The system (2) is said symmetric if all the matrices Aα are symmetric, and moreover A0 is positive definite (every symmetric systems are hyperbolic). 

T. Ruggeri (B) Department of Mathematics and AM2 - Research Center of Applied Mathematics, Alma Mater University of Bologna, Bologna, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_42

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Godunov in a well-known paper of 1961 [1] was able to find a privileged field for which any system coming from a variational principle can be put in a symmetric form, and also he found the field for which the Euler system of a compressible fluid assumes the symmetric form. Now we suppose that all the solutions of the system of balance laws (1) satisfy also a supplementary scalar law with production term not negative (entropy principle): ∂α h α (u) = (u) ≥ 0,

(3)

where the function −h 0 (u) is a strictly convex function of the field u ≡ F0 in an open domain D ⊆ R N . This kind of problem was first studied by Friedrichs and Lax in 1971, and they were able to prove that there exists a N xN symmetric matrix H(u), such that the new system obtained by multiplying (1) by H : H(u) · {∂α Fα (u) − f(u)} = 0

(4)

is symmetric [2]. The important results of these authors have, however, the disadvantage that in the new system (4) the structure of balance laws is lost. Therefore, it is impossible to define the usual weak solutions and to study shocks in particular. In 1974, Boillat [3] proved that the original system (1) becomes symmetric if we choose the field u = ∂h 0 /∂u, with u ≡ F0 and h 0 convex function of u. In this way, he extended and completed Godunov’s idea to general systems which are compatible with a supplementary law (3) and which are not necessarily coming from a variational Euler–Lagrange system. In 1981, Ruggeri and Strumia [4] were interested in extending this technique to a relativistic case by using a covariant formulation. But they realized that it is impossible to define u in the same manner as before because h 0 is the temporal part of the four vector h α and therefore is not a scalar invariant. Also, F0 is not a vector. They observed that (1) and (3) form an overdetermined quasi-linear hyperbolic system. Thus, in order for any smooth solution of (1) to satisfy the entropy law (3), the equation (3) must be obtained as a linear combination of the equations of system (1): there exists a set of Lagrange multipliers, u ∈ R N such that ∂α h α −  ≡ u · (∂α Fα − f).

(5)

Since (5) is an identity, by comparing the differential terms and inhomogeneous term, we have  = u · f ≥ 0. (6) dh α = u · dFα , α = 0, . . . , 3, If we introduce the potentials h α defined as (h 0 is the Legendre transform of h 0 ) h α = u · Fα − h α ,

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then it follows from (6) that dh α = du · Fα . Choosing as a new field u , we have Fα =

∂h α , ∂u

(7)

and inserting (7) into (1), the original system (1) becomes a special symmetric one: 

∂ 2 h α ∂u ∂u



∂α u = f(u ).

(8)

As u is the privileged field for which the original system becomes symmetric, such field was proposed by Ruggeri and Strumia to call main field. This approach reduces to the one obtained by Boillat in a classical context. The system in the form (8) is known as a Godunov System. The interested readers can find in Chap. 2 of Ruggeri– Sugiyama book [5] a brief history of the symmetrization problem. Symmetric systems have the advantage to have the local Cauchy problem wellposed. Moreover, for symmetric systems compatible with an entropy law that satisfy also the so-called K- condition (Kawashima and Shizuta [6]), it is possible to prove the global existence for the smooth solution provided initial data are sufficiently small [7–9] and the equilibrium states are stable [10].

2 Rational Extended Thermodynamics and the Kinetic Theory An example of the Godunov system is the one given by the Rational Extended Thermodynamics (RET) theory, which describes the processes of nonequilibrium gases. RET is motivated at mesoscopic scale by the kinetic theory that describes the state of a rarefied gas through the phase density f (x, t, c), where f (x, t, c)dc is the number density of atoms at point x and time t that has velocities between c and c + dc. The phase density obeys the Boltzmann equation: ∂f ∂f + ci = Q, ∂t ∂ xi

(9)

where Q represents the collisional term. Most macroscopic thermodynamic quantities are identified as moments of the phase density  F=

 f dc, R3

Fk1 k2 ···k j =

R3

f ck1 ck2 · · · ck j dc.

We can define the moments (10) using a multi-index: FA = F for

A = 0,

FA = Fk1 k2 ···k A for

1 ≤ A ≤ N.

(10)

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Due to the Boltzmann equation (9), the moments satisfy an infinite hierarchy of balance laws in which the flux in one equation becomes the density in the next one: ∂t FA + ∂i Fi A = PA ,

A = 0, . . .

By taking into account that P, Pi and Pkk = 0, the first five equations are conservation laws and coincide with the mass, momentum, and energy conservation, respectively, while the remaining ones are balance laws. When we cut the hierarchy at the density with the tensor of rank N , we have the problem of closure because the last flux and the production terms are not in the list of the densities. The first idea of RET (see Müller and Ruggeri [11, 12]) was to view the truncated system as a phenomenological system of continuum mechanics and then we consider the last flux and the production terms as local constitutive functions of the previous moments. According to the continuum theory, the restrictions on the constitutive equations come from universal principles, i.e., Entropy principle, Objectivity Principle, and Causality and Stability (convexity of the entropy). If the number of moments increases, it is too difficult to adopt the pure continuum approach for a system with such a large number of field variables. Therefore, it is necessary to recall that the field variables are the moments of a distribution function truncated at some order. And then the closure of the balance equations of the moments, which is known as the Maximum Entropy Principle (MEP), should be introduced. This is the procedure of the so-called Molecular Extended Thermodynamics (MET) [11, 12]. The principle of maximum entropy has its root in statistical mechanics. It is developed by Jaynes in the context of the theory of information basing on the Shannon entropy. MEP states that the probability distribution that represents the current state of knowledge in the best way is the one with the largest entropy. Concerning the applicability of MEP in nonequilibrium thermodynamics, this was originally given by Kogan (1967). The equivalence between MEP and RET with 13 moments was proved for the first time by Dreyer in 1987 [13]. In this way, the 13-moment theory can be obtained in three different ways: RET, Grad, and MEP. A remarkable point is that all closures are equivalent to each other! The MEP procedure was then generalized by Müller and Ruggeri to the case of any number of moments in the first edition of their book (1993) [11], proving as first that the closed system is symmetric if one chooses as field the Lagrange multipliers. The complete equivalence between the entropy principle and the MEP was finally proved by Boillat and Ruggeri in 1997 by the identification between the Lagrange multipliers and the main field [14]. The variational problem, from which the distribution function f renders the entropy  h = −k B

f log f dc R3

(k B is the Boltzmann constant) maximum for the prescribed moments, is obtained through the functional (we omit the symbol of sum from 0 to N for the repeated capital indexes A, B, . . .):

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 L N ( f ) = −k B

R3

  f log f dc + u A FA − m

R3

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 c A f dc ,

where u A are the Lagrange multipliers: u A = u  for A = 0, u A = u k1 k2 ···k A for

1 ≤ A ≤ N.

The distribution function f N that maximizes the functional L N is given by  m  f N = exp −1 − χ N , k

χ N = u A c A .

(11)

In an equilibrium state, (11) reduces to the Maxwellian distribution function f (M) . Then, the system may be rewritten as follows: J AB ∂t u B + Ji AB ∂i u B = PA (u C ),

A = 0, . . . , N

(12)

where   m2 J AB u C = − kB

 R3

f N c A c B dc,

  m2 Ji AB u C = − kB

 R3

f N ci c A c B dc.

Because of the fact that the matrices J AB , Ji AB are symmetric with respect to the multi-indexes A, B and J AB is definite negative, the system (12) is symmetric hyperbolic and the Lagrange multipliers coincide with the main field according to the general theory of systems of balance laws with a convex entropy density that we discussed before. We observe that f N is not a solution to the Boltzmann equation. But we have the conjecture (open problem) that, for N → ∞, f N tends to a solution of the Boltzmann equation. The RET has been successful because it explains experimental data well (sound waves in high frequencies, light scattering, shock waves) [12]. Nevertheless, previous RET has two limitations: the theory is valid only near equilibrium and as the kinetic theory, is valid only for monatomic rarefied gas.

3 Polyatomic Gas and Recent Results The modern RET theory of rarefied polyatomic gases has been developed starting by the paper of Arima, Taniguchi, Ruggeri and Sugiyama [15]. This theory adopts the balance equations with binary hierarchy structure, where the dynamic pressure (nonequilibrium pressure) is properly taken into account. Note that the dynamic pressure vanishes in rarefied monatomic gases. The number of independent fields is now 14: the mass density, velocity, internal energy, dynamic pressure, shear stress, and heat flux. The first hierarchy consists of the balance equations for mass density, momentum density, and momentum flux, and the second hierarchy consists of the

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balance equations for energy density and energy flux. The hierarchy was justified and also derived from MET [16] by using the generalized kinetic theory where the distribution function depends also on an extra variable that represents the molecular internal degrees of freedom such as rotation and vibration. The validity of the ET14 theory of rarefied polyatomic gases has been successfully confirmed by comparing the theoretical predictions to the experimental data of linear waves, shock waves, and light scattering in the region where the Navier–Stokes and Fourier theory is no longer valid. These results are summarized in the book of Ruggeri and Sugiyama [5]. Quite recently, a RET of polyatomic gases with molecular relaxation processes was studied [17], and now RET of dense gas is coming within the range of our next study. We want to finish this brief survey to mention that other authors consider nonequilibrium theory with Godunov structure in a different approach from RET. In particular, we quote [18, 19].

References 1. Godunov, S.K.: An interesting class of quasilinear systems. Sov. Math. 2, 947 (1961) 2. Friedrichs, K.O., Lax, P.D.: Systems of conservation equation with a convex extension. Proc. Nat. Acad. Sci. USA 68, 1686 (1971) 3. Boillat, G.: Sur l’existence et la recherche d’équations de conservation supplémentaires pour les systémes hyperboliques. C. R. Acad. Sci. Paris A 278, 909 (1974) 4. Ruggeri, T., Strumia, A.: Main field and convex covariant density for quasi-linear hyperbolic systems. Relativistic fluid dynamics. Ann. Inst. H. Poincaré Section A 34, 65 (1981) 5. Ruggeri, T., Sugiyama, M.: Rational Extended Thermodynamics beyond the Monatomic Gas. Springer, Heidelberg (2015) 6. Shizuta, Y., Kawashima, S.: Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation. Hokkaido Math. J. 14, 249 (1985) 7. Hanouzet, B., Natalini, R.: Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Arch. Rational Mech. Anal. 169, 89 (2003) 8. Yong, W.-A.: Entropy and global existence for hyperbolic balance laws. Arch. Rational Mech. Anal. 172, 247 (2004) 9. Bianchini, S., Hanouzet, B., Natalini, R.: Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Comm. Pure Appl. Math. 60, 1559 (2007) 10. Ruggeri, T., Serre, D.: Stability of constant equilibrium state for dissipative balance laws system with a convex entropy. Quart. Appl. Math. 62, 163 (2004) 11. Müller, I., Ruggeri, T.: Extended Thermodynamics. Springer Tracts in Natural Philosophy, I edn. Springer, New York (1993) 12. Müller, I., Ruggeri, T.: Rational Extended Thermodynamics, 2nd edn. Springer, New York (1998) 13. Dreyer, W.: Maximization of the entropy in non-equilibrium. J. Phys. A: Math. Gen. 20, 6505 (1987) 14. Boillat, G., Ruggeri, T.: Moment equations in the kinetic theory of gases and wave velocities. Continuum Mech. Thermodyn. 9, 205 (1997) 15. Arima, T., Taniguchi, S., Ruggeri, T., Sugiyama, M.: Extended thermodynamics of dense gases. Continuum Mech. Thermodyn. 24, 271 (2011) 16. Pavi´c, M., Ruggeri, T., Simi´c, S.: Maximum entropy principle for rarefied polyatomic gases. Phys. A 392, 1302 (2013)

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17. Arima, T., Ruggeri, T., Sugiyama, M.: Extended thermodynamics of rarefied polyatomic gases: 15-field theory incorporating relaxation processes of molecular rotation and vibration. Entropy 20, 301 (2018) 18. Godunov, S.K., Romenskii, E.I.: Elements of Continuum Mechanics And Conservation Laws. Kluwer Academic/Plenum Publishers, New York (2003) 19. Peshkov, I., Pavelka, M., Romenski, E., Grmela, M.: Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations. Continuum Mech. Thermodyn. 30, 1343–1378 (2018)

Variational Inequalities in the Dynamics of Elastic-Plastic Media: Thermodynamic Consistency, Mathematical Correctness, Numerical Implementation Vladimir M. Sadovskii and Oxana V. Sadovskaya Abstract Mathematical models of the dynamics of elastic-plastic, granular, and porous media are formulated as variational inequalities for hyperbolic operators with constraints describing the transition to a plastic state and different resistance of a material to tension and compression. Based on this, the problem of generalized solutions with plastic shock waves is analyzed, integral estimates in characteristic cones of an operator are constructed, which guarantee the uniqueness and continuous dependence on initial data of the solutions of the Cauchy problem and boundary value problems with dissipative boundary conditions. Shock-capturing algorithms, suitable for computation of solutions with singularities such as strong discontinuities and discontinuities of displacements of a medium, are developed that satisfy the properties of monotonicity and dissipativity at the discrete level.

1 Introduction The elementary notion about variational inequalities follows from the consideration of the minimization problem of a convex scalar function f (u) on a convex and closed subset F of the Euclidean space. In the case of a differentiable function, the necessary and sufficient condition of minimum is fulfilled: ˜ f (u) = min f (u) u∈F ˜

⇐⇒

  u˜ − u · ∇ f (u)  0 , u, ˜ u ∈ F.

Here dot means the scalar product of vectors, tilde refers to an arbitrary variation. The simplest example is the determination of the projection u =π(u) ¯ of the  vector u¯ onto F. In this case, f (u) = |u − u| ¯ 2 and, therefore, u˜ − u · u − u¯  0. In applications, more general variational inequalities arise in which the operator is V. M. Sadovskii (B) · O. V. Sadovskaya Institute of Computational Modeling SB RAS, Akademgorodok 50/44, Krasnoyarsk, Russia e-mail: [email protected] O. V. Sadovskaya e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_43

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not potential, that is, it is not a gradient of any function. In the theory of plasticity, the fundamental von Mises principle is formulated as a variational inequality: σ : e p , σ,  σ ∈F σ : ep  

⇐⇒

 −  σ − σ ) : ep  0 ,

which means that a power of plastic dissipation achieves maximum on actual stresses. Here σ is the stress tensor,  σ is an arbitrary admissible variation, e p is the plastic strain rate tensor, colon denotes the operation of double convolution. The boundary of F in a stress space represents the yield surface of an elastic-plastic material. Von Mises principle forms a closed mathematical model of the Prandtl–Reuss elastic-plastic flow together with equations of motion and kinematic equations: ρ

 ∂σ 1 ∂v = ∇ · σ + ρ g , e p = ∇v + ∇v∗ − a : . ∂t 2 ∂t

Since the fourth-rank tensor a of elastic compliance is positive definite and has a special symmetry, this model can be represented in a matrix form:   n ∂U  i ∂U  , U ∈ F, (U − U ) · A − B − QU −G  0 , U ∂t ∂ x i i=1

(1)

where n is the space dimension, U is an unknown vector composed of the velocities and stresses, A and B i are the symmetric matrices ( A is positive definite), matrix Q is equal to zero or antisymmetric, depending on the coordinate system (Cartesian, cylindrical, etc.). If the vector U lies inside F, then, due to the arbitrariness of the variation, from (1) follows the system of equations of dynamic elasticity in the Godunov’s form [1, 2]. If U is a boundary point of F, then it follows the commonly used associated rule, in accordance with which the vector of plastic strain rate is directed along the outward normal to the boundary. Besides the Prandtl–Reuss model, the equations of the theory of linear isotropic and kinematic hardening, as well as the theory of elastic-plastic Cosserat continuum and of Timoshenko-type plates and shells can be reduced to the variational inequality in a matrix form (see, [3–5]). In the case of nonlinear hardening, a more general variational inequality arises for a quasilinear differential operator, written in terms of generating potentials (U ) and i (U ):  n ∂ϕ(U )  ∂ψi (U ) , U ∈ F, − −G 0, U ∂t ∂ x i i=1 ∂(U ) ∂i (U ) , ψi (U ) = . ϕ(U ) = ∂U ∂U

 − U) · (U



(2)

Some models of elastoplasticity under finite strains are reduced to the same form.

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Mathematical models of granular and porous media, having different resistance to tension and compression, are constructed by means of a generalized rheological approach [6]. For this, a rheological scheme is considered, which composed of three traditional elements: elastic spring, viscous damper and plastic hinge, and the fourth element—rigid contact, whose constitutive relations are formulated in terms of variational inequalities. By virtue of the equivalent definition of the projection onto a convex set as the solution of variational inequality, equations of elastic media can be reduced to the following system: ∂U  i ∂U π B − Q Uπ = G , − ∂t ∂ x i i=1 n

A

where the projection U π of vector U is calculated√onto the cone K of admissible stresses with respect to the weighted norm |U | = U · A U . The cone K does not include, for example, tensile stresses, which are impossible in an ideal granular medium. In the case of elastic-plastic material, this system must be replaced by the variational inequality   n  ∂V , V = U π ∈ F  − V ) · A ∂U − Bi −QV −G 0, V (V ∂t ∂ x i i=1

(3)

or by more general inequality for nonlinear differential operator, written in similar form in terms of generated potentials.

2 Principal Results The most important results of application of variational inequalities (1)–(3) in the plasticity theory are associated with the study of weak solutions including elasticplastic shock waves. It was shown that the system of the Prandtl–Reuss equations with the associated flow rule is dynamically incorrect. It is irreducible to a divergent form [7], therefore, it is impossible to generalize this system as a full system of the integral conservation laws. Actually, the relations on plastic shock waves were obtained in [8] on the basis of additional physical assumption about the maximum energy dissipation on the shocks. For simplicity, let us consider the inequality (1). Under certain smoothness requirements, it is reduced to a divergent form:   n  ∂U  · A ∂U − Bi − QU −G  U ∂t ∂ xi i=1 n 1 ∂(U · A U ) 1  ∂(U · B i U )  −U · QU −U ·G . − 2 ∂t 2 i=1 ∂ xi

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Integration by parts after the multiplication of both sides by a smooth finite nonnegative trial function allows to go to the consideration of generalized derivatives. Then using a well-known procedure, one can obtain relations of strong discontinuity in the form of variational inequality [4]:    − U 0 · D [U ]  0 , U

(4)

n + −  ∈ F, D = c A + i=1 where U 0 = U +U , [U ] = U + − U − , U ± , U νi B i , U ± 2 are one-sided limits of the solution on a surface of discontinuity, c is the velocity of a wave, and ν is a directed normal vector to the wave front. If U 0 is an interior point of F, then the system of equations D [U ] = 0 is fulfilled, which means that we have an usual elastic shock wave. By virtue of the convexity of F, U 0 can be a boundary point only if the whole segment with the ends U − and U + lies on the boundary. In this case, a dissipative plastic shock wave is realized. The number of different types of such waves depends on the presence of linear segments on the yield surface. For the von Mises cylinder, there is only one type with the velocity of the bulk waves. For the Treska–Saint-Venant prism, such types are several—in addition to the bulk waves, there are still waves of incomplete and complete plasticity. In the models of linearly hardening media, the number of types does not change, but the wave velocities depend on the hardening parameters [9]. Note that to obtain a full system of relations in the case of quasilinear operator is not so simple. For this, it is necessary to apply the approach of regularization developed in [10]. In granular media, a new type of discontinuities arises, on which the displacement vector changes jumply. A so-called dry boiling is observed—spontaneous formation and collapse of voids in a medium [11, 12]. However, the conditions at such discontinuities are not fully investigated. The next important problem, which can be solved with the help of inequalities (1)– (3), is a problem of mathematical correctness of the elastic-plastic theory. Let U (x, t) and U  (x, t) be two continuously differentiable solutions of the variational inequal = U in analogous  = U  in (1) and U ity (1) defined in the same domain. By setting U  inequality for the solution U , we have   n         ∂   ∂  U −U − U −U · A U − U − Q U  − U − G + G  0 . Bi ∂t ∂ xi i=1

This inequality is the starting point for a priori estimating solutions in characteristic cones of a differential operator [1]. Regardless of the fact that here we consider not a system of equations, but a variational inequality, one can obtain

t1   a(t −t ) U − U (t1 )  U − U (t0 ) e 1 0 + b G  − G (t) ea(t1 −t) dt. t0

(5)

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Here t0 < t1 are the fixed time moments, a and b are the positive constants independent of both solutions. The square of norm is defined as the integral over the cross section ω(t) of a cone:

U 2 =

U · A U dωx . ω(t)

The estimate (5) gives a representation about the correctness of the Cauchy problem and boundary value problems with dissipative boundary conditions for the variational inequality (1). It implies uniqueness in the small on time (for small time intervals) and continuous dependence of the solutions on initial data and “right hand side”, and the statement about the finite region of dependence of the solutions. Significantly, that it is valid even if the solution domain contains one or several discontinuity surfaces, on which the conditions (4) are satisfied. A similar estimate can be obtained for the variational inequalities (2) and (3) (see [4, 6]). An equally important result is that the inequalities (2)–(3) allow an efficient numerical implementation, and on their basis, one can construct robust shock-capturing methods and algorithms. General approach is reduced to two stages: approximation of the differential operator of a problem and approximation of the constraint, which determines the set of admissible variations. As usual, the operator approximation assumes its replacement by a discrete analog. In the approximation of the constraint, it is actually indicated in what sense the inclusion of the discrete solution into the set F is understood. In the simplest case, the problem is reduced to solving a variational inequality for Uˆ on a new time level:      − Uˆ · A U  − U¯  0, U , Uˆ ∈ F, U

(6)

where U¯ is the solution of the system of semi-discrete equations of elasticity: n  ∂U U¯ − U = Bi + QU +G . A t ∂ xi i=1

Due to the symmetry and positive definiteness of the matrix A, √ solution of (6) is the projection of vector U¯ onto F at the Euclidean norm |U | = U · A U , which in general, can be found using iterative methods. In the case of the Prandtl–Reuss theory with the von Mises yield criterion, it leads to the well-known Wilkins’s procedure of stresses correction [13]. More accurate algorithms of solution correction are developed in [4]. To approximate a differential operator of variational inequalities, we use a variant of the Godunov gap decay method, in which the Riemann problem is solved in the linear approximation for the corresponding system of equations, in combination with the two-cyclic splitting method in spatial variables. Numerous examples of computations in problems of the dynamics of elastic-plastic, granular, and porous media are given in the cited publications of the authors.

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3 Conclusions New formulations of the dynamic models of elastic-plastic materials and constructions, granular and porous media in the form of variational inequalities for hyperbolic operators turn out to be useful in the study of weak solutions with shock waves and discontinuities of displacements. On their basis, integral estimates can be obtained that guarantee the uniqueness and continuous dependence on initial data of solutions of the Cauchy problem and boundary value problems, and questions of the existence of solutions can be investigated. They make it possible to construct effective numerical methods for solving elastic-plastic problems, which retain their monotonicity and thermodynamic consistency at a discrete level. There are many unresolved questions in this direction. First of all, the proof of the existence of solutions for Cauchy problem and boundary value problems. The second one is the characterization of weak solutions with strong discontinuities in the case of variational inequalities for nonlinear operators and with discontinuities of displacements in general case.

References 1. Godunov, S.K.: Equations of Mathematical Physics. Nauka, Moscow (1979) 2. Godunov, S.K., Romenskii, E.I.: Elements of Continuum Mechanics and Conservation Laws. Kluwer Academic/ Plenum Publishers, New York (2003). https://doi.org/10.1007/978-1-47575117-8 3. Sadovskii, V.M.: Hyperbolic variational inequalities in problems of the dynamics of elastoplastic bodies. J. Appl. Math. Mech. 55(6), 927–935 (1991). https://doi.org/10.1016/00218928(91)90146-L 4. Sadovskii, V.M.: Discontinuous Solutions in Dynamic Elastic-Plastic Problems. Fizmatlit, Moscow (1997) 5. Sadovskii, V.M., Guzev, M.A., Sadovskaya, O.V., Qi, Ch.: Modeling of plastic deformation based on the theory of an orthotropic Cosserat continuum. Fiz. Mesomekh. 22(2), 59–66 (2019). https://doi.org/10.24411/1683-805X-2019-12005 6. Sadovskaya, O., Sadovskii, V.: Mathematical modeling in mechanics of granular materials. Series: Advanced Structured Materials, vol. 21. Springer, Heidelberg (2012). https://doi.org/ 10.1007/978-3-642-29053-4 7. Kukudzhanov, V.N.: On the study of the equations of the dynamics of elastoplastic media under finite deformations. In: Nonlinear Deformation Waves, vol. 2, pp. 102–105. Tallinn (1977) 8. Bykovtsev, G.I., Kretova, L.D.: Shock wave propagation in elastic-plastic media. J. Appl. Math. Mech. 36(1), 94–103 (1972). https://doi.org/10.1016/0021-8928(72)90086-X 9. Sadovskii, V.M.: Elastoplastic waves of strong discontinuity in linearly hardening media. Mech. Solids 32(6), 88–94 (1997) 10. Kulikovskii, A.G., Chugainova, A.P.: Study of discontinuities in solutions of the Prandtl-Reuss elastoplasticity equations. Comput. Math. Math. Phys. 56(4), 637–649 (2016). https://doi.org/ 10.1134/S0965542516040102

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11. Maslov, V.P., Myasnikov, V.P., Danilov, V.G.: Mathematical Modeling of the Accident Block of Chernobyl Atomic Station. Nauka, Moscow (1987) 12. Sadovskii, V.M.: On the theory of the propagation of elastoplastic waves through loose materials. Dokl. Phys. 47(10), 747–749 (2002). https://doi.org/10.1134/1.1519321 13. Wilkins, M.L.: Calculation of elastic-plastic flow. In: Methods in Computational Physics, vol. 3, Fundamental Methods in Hydrodynamics, pp. 211–263. Academic, New York (1964)

Estimates of Solutions to a System of Differential Equations with Two Delays M. A. Skvortsova

Abstract We consider a model of cell population dynamics proposed by Professor N. V. Pertsev. The model is described by a system of differential equations with two positive delays. The stability of stationary solutions to the system is studied. Using a modified Lyapunov–Krasovskii functional, estimates of solutions characterizing the stabilization rate at infinity are established.

1 Problem Statement Consider a system of delay differential equations ⎧ d ⎪ ⎪ x(t) = f (u(t − τ1 ), w(t − τ2 )) − μx(t), ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎨ d u(t) = α1 x(t) − μ1 u(t), ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ d w(t) = α2 x(t) − μ2 w(t), dt

(1)

where τ1 , τ2 , α1 , α2 , μ, μ1 , μ2 are positive constants and function f (u, w) is continuous, nonnegative, nondecreasing with respect to u and nonincreasing with respect to w. The system can be considered as a model of cell population dynamics, where x(t) is population size, u(t) is stimulator of population growth, w(t) is inhibitor of population growth. The model is proposed by Professor N. V. Pertsev. For system (1), consider initial conditions M. A. Skvortsova (B) Sobolev Institute of Mathematics, Acad. Koptyug avenue 4, Novosibirsk 630090, Russia e-mail: [email protected] Novosibirsk State University, Pirogov street 1, Novosibirsk, Russia © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_44

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⎧ ⎪ ⎪ x(+0) = γ ≥ 0, ⎪ ⎪ ⎨ u(t) = ϕ(t) ≥ 0, t ∈ [−τ1 , 0], u(+0) = ϕ(0), ⎪ ⎪ ⎪ ⎪ ⎩ w(t) = ψ(t) ≥ 0, t ∈ [−τ2 , 0], w(+0) = ψ(0),

(2)

where ϕ ∈ C([−τ1 , 0]) and ψ ∈ C([−τ2 , 0]) are continuous functions. It is easy to see that a solution to the initial value problem (1)–(2) exists, is unique and is defined for all t > 0. Moreover, x(t) ≥ 0, u(t) ≥ 0, w(t) ≥ 0 for all t > 0. Our aim is the study of stability of stationary solutions to system (1) and obtaining estimates of solutions characterizing the stabilization rate at infinity.

2 Stability of Stationary Solutions In this section, we specify conditions on the coefficients of system (1), under which stationary solutions are asymptotically stable or unstable. α1 Let (x∗ , u ∗ , w∗ ) be a stationary solution to system (1), then u ∗ = x∗ , w∗ = μ1   α2 α1 α2 x∗ , where x∗ is a solution to the equation f x∗ , x∗ = μx∗ . The change μ2 μ1 μ2 of variables y0 (t) = x(t) − x∗ ,

y1 (t) = u(t) − u ∗ ,

y2 (t) = w(t) − w∗

leads to the initial value problem ⎧ d ⎪ ⎪ y0 (t) = f (u ∗ + y1 (t − τ1 ), w∗ + y2 (t − τ2 )) − f (u ∗ , w∗ ) − μy0 (t), ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎨ d y1 (t) = α1 y0 (t) − μ1 y1 (t), ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ d y2 (t) = α2 y0 (t) − μ2 y2 (t), dt ⎧ y0 (+0) = γ − x∗ , ⎪ ⎪ ⎪ ⎪ ⎨ y1 (t) = ϕ(t) − u ∗ , t ∈ [−τ1 , 0], y1 (+0) = ϕ(0) − u ∗ , ⎪ ⎪ ⎪ ⎪ ⎩ y2 (t) = ψ(t) − w∗ , t ∈ [−τ2 , 0], y2 (+0) = ψ(0) − w∗ .

(3)

(4)

Suppose that function f (u, w) has continuous partial derivatives f u (u, w) and In this case, the linearized system has the form

f w (u, w).

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339

⎧ d ⎪ ⎪ y (t) = ay1 (t − τ1 ) − by2 (t − τ2 ) − μy0 (t), ⎪ ⎪ dt 0 ⎪ ⎪ ⎪ ⎪ ⎨ d y1 (t) = α1 y0 (t) − μ1 y1 (t), ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ d y2 (t) = α2 y0 (t) − μ2 y2 (t), dt

(5)

where a = f u (u ∗ , w∗ ) ≥ 0, b = − f w (u ∗ , w∗ ) ≥ 0. System (5) can also be written in the brief form d y(t) = Ay(t) + B1 y(t − τ1 ) + B2 y(t − τ2 ), dt where

⎛

⎞ y0 (t) y(t) = ⎝ y1 (t) ⎠ , y2 (t) ⎛

⎞ −μ 0 0 A = ⎝ α1 −μ1 0 ⎠ , 0 −μ2 α2

⎞ 0 a 0 B1 = ⎝ 0 0 0 ⎠ , 0 0 0

⎛

⎛

⎞ 0 0 −b B2 = ⎝ 0 0 0 ⎠ . 0 0 0

The characteristic quasi-polynomial of system (5) has the form Q(λ) = det(λE − A − e−λτ1 B1 − e−λτ2 B2 ) = (λ + μ)(λ + μ1 )(λ + μ2 ) + α2 b(λ + μ1 )e−λτ2 − α1 a(λ + μ2 )e−λτ1 . By analyzing the location of roots of the characteristic quasi-polynomial and using the theorem on stability by linear approximation (see, for example, [1]), it is easy to establish the validity of the following theorem. Theorem 1 (I) If α1 aμ2 + α2 bμ1 < μμ1 μ2 , then the stationary solution (x∗ , u ∗ , w∗ ) to system (1) is asymptotically stable. (II) If α2 bμ1 < α1 aμ2 − μμ1 μ2 , then the stationary solution (x∗ , u ∗ , w∗ ) to system (1) is unstable. Proof (I) Let α1 aμ2 + α2 bμ1 < μμ1 μ2 . At first, we will show that all the roots of the characteristic quasi-polynomial Q(λ) are located in the left half-plane {λ : Re λ < 0}. Indeed, if a root λ0 is such that Re λ0 ≥ 0, then we have α1 ae−λ0 τ1 α2 be−λ0 τ2 α1 a α2 b ≤ μ ≤ |λ0 + μ| = − + . λ0 + μ1 λ0 + μ2 μ1 μ2 This inequality contradicts the condition of the theorem. Consequently, by the theorem on stability by linear approximation, the zero solution to system (3) is asymp-

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totically stable. Therefore, the stationary solution (x∗ , u ∗ , w∗ ) to system (1) is also asymptotically stable. (II) Let α2 bμ1 < α1 aμ2 − μμ1 μ2 . We will show that there exists a positive root λ∗ > 0 of the characteristic quasi-polynomial Q(λ). Indeed, consider the function F(λ) = λ + μ −

α2 be−λτ2 α1 ae−λτ1 + , λ ≥ 0. λ + μ1 λ + μ2

Since F(0) < 0, F(λ) → +∞ as λ → +∞, and F is continuous function, then there exists λ∗ > 0 such that F(λ∗ ) = 0. Consequently, by the theorem on stability by linear approximation, the zero solution to system (3) is unstable. Therefore, the stationary solution (x∗ , u ∗ , w∗ ) to system (1) is also unstable. Theorem is proved. Remark 1 The case α1 aμ2 − α2 bμ1 ≤ μμ1 μ2 ≤ α1 aμ2 + α2 bμ1 should be considered additionally. In this case, results on stability of stationary solutions to system (1) depend on the delay parameters τ1 and τ2 .

3 Estimates of Solutions In this section, we assume that α1 aμ2 + α2 bμ1 < μμ1 μ2 . In this case, by Theorem 1, the stationary solution (x∗ , u ∗ , w∗ ) to system (1) is asymptotically stable. We will establish estimates of solutions characterizing the stabilization rate at infinity. Remark 2 For simplicity, we establish estimates of solutions to linearized system (5). For original nonlinear system (1), estimates for attraction domains and estimates of solutions can also be obtained. We obtain estimates of solutions to system (5) by using a modified Lyapunov– Krasovskii functional [2]. At first, we introduce the notation. Since α1 aμ2 + α2 bμ1 < μμ1 μ2 , we choose k1 > 0 and k2 > 0 such that the following inequality holds: α1 a k1 τ1 /2 α2 b k2 τ2 /2 e + e < μ. μ1 μ2 Let δ (0 < δ ≤ 2μ) be the minimal positive solution to the following equation: α2 b α1 a ek1 τ1 /2 + ek2 τ2 /2 = (μ − (δ/2)). (μ1 − (δ/2)) (μ2 − (δ/2)) Denote h 0 = 1, h 1 = β1 =

a k1 τ1 /2 b e , h 2 = ek2 τ2 /2 , α1 α2

(μ1 − (δ/2))a k1 τ1 /2 (μ2 − (δ/2))b k2 τ2 /2 e , β2 = e . α1 α2

(6)

(7)

(8)

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Let y(t) be a solution to system (5). Consider modified Lyapunov–Krasovskii functional of the following form: V (t, y) = h 0 y02 (t) + h 1 y12 (t) + h 2 y22 (t)

t +

β1 e−k1 (t−s) y12 (s)ds

t

β2 e−k2 (t−s) y22 (s)ds.

+

t−τ1

(9)

t−τ2

Now we obtain an estimate for functional (9). Theorem 2 Let α1 aμ2 + α2 bμ1 < μμ1 μ2 . Then for the solution to the initial value problem (4)–(5), the estimate holds: V (t, y) ≤ V (0, y)e−εt , t > 0,

(10)

where V (t, y) is defined in (9), ε = min{δ, k1 , k2 }.

(11)

Proof Differentiate functional (9) along a solution to system (5):   d V (t, y) = 2h 0 y0 (t) ay1 (t − τ1 ) − by2 (t − τ2 ) − μy0 (t) dt     +2h 1 y1 (t) α1 y0 (t) − μ1 y1 (t) + 2h 2 y2 (t) α2 y0 (t) − μ2 y2 (t)

+β1 y12 (t)

−

t

β1 e−k1 τ1 y12 (t

− τ1 ) − k1

β1 e−k1 (t−s) y12 (s)ds

t−τ1

+β2 y22 (t)

−

β2 e−k2 τ2 y22 (t

t − τ2 ) − k2

β2 e−k2 (t−s) y22 (s)ds.

t−τ2

From this identity, it is easy to obtain d V (t, y) ≤ −2h 0 μy02 (t) + h 20 dt



a2 b2 + −k τ β1 e 1 1 β2 e−k2 τ2

 y02 (t)

    +2h 1 y1 (t) α1 y0 (t) − μ1 y1 (t) + 2h 2 y2 (t) α2 y0 (t) − μ2 y2 (t)

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t +β1 y12 (t)

+

β2 y22 (t)

− k1

β1 e−k1 (t−s) y12 (s)ds

t − k2

t−τ1

β2 e−k2 (t−s) y22 (s)ds.

t−τ2

By using the notation (6)–(8), we establish   d V (t, y) ≤ −δ h 0 y02 (t) + h 1 y12 (t) + h 2 y22 (t) dt

t −k1

β1 e−k1 (t−s) y12 (s)ds

t − k2

t−τ1

β2 e−k2 (t−s) y22 (s)ds.

t−τ2

Taking into account the definitions (9) and (11), we have d V (t, y) ≤ −εV (t, y). dt From this inequality, it is easy to obtain (10). Theorem is proved. Finally, we obtain estimates of solutions to system (5). Theorem 3 Let α1 aμ2 + α2 bμ1 < μμ1 μ2 . Then for the components of the solution to the initial value problem (4)–(5), the estimates hold: |y0 (t)| ≤ |y1 (t)| ≤ |y1 (0)|e

−μ1 t



V (0, y)e−εt/2 , t > 0,



+ α1 V (0, y)e

−μ1 t

t

(12)

e(μ1 −(ε/2))s ds, t > 0,

(13)

e(μ2 −(ε/2))s ds, t > 0,

(14)

0

|y2 (t)| ≤ |y2 (0)|e

−μ2 t



+ α2 V (0, y)e

−μ2 t

t 0

where V (0, y) is defined in (9), ε > 0 is defined in (11).

√ Proof Estimate (12) follows from (10) and the inequality |y0 (t)| ≤ V (t, y). Estimates (13) and (14) follow from the second and the third equations of system (5) and estimate (12). Theorem is proved. Acknowledgements The author is grateful to Professor G. V. Demidenko for the attention to the research and Professor N. V. Pertsev for the problem statement. The author is supported by the Russian Foundation for Basic Research (project no. 18-29-10086).

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References 1. Krasovskii, N.N.: Stability of Motion. Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay. Stanford University Press, Stanford (1963) 2. Demidenko, G.V., Matveeva, I.I.: Asymptotic properties of solutions to delay differential equations (in Russian). Vestn. Novosib. Gos. Univ. Ser. Mat. Mekh. Inform. 5(3), 20–28 (2005)

Moving-Coordinate Method and Its Applications Y. Takakura

Abstract The moving-coordinate method presented by the authors is a method for computing flows about the moving body, where the coordinate systems are attached to individual moving bodies, so that each body could stand still with stationary grid. Here, as numerical methods for flows including moving boundaries, first, a review is given about the conservative form in general coordinates with implication of geometric conservation laws and the ALE (Arbitrary Lagrangean–Eulerian) formulation, to show that the governing equations are derived from the former to the latter systematically. Then, the moving-coordinate method is derived. From the perspective given above, the advantages and disadvantages of this method are discussed. Finally, applications confirmed availability of the moving-coordinate method.

1 Introduction In most methods to numerically compute unsteady flows about moving bodies, the bodies are moved. In contrast, in the moving-coordinate method presented by the authors [1–3], where the coordinate systems are attached to individual moving bodies, each moving body is stationary when viewed from the respective moving-coordinate system, and therefore there is no need to move the grids. In this paper, as numerical methods for flows including moving boundaries, first, a review is given about the general coordinate system in conservative form [4, 5] with implication of geometric conservation laws [6] and the ALE (Arbitrary Lagrangean– Eulerian) formulation [7], and it is shown that the governing equations are derived from the former to the latter systematically. Then, the moving-coordinate method is taken with its derivation. From the perspective given above, the advantages and disadvantages of this method will be discussed. Finally, applications of the movingcoordinate method will be shown.

Y. Takakura (B) Tokai University, Kitakaname, Hiratsuka, Kanagawa 259-1292, Japan e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_45

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2 Governing Equations of Flow Including Moving Boundaries In this section, the inviscid flows without dissipation are considered for simplicity. The conservation laws in Cartesian coordinates (t, x 1 , x 2 , x 3 ) can be written in derivative form as ∂f k ∂q + = 0, (1) ∂t ∂xk ⎡ ⎡ ⎤ ⎤ ρ ρu k ⎢ ρu 1 ⎥ ⎢ ρu 1 u k + δ 1,k p ⎥ ⎢ 2⎥ ⎢ 2 k ⎥ k ⎢ ⎥ ρu u + δ 2,k p ⎥ (2) q = ⎢ ρu ⎥ , f = ⎢ ⎢ ⎥ , ⎣ ρu 3 ⎦ ⎣ ρu 3 u k + δ 3,k p ⎦ E (E + p)u k where physical quantities are indicated by ordinary  nomenclature, and related by the equation of state p = (γ − 1) E − (1/2)ρu k u k .

2.1 Derivation of Conservative Form in General Coordinate System The transformation from Cartesian coordinates (t, x 1 , x 2 , x 3 ) to general coordinates (τ, ξ 1 , ξ 2 , ξ 3 ) is described as τ = t, ξ i = ξ i (t, x 1 , x 2 , x 3 ), i = 1, 2, 3. The conservation laws in Cartesian coordinates (1), (2) are transformed to a nonconservative form in general coordinates by the chain rule, k ∂q ∂q i ∂f + ξti + ξ = 0. (3) k x ∂τ ∂ξ i ∂ξ i

1 2 3

1 2 3

∂(ξ ,ξ ,ξ )

,ξ ,ξ ) = ∂(x Define Jacobian with J ≡ ∂(τ,ξ 1 2 3 1 ,x 2 ,x 3 ) and multiply equation (3) by ∂(t,x ,x ,x )

1/J , then with the metrics into the partial differential operators, we obtain

∂ ∂τ





i   i    ξx k ξxi k k ξti ξt 1 1 ∂ ∂ ∂ k ∂ + i −f q + i q+ f −q = 0. J ∂ξ J J ∂τ J ∂ξ J ∂ξ i J

Viviand [4] derived the conservative form, ∂ q ∂ Fi + ∂τ ∂ξ i  q=

1 q, J

= 0,

ξik ξi  Fi = t q + x f k . J J

(4)

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with the following relations for metrics: ∂ ∂τ

 i  ξt 1 ∂ + i =0, J ∂ξ J

 ξxi k ∂ = 0. ∂ξ i J

(5) (6)

Pulliam and Steger [5] introduced the contravariant velocities U i = ξti + ξxi k u k

(7)

and concretely described the flux tensors in (4), including viscous stress tensors, ∂ Fi ∂ q + ∂τ ∂ξ i

= 0,

⎤ ⎡ ⎤ ρU i ρ ⎢ ρu 1 U i + ξ i 1 p ⎥ ⎢ ρu 1 ⎥ x ⎥ ⎢ ⎥ 1⎢ 1 ⎥ ⎢ ρu 2 ⎥ ,  Fi = ⎢ ρu 2 U i + ξxi 2 p ⎥ ,  q= ⎢ ⎥ ⎢ ⎥ J ⎣ ρu 3 ⎦ J ⎢ ⎣ ρu 3 U i + ξxi 3 p ⎦ E (E + p)U i − ξti p

(8)

⎡

(9)

with metrics formulations by the chain rule. Later, metrics relations came to be called geometric conservation laws [6], with volume conservation law (5) which moving grids should satisfy and area conservation laws (6) which any grid stationary or moving should satisfy [8, 9].

2.2 ALE (Arbitrary Lagrangean–Eulerian) Formulation Hirt [7] proposed a method called ALE to observe fluid motion on a control volume moving and deforming at an arbitrary speed. Although this was derived from physical considerations, here it is shown that this can be derived from the general coordinate system in conservative form (8), (9). Assign general coordinates to the mesh moving and deforming, and define moving velocity v of a grid point with v = (v1 , v2 , v3 ) where vk = (x k )τ , then, the chain rule on metrics, ξτi = ξti tτ + ξxi k (x k )τ = 0, leads to ξti = −ξxi k (x k )τ = −ξxi k vk ,

(10)

and therefore the contravariant velocities are represented as U i = ξti + ξxi k u k = ξxi k (u k − vk ).

(11)

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Substituting Eqs. (10), (11) into fluxes (9) of the conservative form in the general coordinates and eliminating ξti leads to ALE formulation in the general coordinates: ∂ q ∂ Fi + ∂τ ∂ξ i

= 0,

ξik  Fi = x gk , J ⎡ ⎤ ρ(u k − vk ) ⎢ ρu 1 (u k − vk ) + δ 1,k p ⎥ ⎢ ⎥ k 2 k k 2,k ⎥ g =⎢ ⎢ ρu 3 (u k − vk ) + δ 3,k p ⎥ , ⎣ ρu (u − v ) + δ p ⎦ (E + p)(u k − vk ) + pvk

(12)

(13)

(14)

which can be written in Cartesian coordinates (x 1 , x 2 , x 3 ) as ∂gk ∂q + = 0. ∂τ ∂xk

(15)

3 Moving-Coordinate Method In the moving-coordinate method proposed by the authors [1–3], each coordinate system is attached to individuals of arbitrary n regions moving acceleratedly to the inertial frame. For simplicity, consider an accelerating frame, then the governing equations can be derived as follows (the case n > 1 can be similarly handled). Consider coordinate system I belonging to the inertial frame and movingcoordinate system A accelerating to system I with translational velocity V0 of origin O A to origin O I and rotational angular velocity  around O A , let r be a position vector of O A in system I and x∗ be an arbitrary position vector in system A, and denote the flow velocity as u at position r+x∗ observed in system I, and u∗ at position x∗ observed in system A. Then the velocity relation holds, u − v = u∗ , v = V0 +  × x∗ ,

(16)

where v is the relative velocity of arbitrary position x in system A to system I. Continuity equation Eliminating u from the continuity equation in the ALE formulation (12), (13), (14) by use of velocity relation (16) and setting ξ i = x∗i leads to ∂  ∂ρ + i ρu i∗ = 0, (17) ∂τ ∂ x∗ ρu i∗ = (x∗i )kx (ρu k∗ | I ),

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where u i∗ and u i∗ | I are the x∗i - and x i -directional compounds of u∗ , respectively. Momentum equations With rotation of the coordinate system, time derivatives of an arbitrary vector b have relation between the inertial and accelerating frames [10].     Db Db = +×b (18) Dτ I Dτ A Substituting velocity relation (16) into the motion equations in the inertial system: ρ

Du = −∇ p + ∇ · T Dτ

(19)

together with relation (18) leads to those in the moving-coordinate system [3, 10].  ρ

Du∗ Dτ



dV0 d − 2ρ×u∗ − ρ×(×x∗ ) − ρ ×x∗ dτ dτ (20) translational Coriolis centrifugal force force force

= −∇ p + ∇ · T∗ − ρ A

Energy equation From the total energy expressed in each coordinate system E=

1 p + ρu · u , γ −1 2

E∗ =

1 p + ρu∗ · u∗ γ −1 2

and velocity relation (16), we obtain 1 E = E ∗ + ρu∗ · (V0 +  × x∗ ) + ρ(V0 +  × x∗ ) · (V0 +  × x∗ ). 2

(21)

Substituting the above into the energy equations in the inertial system: ρ

D(E/ρ) ˙ , = −∇ · ( pu) + ∇ · (T · u) − ∇ · Q Dτ

(22)

differentiating it in the inertial system and organizing it leads to the energy equation in the moving-coordinate system [3]. ρ

  D(E ∗ /ρ) ˙ − ρu∗ · dV0 + ×(×x∗ ) + d ×x∗ = −∇ ·( pu∗ ) + ∇ ·(T∗ ·u∗ ) − ∇ · Q Dτ dτ dτ

(23) Since continuity equation (17) holds in the moving-coordinates, each left hand side of the momentum and energy equations can be rewritten in a conservative form [11]. Therefore, the governing equations in the moving-coordinate system are represented in the non-dimensional, conservative form as

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∂(f∗ k − Re−1 fv∗ k ) ∂q∗ + = s∗ , ∂τ ∂ x∗k

(24)

⎡

⎡ ⎡ ⎤ ⎤ ⎤ 0 ρ ρu k∗ 1,k ⎢ρu 1 ⎥ ⎢ 1 k 1,k ⎥ ⎢ ⎥ τ∗ ⎢ ∗2 ⎥ k ⎢ρu 2∗ u k∗ +δ 2,k p⎥ ⎢ ⎥ 2,k k ⎢ ⎢ ⎢ ⎥ ⎥ ⎥, τ q∗ = ⎢ρu ∗ ⎥ , f∗ = ⎢ρu ∗ u ∗ +δ p⎥ , fv∗ = ⎢ ∗ ⎥ ⎣ρu 3∗ ⎦ ⎣ρu 3∗ u k∗ +δ 3,k p⎦ ⎣ ⎦ τ∗3,k j j,k k ˙ E∗ (E ∗ + p)u ∗ u ∗ τ∗ − Q k ⎡

0 ⎫ ⎢⎬ ⎢  dV0 s∗ = ⎢ ⎢ ⎭ − ρ dτ + 2×u∗ + ×(×x∗ ) + ⎣   0 −ρu∗ · dV + ×(×x∗ ) + d ×x∗ dτ dτ

⎤ d ×x∗ dτ

⎥ ⎥ ⎥. ⎥ ⎦

(25)

Equations (24), (25) are interpreted as the conservation laws in the inertial frame with the inertial effect due to acceleration of the moving region added as source terms. In the moving-coordinate method, it is necessary to exchange physical quantities observed between coordinate systems in order to set outer boundary conditions. Momentum and energy are transformed from system I to system A as (ρu∗ ) A = (ρu) I − ρ(V0 +×x∗ )

(26) 1 (E ∗ ) A = (E) I −ρu·(V0 +×x∗ ) + ρ(V0 +×x∗ )·(V0 +×x∗ ) , (27) 2

and simultaneously, from system A to system I as (ρu) I = (ρu∗ ) A + ρ(V0 +×x∗ ) 1 (E) I = (E ∗ ) A +ρu∗ ·(V0 +×x∗ ) + ρ(V0 +×x∗ )·(V0 +×x∗ ) 2

(28) (29)

In the moving-coordinate method, since the coordinate systems are attached to individual moving bodies, each moving body is stationary when viewed from the respective moving-coordinate system. Therefore, there is no need to move the grids, with only necessity to impose the area conservation laws out of the geometric conservation laws that the metrics should satisfy. The advantages of the moving-coordinate method are that the boundary conditions about bodies become identical as those at the time of bodies stationary, and there are no grid generation problems or errors that occur during usual body movement. Therefore, this effect is considered to be noticeable when bodies move for a long distance/long time. The disadvantage is that the source terms should be added to the governing equations. Applications of the moving-coordinate method are shown in [12, 13] for translational motion, and in [14] for rotational motion.

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4 Summary As numerical methods for flows including moving boundaries, a review was given about the conservative form in general coordinates and the ALE formulation, to show that the governing equations are derived systematically. Then, the moving-coordinate method proposed by the authors was derived dynamically with part from the ALE formulation. After the advantages and disadvantages of this method were discussed, application examples confirmed its availability.

References 1. Takakura, Y., Higashino, F., Ogawa, S.: Unsteady flow computations on a flying projectile within a ballistic range. Comput. Fluids 27(5–6), 645–650 (1998) 2. Takakura, Y.: Methods for computing flows with moving boundaries: governing equations, geometric conservation laws, and applications of moving-coordinate method, bulletin of computational. Fluid Dyn. 8(3), 117–129 (2000). (in Japanese) 3. Nomura, M., Takakura, Y.: Investigation of Moving-coordinate Method for Supersonic Flows around a Concave Body, JAXA-SP-16-007, pp. 57–62 (2016) (in Japanese) 4. Viviand, H.: Bréves informations. La Rech. Aérospatiale 1, 65–66 (1974) 5. Pulliam, T.H., Steger, J.L.: Implicit finite-difference simulations of three-dimensional compressible flow. AIAA J. 18(2), 159–167 (1980) 6. Thomas, P.D., Lombard, C.K.: Geometric conservation law and its application to flow computations on moving grids. AIAA J. 17(10), 1030–1037 (1979) 7. Hirt, C.W., Amsden, A.A., Cook, J.L.: An Arbitrary Lagrangian-Eulerian computing method for all flow speeds. J. Comput. Phys. 14, 227–253 (1974) 8. Zhang, H., Reggio, M., Trépanier, Y., Camarero, R.: Discrete form of the GCL for moving meshes and its implementation in CFD schemes. Comput. Fluids 22, 9–23 (1993) 9. Tamura, Y., Fujii, K.: Conservation law for moving and transformed grids. AIAA Paper 93– 3365 (1993) 10. Spurk, J.H.: Fluid Mechanics. Springer, New York (1997) 11. Liepmann, H.W., Roshko, A.: Elements of Gasdynamics. Wiley, Hoboken (1960) 12. Takakura, Y., Higashino, F.: Numerical study on the sonic boom simulator using a ballistic range. In: Houwing, A.F.P. (ed.) Proceedings of the 21st ISSW (International Symposium on Shock Waves), pp. 1085–1090. Pnther Publishing & Printing (1997) 13. Takakura, Y.: A method for computing flows about complex configurations with moving boundaries. ICCFD1 (1st International Conference on Computational Fluid Dynamics) (2000) 14. Nomura, M., Takakura, Y.: Highly-accurate computation of supersonic flows around a concave body (II. motion analysis by moving-coordinate method). ICCFD9 (9th International Conference on Computational Fluid Dynamics), ICCFD9-2016-326 (2016)

Numerical Methods for Model Kinetic Equations and Their Application to External High-Speed Flows Vladimir A. Titarev

Abstract The paper is a short review of the recent results of the author in the development of numerical methods and parallel computer code Nesvetay to solve kinetic equations with approximate (model) collision integrals. The efficiency and robustness of the methods are demonstrated by computing M∞ = 25 rarefied gas flow over a model winged geometry of a spacecraft.

1 Introduction At present, most of the numerical modeling of high-speed rarefied gas flows at high altitudes is carried out using the direct simulation Monte Carlo approach, implemented in various codes, such as SMILE [1], CATOD [2] and PICLas [3] codes. An alternative, which has been gaining popularity in recent years, is to solve the Boltzmann kinetic equation with either exact or approximate (model) collision integrals. The attractive feature of this approach is the possibility to construct modern high-order methods for both steady and transient applications. The corresponding three-dimensional codes have been developed by a number of authors, e.g. [4–7]. In particular, model kinetic equations seem attractive due to the possibility to construct more efficient numerical methods, resulting in significantly lower computational cost. It has been recently shown [8–11] that the use of the velocity-independent collision frequency in the construction of model collision integrals does not significantly affect the accuracy of the computed aerodynamics and heat transfer in rarefied flows, paving the way to the practical applications. The author has been developing Godunov-type [12] numerical methods to solve S-model kinetic equation [13] in three space dimensions and corresponding solver Nesvetay for a number of years [8, 14–16]. The distinguishing features of Nesvetay solver include its ability to use full unstructured arbitrary meshes in both physical and velocity spaces, implicit high-order accurate method of solution and two-level V. A. Titarev (B) Federal Research Center “Computer Science and Control”, Moscow, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_46

353

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OpenMP+MPI parallel implementation on the computers with tens of thousands of x86 cores. The purpose of the present paper is to present a short review of the current status of the numerical capabilities of Nesvetay as applied to three-dimensional external high-speed flows of monatomic rarefied gases.

2 Governing Equations Consider external flow of a rarefied monatomic gas over a three-dimensional body, described by free-stream values of pressure p∞ , temperature T∞ , velocity modulus u ∞ and angle of attack α. A three-dimensional state of the rarefied gas is determined by the velocity distribution function f (t, x, ξ), where t is physical time, ξ = (ξx , ξ y , ξz ) are the components of the molecular velocity vector in the spatial directions x = (x1 , x2 , x3 ) = (x, y, z). Macroscopic quantities are defined as the integrals of the velocity distribution function with respect to the molecular velocity:   1 1 3 mn Rg T + mnu 2 = m ξ 2 f dξ, f dξ, nu = ξ f dξ, 2 2 2  1 2 q = m vv f dξ, v = ξ − u, ρ = mn, p = ρRg T, 2

 n=

(1)

where m is molecular mass, k is the Boltzmann constant and Rg -gas constant. The distribution function is found by solving the kinetic equation with the S-model collision integral [13]: ∂f p ∂f + ξα = J, J = ν( f + − f ), ν = , ∂t ∂x  μ α   4 5 1 + 2 , Si = f = f M 1 + (1 − Pr)Sα cα c − ci c2 f dξ, 5 2 n n v 2 fM = e−c , c =  . 3/2 (2π Rg T ) 2Rg T

(2)

Here Pr = 2/3 is Prandtl number, setting Pr = 1, one recovers the simpler BGK kinetic model [17]. The power-law intermolecular interaction μ(T ) = μ∞ (T /T∞ )ω is assumed. The kinetic equation (2) has to be augmented with the boundary conditions. At the solid surfaces, the diffuse molecular scattering boundary condition with complete thermal accommodation to the surface temperature Tw is assumed so that the distribution function of reflected molecules is given by fw =

  nw ξ2 . exp − (2π Rg Tw )3/2 2Rg Tw

(3)

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The number density of reflected molecules n w is found from impermeability condition. At the inflow and outflow boundaries, the distribution function of the incoming molecules is prescribed as the locally Maxwellian one with the corresponding values of the macroscopic variables.

3 Basic Parallel Method of Solution The numerical Godunov-type [12] method is a discrete velocity scheme conservative with respect to the S-model collision integral. Unlike most of the reported schemes, it uses fully unstructured meshes with general elements in both physical and molecular velocity spaces. For each cell Vi in the physical space, define the vector f , corresponding to the values of the distribution function at all velocity nodes. Integrating Eq. (2) over the four-dimensional control volume Vi × [t n , t n+1 ], we obtain the following implicit time marching scheme: 1  n+1 Δ fi =− Φ + J in+1 , Δ f i = f in+1 − f in . Δt |Vi | l=1 li

(4)

Details of the calculation of high-order numerical fluxes Φ li for the face l of a cell Vi on arbitrary unstructured mesh can be found in [8, 14–16]. Either directional (TVD1D) or least-square type (TVD3D) reconstructions can be used. A minmodtype limiter is applied to suppress spurious oscillations. The conservative evaluation of the macroscopic parameters in the collision term J in+1 for the S-model equation [18] is based on the direct discretization of the equations used to derive the S-model:  (5) (1, ξ, ξ 2 , mvv 2 )ν( f + − f ) = (0, 0, 0, −(4/3)νq). In the special case Prandtl = 1, the last three equations can be omitted and system (5) reduces to the use of the conservation conditions only as was proposed in [19]. Finally, the system (4) for time increments of the distribution function Δ f i is linearized approximately and then solved using LU-SGS method [20]. In order to speed up calculations, the numerical method is extended to run on multiprocessor computers using a two-level OpenMP+MPI parallel model [21]. In the present work, velocity mesh decomposition is used to distribute work between MPI processes whereas spatial mesh decomposition is used for OpenMP parallelization within the node, including a multi-threaded implementation of the implicit time marching scheme.

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4 Modifications to High-Speed Flows The following modification to the general numerical method is implemented when applied to high-speed flows over complex three-dimensional geometries. Firstly, a highly nonuniform mesh is used in the velocity space [8, 16], with relatively fine uniform spacing near peaks of the distribution function ξ = u∞ and ξ = 0 and large cells away from these points. Such a mesh construction results in only linear dependence on the number of cells on the free-stream Mach number and thus greatly reduces the computing cost. On the other hand, it is much simpler than octree velocity mesh developed in [5, 6] and does not require complicated changes to the code. Secondly, the spatial discretization scheme is modified. Note that block-structured spatial meshes for such geometries may contain some poor quality cells. Our experience shows that TVD3D scheme is robust on such meshes, but may struggle to achieve strict convergence to steady state, whereas TVD1D method is good in all aspects apart from behaving erratically on bad cells. To improve the numerical method, in the present work, we introduce an adaptive switching between TVD1D and TVD3D reconstruction methods. As time marching advances to steady state, the solution procedure marks trouble cells and then switches to TVD3D method for those cells. The resulting nonuniform numerical scheme converges to steady state. The general solution procedure to compute the external flow is as follows. First, initialize flow field by the free-stream values or by running a compressible Navier– Stokes solver [22]. Next, the preliminary kinetic solution is provided by using a first-order numerical method and BGK model. The final result is then obtained by restarting the calculation and using the second-order TVD scheme with S-model equation. Such a procedure allows to compute the flows for practically arbitrarily large Mach numbers without any difficulties.

5 Example To demonstrate the robustness and efficiency of Nesvetay, a high-speed rarefied flow of air over the generic winged re-entry space vehicle (RSV) geometry was computed. The total length of the RSV with the flap is 10 m. The flow conditions correspond to air at the altitude H = 100 km, free-stream velocity U∞ = 7091 m/s (M∞ = 25) and angle of attack α = 25◦ ; p∞ = 0.0319 Pa, T∞ = 199.34 K. The surface temperature is Tw = 300 K. Parameters μ∞ = 1.241 × 10−5 Pa · s, ω = 0.734 are used in the viscosity law. Since we use the S-model equation, internal degrees of freedom are ignored. The multi-block hexahedral mesh consists of 159 blocks and contains 568 × 103 cells, the height of the first cell ≈1 mm. The velocity mesh contains 32144 nodes. The total number of degrees of freedom is approximately 18 billions. Calculations were run on a Dell HPC machine with Intel Xeon E5-2697v2 processors. Typically, 6 dual-CPU cluster nodes (144 cores) were used.

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Fig. 1 Non-dimensional pressure p/ p∞ contour lines computed by Nesvetay code, altitude H = 100 km, free-stream Mach number M∞ = 25, angle of attack α = 25◦

The initial first-order BGK calculation starting from uniform flow required 6000 core-hours to reduce residuals in both maximum (point-wise) and integral norms by at least 7 orders of magnitude. The S-model calculation using TVD1D scheme required additional 2000 core-hours to achieve the same reduction of the residual in both norms. However, the temperature dropped to unrealistic values in 33 spatial cells (0.05%) near flap and trailing edge. The final calculation with hybrid TVD1D/3D solver rectified this problem requiring only 500 additional core-hours. However, the strict convergence was achieved in the integral norm only, whereas maximum norm residual oscillated around a small value at some of the troubled cells. Figure 1 shows the final computed non-dimensional pressure contour lines on the symmetry plane and the surface. A typical flow pattern is observed, with good solution quality. Acknowledgements Calculations were run on supercomputers, installed at Joint Supercomputing Center of the Russian Academy of Sciences and Peter the Great St. Petersburg Polytechnic University, Russia.

References 1. Kashkovsky, A.V., Bondar, Ye.A., Zhukova, G.A., Ivanov, M.S., Gimelshein, S.F.: Objectoriented software design of real gas effects for the DSMC method. In: 24th International Symposium on Rarefied Gas Dynamics. AIP Conference Proceedings, vol. 762, pp. 583–588 (2004) 2. Kusov, A.L.: On the relaxation of molecules’s rotational energy in the direct simulation Monte Carlo method. Math. Model. Comput. Simul. 10(2), 237–248 (2018) 3. Fasoulas, S., Munz, C.-D., Pfeiffer, M., Beyer, J., Binder, T., Copplestone, S., Mirza, A., Nizenkov, P., Ortwein, P., Reschke, W.: PICLas – combining PIC and DSMC for the simulation of reactive plasma flows. Phys. Fluids 31, 072006 (2019)

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4. Anikin, Yu.A., Kloss, Yu.Yu., Martynov, D.V., Tcheremissine, F.G.: Computer simulation and analysis of the Knudsen experiment of the 1910 year. J. Nano Microsyst. Tech. 8, 6–14 (2010) 5. Arslanbekov, R.R., Kolobov, V.I., Frolova, A.A.: Kinetic solvers with adaptive mesh in phase space. Phys. Rev. E 88, 063301 (2013) 6. Baranger, C., Claudel, J., Herouard, N., Mieussens, L.: Locally refined discrete velocity grids for stationary rarified flow simulations. J. Comput. Phys. 257, 572–593 (2014) 7. Dimarco, G., Loubere, R., Narski, J.: Towards an ultra efficient kinetic scheme. Part III: highperformance-computing. J. Comput. Phys. 284, 22–39 (2015) 8. Titarev, V.A.: Application of model kinetic equations to hypersonic rarefied gas flows. Comput. Fluids, Special issue Nonlinear Flow Transp. 169, 62–70 (2018) 9. Frolova, A.A., Titarev, V.A.: Recent progress on supercomputer modelling of highspeed rarefied gas flows using kinetic equations. Supercomput. Front. Innov. 5(3), 117–121 (2018) 10. Titarev, V.A., Frolova, A.A.: Application of model kinetic equations to computing of superand hypersonic flows of molecular gas. Fluid Dyn. 53(4), 536–551 (2018) 11. Titarev, V.A., Frolova, A.A., Rykov, V.A., Vashchenkov, P.V., Shevyrin, A.A., Bondar, Ye.A.: Comparison of the Shakhov kinetic equation and DSMC method as applied to space vehicle aerothermodynamics. J. Comput. Appl. Math. 364, 1–12 (2020) 12. Godunov, S.K.: A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb. 47, 357–393 (1959) 13. Shakhov, E.M.: Generalization of the Krook kinetic relaxation equation. Fluid Dyn. 3(5), 95–96 (1968) 14. Titarev, V.A.: Implicit numerical method for computing three-dimensional rarefied gas flows using unstructured meshes. Comput. Math. Math. Phys. 50(10), 1719–1733 (2010) 15. Titarev, V.A., Dumbser, M., Utyuzhnikov, S.V.: Construction and comparison of parallel implicit kinetic solvers in three spatial dimensions. J. Comput. Phys. 256, 17–33 (2014) 16. Titarev, V.A.: Numerical modeling of high-speed rarefied gas flows over blunt bodies using model kinetic equations. Eur. J. Mech./B Fluids, Special Issue on Non-Equilib. Gas Flows 64, 112–117 (2017) 17. Bhatnagar, P.L., Gross, E.P., Krook, M.: A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94(511), 1144– 1161 (1954) 18. Titarev, V.A.: Conservative numerical methods for model kinetic equations. Comput. Fluids 36(9), 1446–1459 (2007) 19. Mieussens, L.: Discrete-velocity models and numerical schemes for the Boltzmann-BGK equation in plane and axisymmetric geometries. J. Comput. Phys. 162(2), 429–466 (2000) 20. Men’shov, I.S., Nakamura, Y.: An implicit advection upwind splitting scheme for hypersonic air flows in thermochemical nonequilibrium. In: A Collection of Technical Papers of 6th International Symposium on CFD. Lake Tahoe, Nevada, vol. 2, p. 815 (1995) 21. Titarev, V.A., Utyuzhnikov, S.V., Chikitkin, A.V.: OpenMP + MPI parallel implementation of a numerical method for solving a kinetic equation. Comput. Math. Math. Phys. 56(11), 1919–1928 (2016) 22. Petrov, M.N., Tambova, A.A., Titarev, V.A., Utyuzhnikov, S.V., Chikitkin, A.V.: FlowModellium software package for calculating high-speed flows of compressible fluid. Comput. Math. Math. Phys. 58(11), 1865–1886 (2018)

The ADER Path to High-Order Godunov Methods Eleuterio Toro

Abstract Sixty years ago, Godunov introduced his method to solve the Euler equations of gas dynamics, thus creating the Godunov school of thought for numerical approximation of hyperbolic conservation laws. The building block of the original first-order Godunov upwind method is the piece-wise constant data Riemann problem. Modern computational science, however, requires high-order accurate schemes; these may be orders-of-magnitude cheaper than first-order methods for attaining a prescribed error, and are thus mandatory in ambitious scientific and technological applications. Here we briefly review the ADER approach to construct one-step, fully discrete high-order Godunov methods for solving hyperbolic balance laws, whose building block is now the generalized Riemann problem, a piece-wise smooth data Cauchy problem including stiff or non-stiff source terms.

1 Introduction Contributions to the development of high-order non-linear generalizations of the original Godunov method [6] have seen astonishing developments in the last four decades or so. Two main frameworks are available, finite volume and discontinuous Galerkin finite elements [4], both requiring a numerical flux, which in turn is associated with a Riemann problem [14]. Then there are two approaches to deal with space/time discretization, the semi-discrete and the fully discrete approaches. The former separates space and time, usually requiring solvers for ordinary differential equations in time, typically Runge–Kutta methods [13], resulting in multi-level multi-step methods. The fully discrete methodology keeps space and time unified in the discretization, usually leading to one-step two-level methods. Here we review the fully discrete ADER approach to construct one-step, high-order methods in both space and time for solving hyperbolic balance laws. Central to this methodology is the solution of the generalized Riemann problem; this is a local Cauchy problem in E. Toro (B) Laboratory of Applied Mathematics, University of Trento, Trento, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_47

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which the initial conditions are piece-wise smooth functions, typically polynomials of any degree and the governing equations include source terms. Current versions of the ADER methodology admit stiff or non-stiff source terms. The rest of this paper is structured as follows: Sect. 2 defines Godunov methods and Riemann problems; Sect. 3 deals with solvers for the generalized Riemann problem. Conclusions are found in Sect. 4.

2 Godunov Methods and Riemann Problems Consider a general one-dimensional system of hyperbolic balance laws ∂t Q(x, t) + ∂x F(Q(x, t)) = S(Q(x, t)) .

(1)

The vectors Q, F(Q) and S(Q) define conserved variables, fluxes and sources, respectively. Assuming the space/time domain to be discretised by finite volumes Vin = [xi− 21 , xi+ 21 ] × [t n , t n+1 ], exact integration of (1) in Vin gives Qn+1 = Qni − i

t (F 1 − Fi− 21 ) + tSi , x i+ 2

(2)

with Qni

1 = x



1 Si = tx

xi+ 1 2

xi− 1



2

Q(x, t )dx , Fi+ 21

t n+1

n



xi+ 1 2

tn

1 = t



t n+1 tn

⎫ ⎪ ⎪ F(Q(xi+ 21 , t))dt , ⎪ ⎪ ⎬

S(Q(x, t))dxdt .

xi− 1

⎪ ⎪ ⎪ ⎪ ⎭

(3)

2

Numerically, Eqs. (2)–(3) motivate the construction of finite volume methods to solve (1), in which suitable approximations to the integrals (3) determine the numerical flux Fi+ 21 and the numerical source Si in the approximating formula (2). The classical Godunov method [6] results from (2)–(3) by assuming S(Q) = 0 and ˜ i+ 1 (x/t) to be the similarity solution of the piece-wise constant data, Q(xi+ 21 , t) = Q 2 homogeneous, Riemann problem ⎫ PDEs: ∂t Q + ∂x F(Q)  = 0 , n x ∈ (−∞, ∞) , t > 0 , ⎬ if x < 0 , QL = Qi ICs: Q(x, 0) = ⎭ QR = Qni+1 if x > 0 .

(4)

Figure 1, top frame, depicts the initial conditions and structure of the solution of (4). In this simplified case, the numerical flux Fi+ 21 in (2) results from exact integration in (3), namely

The ADER Path to High-Order Godunov Methods

qL

361

q(x, 0)

t

qR

x=0

0

x

x

q(x, 0) qL (x) qR (x)

x=0

x

Fig. 1 Riemann problems. Top frame: classical Riemann problem for the first-order Godunov method. Bottom frame: generalized Riemann problem for the ADER method. Left-hand side pictures depict initial conditions and right-hand side pictures depict the emerging wave structures in the x-t plane

Godunov = Fi+ 1 2

1 t



t n+1

tn

˜ ˜ i+ 1 (0)) . F(Q(0))dt = F(Q 2

(5)

For exact and approximate Riemann solvers for the first-order Godunov method, see [14]. The ADER extension of Godunov’s method [17] includes three steps: 1. Replacing piece-wise constant data by piece-wise smooth data, as resulting from reconstruction procedures, for example, see Fig. 1, bottom frame. This leads naturally to the generalized Riemann problem ⎫ PDEs: ∂t Q + ∂x F(Q)  = S(Q) , x ∈ (−∞, ∞) , t > 0 , ⎬ QL (x) if x < 0 , ICs: Q(x, 0) = ⎭ QR (x) if x > 0 .

(6)

2. Computing QLR (τ ), the time-dependent solution of (6) at the interface, followed by evaluation of the numerical flux Fi+ 21 =

1 t



t n+1

tn

F(QLR (τ ))dt .

(7)

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3. Defining a function Qi (x, t) within each volume Vin , followed by evaluation of the numerical source 1 Si = tx



t n+1



xi+ 1 2

tn

S(Qi (x, t))dxdt .

(8)

xi− 1 2

Early communications on the ADER methodology include [12, 17]. For an introduction to ADER methods see Chaps. 19 and 20 of [14]. The next section deals with solvers for the generalized Riemann problem (6).

3 Solvers for the Generalized Riemann Problem GRPk Several methods for solving the generalized Riemann problem are currently available. Here we limit ourselves to brief descriptions for two of them and give appropriate reference to the others.

3.1 The Toro–Titarev GRPk solver This solver [16] seeks an approximate solution QLR (τ ) of (6) expressed as QLR (τ ) = Q(0, 0+ ) +

K

τk  , ∂t(k) Q(0, 0+ ) k!

(9)

k=1

following [10]. The leading term is defined as Q(0, 0+ ) = lim Q(0, t) t→0+

(10)

and is computed by solving a classical Riemann problem. The challenge is to determine the coefficients ∂t(k) Q(0, 0+ ) in (9) for the higher order terms. A precursor to the Toro–Titarev solver [16] results from a modification of the BenArtzi/Falcovitz second-order GRP solver [1] that departs from QLR (τ ) = Q(0, 0+ ) + τ ∂t Q(0, 0+ ), with Q(0, 0+ ) defined as in (10). See Chap. 14 of the 1997 edition of [14]. The Cauchy–Kovalevskaya procedure is then used to determine ∂t Q(0, 0+ ), that is (11) ∂t Q + ∂x F(Q) = S(Q) → ∂t Q = −A(Q)∂x Q + S(Q)

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and the solution then reads QLR (τ ) = Q(0, 0+ ) + τ [−A(Q(0, 0+ ))∂x Q(0, 0+ ) + S(Q(0, 0+ ))] .

(12)

The pending problem in (12) is to compute the spatial derivative ∂x Q(0, 0+ ). First it is noted that for the linear homogeneous case ∂t Q + A∂x Q = 0, the following evolution equations for all spatial derivatives are valid ∂t (∂x(k) Q) + A∂x (∂x(k) Q) = 0 .

(13)

One can then define the classical Riemann problem for (13), with k = 1, to find ∂x Q(0, 0+ ) in (12). The numerical flux (7) then follows. The Toro–Titarev solver for the GRPk [14] departs from the LeFloch–Raviart expansion (9) and the Cauchy–Kovalevskaya procedure, leading to ∂t(k) Q(x, t) = P(k) (∂x(0) Q, ∂x(1) Q, . . . , ∂x(k) Q) .

(14)

The functionals P(k) are specific to the particular system (1); its arguments are spatial derivatives, yet to be found. To determine these, evolution equations are invoked ∂t (∂x(k) Q(x, t)) + A(Q)∂x (∂x(k) Q(x, t)) = H(k) .

(15)

Then, simplified classical Riemann problems for spatial derivatives are posed ⎫ (k) PDEs: ∂t (∂x(k) Q(x, t)) + A(0) LR ∂x (∂x Q(x, t)) = 0 , ⎪ ⎬ Q(k) (0) if x < 0 , (k) L ⎪ ICs: ∂x Q(x, 0) = ⎭ Q(k) R (0) if x > 0 .

(16)

The complete solution (9) has then been determined and the numerical flux results from (7). A similar but simpler procedure is used to determine the source term in (8), see [16].

3.2 The Montecinos–Toro Implicit GRP Solver This solver [11, 15] is implicit and can deal with stiff source terms. The key step is the following lemma: Lemma: Let Q(x, τ ) be an analytical function of τ , then Q(x, τ ) = Q(x, 0+ ) −

∞  (−τ )k k=1

k!

∂t(k) Q(x, τ ) .

(17)

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This expression is the implicit counterpart of the explicit Raviart–LeFloch expansion (9). Now the implicit Cauchy–Kovalevskaya procedure leads to ∂t(k) Q(0, τ ) = P(k) (Q(0, τ ), . . . , ∂x(k) Q(0, τ )) .

(18)

As for the Toro–Titarev solver, a key step is to determine the arguments of functionals P(k) in (18). For details, see [15] for the general case. Example: Second-order scheme. In this special case, see [11] for details, we have ∂t Q + ∂x F(Q) = S(Q) ,

Q(0, τ ) = Q(0, 0+ ) + τ ∂t Q(0, τ ) .

(19)

The implicit Cauchy–Kovalevskaya procedure gives Q(0, τ ) = Q(0, 0+ ) + τ [−A(Q(0, τ ))∂x Q(0, τ ) + S(Q(0, τ ))] .

(20)

Then, applying the implicit expansion (17), we have ∂x Q(0, τ ) = ∂x Q(0, 0+ ) + τ ∂t (∂x Q(0, τ ))

(21)

and ∂x Q(0, τ ) = ∂x Q(0, 0+ ) + τ ∂x [−A(Q(0, τ ))∂x Q(0, τ ) + S(Q(0, τ ))] . (22) Therefore ∂x Q(0, τ ) = [I − τ B(Q(0, τ ))]−1 ∂x Q(0, 0+ ) .

(23)

Finally, the solution Q(0, τ ) = Q∗ (τ ) satisfies the non-linear algebraic system

Q∗ = Q(0, 0+ ) + τ −A(Q∗ ) [I − τ B(Q∗ )]−1 ∂x Q(0, 0+ ) + S(Q∗ )

(24)

and the numerical flux follows. This gives a second-order, locally implicit method suitable for balance laws with stiff source terms.

3.3 Other GRPk Solvers We have described schemes in the class of GRP solvers that seek the solution right at the interface and extend the Ben-Artzi/Falcovitz second-order method. Another class of solvers emerges from a re-interpretation [2] of the high-order method of Harten et al. [9] that extends the MUSCL-Hancock second-order method. In the latter approach, the initial data is evolved in time and interactions at the interface are sought at selected time-integration points via classical Riemann problems. The

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method of Dumbser, Enaux and Toro [5] evolves the data numerically via a space– time discontinuous Galerkin approach and is suitable for stiff source terms. Other solvers are due to Götz and Iske [8]; Götz and Dumbser [7] and Dematté et al. [3].

4 Conclusions We have briefly reviewed the ADER approach to extend Godunov’s method to high order of accuracy. Very high-order methods are essential for the following reasons: (i) there is a growing trend to use PDEs to understand the physics they embody; (ii) only very accurate solutions of PDEs will achieve this and also reveal limitations of mathematical models (PDEs themselves) and uncertainty on parameters of the problem; (iii) efficiency: given an error deemed acceptable, then high-order methods attain that error, if small, much more efficiently than low-order methods on fine meshes, by orders of magnitude. For very long time evolution simulations, highorder methods are essential.

References 1. Ben-Artzi, M., Falcovitz, J.: A second order Godunov-type scheme for compressible fluid dynamics. J. Comput. Phys. 55, 1–32 (1984) 2. Castro, C.E., Toro, E.F.: Solvers for the high-order Riemann problem for hyperbolic balance laws. J. Comput. Phys. 227, 2481–2513 (2008) 3. Dematté, R., Titarev, V.A., Motecinos, G.I., Toro, E.F.: ADER methods for hyperbolic equations with a time-reconstruction solver for the generalized Riemann problem. The scalar case. Commun. Appl. Math. Comput. To appear (2019) 4. Dumbser, M., Balsara, D., Toro, E.F., Munz, C.D.: A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes. J. Comput. Phys. 227, 8209–8253 (2008) 5. Dumbser, M., Enaux, C., Toro, E.F.: Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J. Comput. Phys. 227(8), 3971–4001 (2008) 6. Godunov, S.K.: A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb. 47, 357–393 (1959) 7. Götz, C.R., Dumbser, M.: A novel solver for the generalized Riemann problem based on a simplified LeFloch-Raviart expansion and a local space-time discontinuous Galerkin formulation. J. Sci. Comput. (2016). https://doi.org/10.1007/s10915-016-0218-5 8. Götz, C.R., Iske, A.: Approximate solutions of generalized Riemann problems for nonlinear systems of hyperbolic conservation laws. Math. Comput. 85, 35–62 (2016) 9. Harten, A., Osher, S.: Uniformly high-order accurate nonoscillatory schemes I. SIAM J. Numer. Anal. 24(2), 279–309 (1987) 10. Le Floch, P., Raviart, P.A.: An asymptotic expansion for the solution of the generalized Riemann problem. Part 1: general theory. Ann. Inst. Henri Poincaré. Analyse non Lineáre 11. Montecinos, G.I., Toro, E.F.: Reformulations for general advection-diffusion-reaction equations and locally implicit ADER schemes. J. Comput. Phys. 275, 415–442 (2014) 12. Schwartzkopff, T., Munz, C.D., Toro, E.F.: ADER: high-order approach for linear hyperbolic systems in 2D. J. Sci. Comput. 17, 231–240 (2002)

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13. Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988) 14. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd edn. Springer, Berlin (2009) 15. Toro, E.F., Montecinos, G.I.: Implicit, semi-analytical solution of the generalised Riemann problem for stiff hyperbolic balance laws. J. Comput. Phys. 303, 146–172 (2015) 16. Toro, E.F., Titarev, V.A.: Solution of the generalised Riemann problem for advection-reaction equations. Proc. R. Soc. Lond. A 458, 271–281 (2002) 17. Toro, E.F., Millington, R.C., Nejad, L.A.M.: Towards very high–order Godunov schemes. In: Toro, E.F. (ed.) Godunov Methods: Theory and Applications. Edited Review, pp. 905–937. Kluwer Academic/Plenum Publishers, New York (2001)

Near-Wall Domain Decomposition for Modelling Turbulent Flows: Opportunities and Challenges S. Utyuzhnikov

Abstract It is well known that resolution of a near-wall region in turbulence modelling can take most computational time. The paper provides a review of the current opportunities and challenges of a novel approach which is based on non-overlapping domain decomposition. The technique can effectively be used for engineering applications where a trade-off between the accuracy and computational time is important while choice is often made in favour of the latter one. In contrast to the traditional approach based on wall functions, which is still very popular for industrial applications, the current technique can be used for a controllable trade-off when the accuracy can be easily increased by a compromise with computational time in the framework of the same model. The key issue of the approach based on non-overlapping domain decomposition is the interface boundary condition. To make the technique efficient, it should be nonlocal on one side and easily applicable on the other side. The approach can be extended to essentially unsteady problems and flows past complex geometries. However, for these purposes, it requires a significant elaboration.

1 Introduction Effective numerical resolution of a near-wall boundary layer is one of the challenges in the turbulence modelling. Due to the no-slip boundary condition and damping effect of the wall, a very thin laminar sublayer is formed near the wall. Despite the relative thickness of the near-wall region being only about 1% or so, its resolution often requires about 90% of the total computational time. Thus, there are at least two scales in the problem. To effectively resolve it, domain decomposition can be efficient. A traditional approach, which is usually used for industrial applications, is based on so-called wall functions. The wall functions are applied as off-wall Dirichlet boundary conditions and contain the entire available information on the wall since S. Utyuzhnikov (B) The University of Manchester, Manchester M13 8PL, UK e-mail: [email protected] Moscow Institute of Physics and Technology, Dolgoprudny, Russia © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_48

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the governing equations used in this case are isotropic. It is clear that such boundary conditions can only be semi-empirical and their application is quite limited. As noted in [1], this approach represents a simplest form of domain decomposition in which the inner region is replaced by a wall function. It is to be noted that there are also attempts to construct a profile of the solution nearest to the wall cell which takes into account the behaviour of the solution outside (see e.g. [2, 20]). This approach demonstrates a notable improvement although the solution is quite sensitive to the approximation of the profile in the nearest to the wall cell and its size. In particular, if the cell is situated in the laminar sublayer, then a special treatment is needed. This is a common problem for the wall-function approach. To resolve it, the scalable wall functions are often used [3]. An alternative approach developed in [4–6] and other publications is based on a transmission of the boundary conditions to the interface boundary. In the exact formulations, the interface boundary conditions (IBC) must be nonlocal. For a practical realisation, they are simplified and reduced to a Robin boundary condition. As demonstrated in [7–9] on a number of test cases, the approach is efficient and reduces computational time needed by one order of magnitude while retaining good enough accuracy. In contrast to the wall functions, the IBC are mesh independent and do not contain any tuning parameter. If the approach is applied to a low-Reynolds number model, then the location of the interface boundary effectively controls the trade-off between the accuracy and computational time [7]. The technique was also applied to laminar–turbulent transition regimes [10]. However, it was recognised that if the model of a thin layer used in the inner region becomes unacceptable, then the accuracy and robustness of the entire approach deteriorate. Thus, for modelling flows past complex geometries and in interface regimes, the approach based on near-wall domain decomposition (NDD) must be significantly modified. In addition, as shown in [11], in the case of essentially unsteady flows, the IBC must be nonlocal in time.

2 Approximate Near-Wall Domain Decomposition As shown in [12, 13], the boundary condition can be effectively transferred from the wall to an interface boundary if a thin-layer model is used in the inner region. To demonstrate it, consider a domain decomposition for a 1D second-order equation in an interval  := [y0 , ye ]:   dU d μ = f, (1) dy dy where μ is the efficient viscosity coefficient. U (y0 ) = U0 , U (ye ) = U1 .

(2)

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369

Then the IBC is reduced to a Robin boundary condition set at y ∗ [13]: U = where I1 =

 y∗ y0

μ∗ dy, μ

I S1 =

 y∗ y0

U − U0 I1 I S1 − I S2 + , I1 μ∗ I1 f dy, I S2 =

 y∗ y0

μ∗ μ

 y y0

(3)

f dydy  , μ∗ = μ(y ∗ ).

It is easy to prove that the IBC (3) is exact in one-dimensional case, and it does not depend on the solution in the outer region + := [y ∗ , ye ] provided that coefficient μ is known. A theoretical justification of the approach in a multidimensional case is provided in [13, 14]. The entire algorithm is described step by step in [15]. Despite a relatively good performance of the IBC in application to flows with a clearly formed boundary layer, the approach should be modified for complex geometries. One possible modification can be related to nonlocal IBC as demonstrated in [5]. Another approach can be related to the exact domain decomposition, in which an approximate NDD is used to invert a preconditioning operator [16]. As noted above, in the case of essentially unsteady problems, the IBC should be nonlocal with respect to time [11]. Modified IBC with nonlocal temporal terms is considered in the next section.

3 Unsteady Near-Wall Domain Decomposition The IBC with unsteady terms was first formulated in [11] for a model equation. Then, they were derived in [17] for the Navier–Stokes equations under assumption of steady forces and source terms. In the current study, they are formulated in the most general form [18]. It takes into account the unsteadiness of a driving force. Next, consider the following parabolic equation with respect to a variable φ formulated in domain : φt = L y φ + R h , where L y := ∂∂y μ ∂∂y , Rh = Rh (y, t). It is to be noted that in , there is the full orthogonal system of eigenfunctions  p of operator L y : L y  p = −λ p  p , p = 1, 2, . . . with negative eigenvalues. For the sake of simplicity, suppose that φ(y0 ) = 0. Next, we exactly transfer this boundary condition to the interface boundary at y = y ∗ . For this purpose, we introduce two auxiliary boundary-value problems (BVPs).

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BVP 1: L y V0 + Rh = 0, V0 (y0 ) = 0, V0 (y ∗ ) = 0, and BVP 2: L y V = 0, V (y0 ) = 0, V (y ∗ ) = 1. Then, the unsteady IBC (UIBC) reads φ y = V y φ + Rh V0|y +

∞  (k + Dk )k|y ,

(4)

1

where V =V−

∞ 

C k k ,

1

and C p = (V,  p ),



 p = (V,  p ) exp(−λ p t)[−φ y∗ (0) + λ p

t

exp(λ p τ )φ y ∗ (τ )dτ ],  t exp(λ p τ )b p (τ )dτ, D p = −b p (t) + b p (0) exp(−λ p t) + λ p exp(−λ p t) 0

0

b p = (V0 ,  p ), p = 1, 2, . . . Here the following inner project is used for any functions a and b from R[y0 y ∗ ]: (a, b) =

1 y∗



y∗

abdy. y0

It is to be noted that in contrast to UIBC formulated in [11, 17], UIBC (4) contain additional terms Dk which are caused by the variation of the driving force Rh in time. Overall, UIBC (4) contains a memory term which is nonlocal in time and contains some history of the flow related to the unsteadiness of both the solutions at the interface boundary and driving force in the inner region. It is clear that without the memory term, the UIBC is reduced to the IBC for a steady BVP, given in the previous section.

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4 Exact Domain Decomposition It seems attractive to realise an exact NDD in such a way that a thin-layer model is used to construct a preconditioning operator. Then, the approximate NDD (ANDD) can effectively be used to invert this operator. To analyse this idea, consider pseudo-differential Steklov–Poincaré operators S1 and S2 for the inner and outer regions, respectively [19]. In addition, in the inner region, we introduce the Steklov–Poincaré operator S1∗ for the inner region corresponding to the ANDD. Then, the IBC for full governing equations can be formulated as follows [16]: ∂u m+1 ∂u m+1 2 = 1 + (u m+1 − u m+1 )S1∗ I, 2 1 ∂n ∂n

(5)

where I is the one-dimensional unit function, indexes 1 and 2 correspond to the inner and outer regions, respectively, and m is a number of iteration. In the case of a linear second-order equation, the iterative process can be reduced to the following pseudo-differential equation: [S1∗ + S2 ](λm+1 − λm ) = θ (χ − Sλm ), where S = S1 + S2 , χ = ∂G∂n2 f − ∂G∂n1 f , G 1 and G 2 are Green’s operators in the inner and outer regions, respectively. Thus, we arrive at a Richardson procedure with operator S1∗ + S2 used as a preconditioner. Such operator can effectively be inverted with the use of NDD. It is almost optimal since condition number of operator (S1∗ + S2 )−1 S  O(1). If operator L is symmetric, then the convergence of the algorithm immediately follows from Theorem 4.2.7 in [19] if 0 0. In application to the Cauchy problem, we consider the method of successive approximations and estimate the successive approximations quality in dependence on the number of the iterated solution.

1 Introduction In the present article, we consider the following ordinary integrodifferential equation with quadratic nonlinearity on the right-hand side:

V. L. Vaskevich (B) Sobolev Institute of Mathematics, Novosibirsk, Russia e-mail: [email protected] A. I. Shcherbakov Novosibirsk State University, Novosibirsk, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_49

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du (t, k) + a(t)u(t, k) = dt

 W (k, k1 , k2 )u(t, k1 )u(t, k2 )dk1 dk2 . P(k)

The independent variables in this equation and also the integration variables on the right-hand side are positive: t > 0, k > 0, k1 > 0, k2 > 0. Here the function a(t) is continuous on the positive half-axis and the domain of integration P(k) is a semibounded strip in the first quadrant of the plane (k1 , k2 ) defined as follows: P(k) = {(k1 , k2 ) | k2 ≥ k1 − k, k2 ≤ k1 + k, k1 + k2 ≥ k}. The properties of the set of solutions to the equation under consideration are largely determined by the weight kernel W (k, k1 , k2 ) of its integral operator and also by conditions on the behavior of the solution u(t, k) as k → +0 and k → +∞. Let W (k, k1 , k2 ) be equal to the function b(k)δ(k1 − k)δ(k2 − k), where δ(k1 − k) and δ(k2 − k) are shifts of the Dirac delta function. Then the initial integrodifferential equation takes on a form du (t, k) + a(t)u(t, k) = b(k)u 2 (t, k). dt For any k > 0 it is the well-known Riccati differential equation. Results obtained under different types of numerical modeling for quadratically nonlinear integrodifferential equations are presented and illustrated graphically in [1–6].

2 Statement of the Problem and Main Results Suppose that the kernel W (k, k1 , k2 ) of the integral operator is a function continuous in the first octant for which  sup |W (k, k1 , k2 )| dk1 dk2 ≤ M < +∞. k≥0 P(k)

Here M is a given nonnegative constant. The set of all possible functions W (k, k1 , k2 ) continuous in the first octant and satisfying assigned conditions will be denoted by K(M). The identically zero function always belongs to K(M), i.e., this class is nonempty. For M > 0, the accompanying class K(M) contains sufficiently many nontrivial elements. Suppose, for example, that W1 (k, k1 , k2 ) is integrable in the first quadrant for any fixed nonnegative k and, moreover, there exists a finite constant M1 such that the following estimates hold uniformly over k:

Quasilinear Integrodifferential Riccati-Type Equations

377

∞∞ |W1 (k, k1 , k2 )| dk1 dk2 ≤ M1 ,

∀ k > 0.

0 0

Then we can take the following quantity as the constant M: ∞∞ |W1 (k, k1 , k2 )| dk1 dk2 < +∞.

M = sup k≥0 0 0

Pose the Cauchy problem with data on the positive half-axis for the initial equation:  ⎧ du ⎪ W (k, k1 , k2 )u(t, k1 )u(t, k2 )dk1 dk2 , ⎨ (t, k) + a(t)u(t, k) = dt P(k) ⎪ ⎩ u(t, k)|t=0 = ϕ(k), k ≥ 0. Here the function ϕ(k) is continuous on the positive half-axis and satisfies the following estimate: |ϕ(k)| ≤ Ce−γ k ∀ k ≥ 0. The exponent γ and the constant C on the right-hand side of this inequality are nonnegative. In particular, as initial data in the Cauchy problem, we can take any function with compact support on the positive half-axis. Define the action of a quadratic nonlinear integral operator at a function U (t, k) continuous in the first quadrant of the plane (t, k) by the relation  W (k, k1 , k2 )U (t, k1 )U (t, k2 ) dk1 dk2 .

W [U ] = W [U ](t, k) = P(k)

Assume that μ(t) = exp

t 

a(τ ) dτ . The function μ(t) is continuously differen-

0

tiable and μ(0) = 1. Multiplying both parts of the initial equation by a function μ(t) write the result in the following form: d [μ(t)u(t, k)] = μ(t)W [u](t, k). dt The class D = D(H, B, γ , ϕ) consists of functions U (t, k) continuous in the strip 0 ≤ t ≤ H , k ≥ 0 and such that the product μ(t)U (t, k) differing in the strip from the initial function ϕ(k) only by a quantity whose modulus does not exceed the product Be−γ k :

 D(H, B, γ , ϕ) = U (t, k) : 0 ≤ t ≤ H, k > 0 ⇒ |μ(t)U (t, k) − ϕ(k)| ≤ Be−γ k .

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Note that the function class D is not empty: for example, it contains the function ϕ(k)/μ(k). In case of μ(t) = 1 the function class D was introduced in [7]. Let a kernel W (k, k1 , k2 ) be of class K(M). Then for any pair of functions U (t, k), V (t, k) in D(H, b, γ , ϕ) we have the following estimate: sup eγ k W [U ](t, k) − W [V ](t, k) ≤

k≥0

L sup eγ ξ |U (t, ξ ) − V (t, ξ )|, 0 ≤ t ≤ H . μ(t) ξ ≥0

Here the constant L is defined by the equality L = 4B M. The operator W [ · ] is finite at any function U (t, k) in D(H, B, γ , ϕ) and

2 4M B 2 e−γ k , 0 ≤ t ≤ H. W [U ](t, k) ≤ Me−γ k sup eγ ξ |U (t, ξ )| ≤ 2 μ (t) ξ ≥0 Theorem 1 Any two solutions U (t, k) and V (t, k) to the equation of class D(H, b, γ , ϕ) satisfy the a priori estimate 

t t

γk

sup e |U (τ, k) − V (τ, k)| dτ ≤

0 k≥0

e0

L μ(τ ) dτ

L

−1

sup eγ k |U (0, k) − V (0, k)|, k≥0

with 0 ≤ t ≤ H . Corollary 1 (uniqueness theorem) Let U (t, k) and V (t, k) be two solutions to the same Cauchy problem belonging to the class D(H, b, γ , ϕ). Then U (t, k) and V (t, k) coincide everywhere in the domain 0 < t < H , k > 0. Define successive approximations to the solution to the Cauchy problem by the recurrent relations u [0] (t, k) =

1 ϕ(k), t ≥ 0, k ≥ 0, μ(t)

and then successively for j = 1, 2, . . .: μ(t)u [ j] (t, k) = ϕ(k) +



t μ(τ ) 0

W (k, k1 , k2 )u [ j−1] (τ, k1 )u [ j−1] (τ, k2 )dk1 dk2 dτ .

P(k)

These recurrent relations correctly define all the functions u [ j] (t, k), j = 1, 2, . . ., on some finite interval 0 ≤ t ≤ T0 . Theorem 2 Let the kernel W (k, k1 , k2 ) of the equation be continuous in the first octant and let W (k, k1 , k2 ) be a member of the class K(M). If the estimate |μ(t)u [ j−1] (t, k) − ϕ(k)| ≤ Be−γ k

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is valid for some positive number B > 0 for all t with 0 ≤ t ≤ H then the inequality ⎛ |μ(t)u [ j] (t, k) − ϕ(k)| ≤ R ⎝

t

⎞ μ(τ ) dτ ⎠ e−γ k

0

also holds for all t, 0 ≤ t ≤ H . Here the constant R is finite and defined by the relation R = M(B + C)2 . T0 Let T0 be a positive number such that R( μ(τ ) dτ ) ≤ B. Under the conditions of 0

Theorem 2, take H = T0 . Then, for all t with 0 ≤ t ≤ T0 and any number j = 1, 2, . . . we have the inequality ⎛T ⎞ 0 |μ(t)u [ j] (t, k) − ϕ(k)| ≤ R ⎝ μ(τ ) dτ ⎠ e−γ k ≤ Be−γ k . 0

Since for j = 0 this estimate is fulfilled, we conclude by induction that all the successive approximations u [ j] (t, k) are certainly defined for all t, 0 ≤ t ≤ T0 . Moreover, each of the functions u [ j] (t, k) belongs to the introduced class D(T0 , B, γ , ϕ). Theorem 3 Suppose that the kernel W (k, k1 , k2 ) of the initial equation is continuous in the first octant and belongs to the class K(M), and μ(t) ≥ μ(0) = 1. Then the successive approximations u [N ] (t, k) converge uniformly in the half-strip 0 ≤ t ≤ T0 , 0 ≤ k < +∞ to a continuous function u(t, k). The limit function u(t, k) belongs to the class D(T0 , B, γ , ϕ) and satisfies the estimates +∞  (Lt) j , μ(t) sup eγ k u(t, k) − u [N ] (t, k) ≤ B j! k≥0 j=N

N = 0, 1, 2, . . ..

Thus, the quantity T0 is the length of the time interval on which the solution u(t, k) to the Cauchy problem certainly exists. Theorem 4 (existence theorem) The limit u(t, k) of the successive approximations is the function continuous for 0 ≤ t ≤ H0 , k ≥ 0 and u(t, k) has continuous first derivative with respect to t on its domain of definition. This function u(t, k) defines the solution to the Cauchy problem on this set, which satisfies the estimates sup

0≤t≤T0 , k≥0

eγ k |u(t, k)| ≤ sup eγ k |ϕ(k)| + B, k≥0

sup |u(t, k)| ≤ (B + C)e−γ k . 0≤t≤T0

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Note that the numerical parameter B is strictly positive and is arbitrary in all other respects. Passage to the limit as B → 0 in these estimates is useless: the length T0 defining the interval of existence in time of the solution to the problem also tends to zero as B → 0. Let μ(t) = 1. Then for given M and C, the guaranteed length T0 of the existence interval is given by B B = . T0 = R M(B + C)2 This function of B attains its maximum for B = C: max T0 (B) = T0 (C) = B≥0

1 . 4MC

For B ≥ C the guaranteed length T0 (B) decreases monotonically from its maximal value to zero at infinity. If μ(t) = 1 and 4MC ≤ 1 then the solution to the Cauchy problem certainly exists for all t with 0 ≤ t ≤ 1. Acknowledgements The authors were partially supported by the Russian Foundation for Basic Research (project no. 19-01-00422).

References 1. Bell, N.K., Grebenev, V.N., Medvedev, S.B., Nazarenko, S.V.: Selfsimilar evolution of Alfven wave turbulence. J. Phys. A: Math. Theor. 50(43), 1–14 (2017) 2. Galtier, S.: Wave turbulence in astrophysics. Advances in Wave Turbulence, vol. 83, pp. 73–111. World Scientific Publishing, New Jersey (2013) 3. Galtier, S., Nazarenko, S., Newell, A.C., Pouquet, A.: A weak turbulence theory for incompressible magnetohydrodynamics. J. Plasma Phys. 63, 447–488 (2000) 4. Galtier, S., Buchlin, E.: Nonlinear diffusion equations for anisotropic magnetohydrodynamic turbulence with cross-helicity. Astrophys. J. 722, 1977–1983 (2010) 5. Nazarenko, S.: Wave Turbulence. Lecture Notes in Physics, vol. 825. Springer, Berlin (2011) 6. Vainberg, M.M., Trenogin, V.A.: Theory of Branching of Solutions of Non-linear Equations. Nauka, Moscow (1969); Noordhoff International Publishing, Leyden (1974) 7. Vaskevich, V.L., Shcherbakov, A.I.: Convergence of the successive approximation method in the Cauchy problem for an integro-differential equation with quadratic nonlinearity. Sib. Adv. Math. 29, 128–136 (2019)

S.K. Godunov and Kinetic Theory in KIAM RAS V. V. Vedenyapin, S. Z. Adzhiev, Ya. G. Batischeva, Yu. A. Volkov, V. V. Kazantseva, I. V. Melikhov, Yu. N. Orlov, M. A. Negmatov, N. N. Fimin and V. M. Chechetkin

Abstract The article describes the history of the development of cooperation between scientists from the Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences with S. K. Godunov. A lot of interesting results have been established in the theory of kinetic equations and computational mathematics in the process of this cooperation.

In 1971, a review [1] was published. The review was the first article in the USSR on discrete models of the Boltzmann equation, contained a study of a model with three speeds. As it turned out later, Boltzmann wrote a similar model in 1872, but now this model has become known as the Godunov–Sultangazin model, although abroad it is called the one-dimensional Broadwell model (d = 1, n = 3): ⎧ ∂ f1 ⎨ ∂t + ⎩

∂ f3 ∂t

∂ f1 ∂x

= f 22 − f 1 f 3 , = 2( f 1 f 3 − f 22 ), ∂ f3 − ∂ x = f 22 − f 1 f 3 .

∂ f2 ∂t

(1)

Here is an example where this name is included in the title of [2]. The relationship between the four-velocity Broadwell model (d = 2, n = 4)

V. V. Vedenyapin (B) KIAM RAS, RUDN University, Moscow, Russia e-mail: [email protected] S. Z. Adzhiev · I. V. Melikhov Chemical Department, Moscow State University, Moscow, Russia Ya. G. Batischeva · Yu. A. Volkov · V. V. Kazantseva · Yu. N. Orlov · N. N. Fimin · V. M. Chechetkin KIAM RAS, Moscow, Russia M. A. Negmatov Central Research Institute of Machine Building, Moscow, Russia © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_50

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⎧ ∂f 1 ⎪ ⎪ ∂t + ⎪ ⎪ ⎪ ⎨ ∂ f2 − ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

∂ f1 = f3 f4 − f2 f1 , ∂x ∂ f2 = f3 f4 − f2 f1 , ∂x ∂ f3 ∂ f3 + ∂ y = f1 f2 − f3 f4 , ∂t ∂ f4 − ∂∂fy4 = f 1 f 2 − f 3 f 4 ∂t

and this Godunov–Sultangazin is also discussed in the review. The review played a significant role in the development of both the theory of discrete models of the Boltzmann equation and for the development of the practice of discrete modeling. In 1971, in the Department of Kinetic Equations, M. V. Maslennikov organized a seminar on the Boltzmann equation, where, in particular, the book of the Swedish mathematician Torstein Carleman, “Mathematical Problems of the Kinetic Theory of Gases”, was analyzed. This book had an application where a model with two velocities  ∂f 1 + v1 ∂∂fx1 = f 22 − f 12 , ∂t ∂ f2 ∂t

+ v2 ∂∂fx2 = f 22 − f 12

was introduced, which is now called Carleman model (d = 1, n = 2). Someone remembered (maybe it was M. V. Maslennikov) that a review of Godunov and Sultangazin had just appeared. After that, they began to analyze this review. One of the authors (V. V.—later I) reported on both works (I was a regular reserve speaker). Therefore, it was necessary to read carefully, and in the review, many questions, tasks, and hints were formulated. One of these hints on the task looked like this. “At first glance, the model system chosen by us is not much more complicated than the Carleman model. However, a detailed analysis of this system shows that it does not have quadratic dissipating integrals, and therefore it is difficult to obtain an existence theorem for it covering all positive times 0 < t < ∞.” I took this question to investigate: what is the difference between the two models, what are the dissipative integrals there, and what will be for the Boltzmann equation? The theorem on the uniqueness of the Boltzmann H-function was obtained: for a gas described by the Boltzmann equation, entropy is the only extensive (additive) functional that increases with time. It also clarified the reason for the uniqueness of the H-function as a dissipative integral for the model of both Broadwell and Godunov–Sultangazin and non-uniqueness for Carleman model. This was the main result of both of my theses and was also published in UMN [3]. Sergei Konstantinovich often appeared in the Keldysh Institute. In one of these visits, he drew attention to the double-divergent form of the hydrodynamic consequences of discrete models (1) of the Boltzmann equation: ∂ Mqi ∂ L qi + = 0, ∂t ∂x L = exp(q1 + q2 − 1) + exp(q1 − 1) + exp(q1 − q2 − 1), M = exp(q1 + q2 − 1) − exp(q1 − q2 − 1),

S.K. Godunov and Kinetic Theory in KIAM RAS

f 1 = exp(q1 + q2 − 1),

383

f 2 = exp(q2 − 1),

f 3 = exp(−q1 − q2 − 1)

(q1 , q2 are new variables). I did not react, and in the next visit, he presented an extract from his review for a simple discrete model on one page. After that, we got down to research: this form was obtained for the Boltzmann equation (which was done, as it turned out, by the review authors themselves) and for its quantum analogs: ∂ L αμ + div Mαi = 0, i = 0, . . . , 4, ∂t

   1 L(α) = − ϕi (v)αi dv, ln 1 − θ exp θ i where ϕ0 = 1, ϕ1 = v1 , ϕ2 = v2 , ϕ3 = v3 , ϕ4 = v2 , and parameters αi are connected with usual macroscopic parameters density ρ, temperature T , and mean velocity u by formulae α4 = −

u2 1 ui 3 , αi = , α0 = ln ρ − ln(2π kT ) − . 2kT kT 2 2kT

This connection is obtained by comparison of two different forms of Maxwellian: ρ f 0 (v) ≡ exp (2π kT )3/2



(v − u)2 − 2kT



 = exp α0 +



αi vi + α4 v

2

=

i

   α2 α2 exp α4 v + , α = (α1 , α2 , α3 ), = exp α0 − 4α4 2α4 where θ is the indicator of statistics: θ → 0 denotes the Boltzmann statistics, θ = −1 denotes Bose statistics, and θ = +1 denotes Fermi statistics. This was published in the book [4] on kinetic equations, where these forms of Godunov were included. But S. K. Godunov did not stop: he proposed to do the same for the Vlasov equation. Here, too, an opportunity turned up: M. A. Negmatov and I began writing an article on the Vlasov equation. The reviewer wrote to us that the main result of the novelty is just the form of Godunov, and even proposed to change the name. We did just that. It turned out even two articles with the name of Godunov [5, 6]. Here is another comment from this review of Godunov and Sultangazin in 1971, which had consequences: “It should be noted that the method of choosing discrete velocities, which provide a description of gas flow with the conservation laws of three

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components of momentum and with energy conservation, has not been developed in the general case. This development is associated with difficulties of a combinatorialgeometric nature.” Now we can firmly say: developed. A somewhat detective story that influenced the development of European education. Going from one foreign country to another, Alexander Vasilyevich Bobylev entered our room and said: “Here, Vitya, look”. And he gave his joint article with Carlo Cercignani. The pathos of the article was in the construction of the first discrete models of the Boltzmann equation for mixtures. I looked at the models, and became suspicious, deciding to check the number of invariants. The fact is that a discrete model must contain exactly as many invariants as the original equation. In the Boltzmann equation, this is energy, the number of particles and components of an impulse, which means in a discrete model it should be the same, otherwise we have a catastrophe: if there is not enough or too many invariants, then a final distribution is not Maxwellian, and the hydrodynamic consequences are not correct. We already had a procedure for calculating invariants, and in the first of the models, there was one extra invariant. In the West, such invariants have the special name spurious invariants. The other model had twenty-five variables (the first twelve), and manually counting the number of invariants was difficult. But we knew that Alexander Dmitrievich Bruno had a special program for calculating Hamiltonian system invariants. In addition, we already knew that the algebra of conservation laws for discrete models of the Boltzmann equation and for chemical kinetics equations is the same as for Hamiltonian systems. I called A. D. Bruno, and he connected me with his graduate student A. Aranson, who did this. He found two extra invariants. I asked Viktor I. Turchaninov to check up, the result was the same. At that time we prepared for publishing an article just on the relation between conservation laws for discrete models of the Boltzmann equation and quantum Hamiltonians, and we included these results there [7]. When A. V. Bobylev returned, I showed the results, he immediately agreed, and even asked what kind of invariants these were. Again we turned to Aranson, he also presented the extra invariants themselves. A. V. Bobylev just at that time got a place in Sweden, in Karlstadt, where his task was to do University “from the Ryazan State Pedagogical College”, as he said. This meant to bring up several candidates of science (Ph.D.). He involved his two post-graduate students, Miriella Vinerian and Nicholas Bernhof, in this topic, and things went forward [8, 9]. Miriella Vinerian and Nicholas Bernhof fulfilled Ph.D., got married, and named their first child Alexander (as Bobylev). So our M. V. Keldysh Institute of Applied Mathematics of RAS with the efforts of several departments made a comprehensive contribution to the development of the Swedish and all-European civilization. These topics, which started in this way from the review of Godunov and Sultangazin [1], went on to develop further: conservation laws for the equations of chemical kinetics and the Liouville equation [10–14]. And the results for the Vlasov equation are now transferred to the Vlasov–Maxwell–Einstein equations [15].

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References 1. Godunov, S.K., Sultangazin, U.M.: On discrete models of the Boltzmann kinetic equation. Russ. Math. Surv. 26(3), 3–51 (1971) 2. Vasilyeva, O.A., Dukhnovsky, S.A., Radkevich, E.V.: On the nature of the local equilibrium of the Carleman and Godunov–Sultangazin equations. Contemp. Math. Fundam. Dir. 60, 23–81 (2016) 3. Vedenyapin, V.V.: Differential forms in spaces without a norm. The Boltzmann H –function uniqueness theorem. Russ. Math. Surv. 43(1), 193–219 (1988) 4. Vedenyapin, V.V.: Boltzmann and Vlasov Kinetic Equations. Fizmatlit, Moscow (2001) 5. Vedenyapin, V.V., Negmatov, M.A.: Derivation and classification of Vlasov-type and magneto hydrodynamics equations: Lagrange identity and Godunov’s form. Theor. Math. Phys. 170(3), 394–405 (2012) 6. Vedenyapin, V.V., Negmatov, M.A.: On the derivation and classification of equations of Vlasov type and magnetic hydrodynamics. Lagrange’s identity, Godunov form and critical mass. J. Math. Sci. (N.Y.) 202(5), 769–782 (2014) 7. Vedenyapin, V.V., Orlov, Yu.N.: On conservation laws for polynomial Hamiltonians and for discrete models of the Boltzmann equation. Theor. Math. Phys. 121(2), 1516–1523 (1999) 8. Bobylev, A.V., Vinerean, M.C.: Construction of discrete models with given invariants. J. Stat. Phys. 132(1), 153–170 (2008) 9. Bernhoff, N., Vinerean, M.: Discrete velocity models for mixtures without nonphysical collision invariants. J. Stat. Phys. 165(2), 434–453 (2016) 10. Batishcheva, Ya.G., Vedenyapin, V.V.: The second law of thermodynamics for chemical kinetics. Mat. Model. 17(8), 106–110 (2005) 11. Vedenyapin, V.V., Adzhiev, S.Z.: Entropy in the sense of Boltzmann and Poincare. Russ. Math. Surv. 69(6), 995–1029 (2014) 12. Adzhiev, S.Z., Vedenyapin, V.V., Volkov, Yu.A., Melikhov, I.V.: Generalized Boltzmann–type equations for aggregation in gases. Comput. Math. Math. Phys. 57(12), 2017–2029 (2017) 13. Vedenyapin, V.V., Adzhiev, S.Z., Kazantseva, V.V.: Entropy in the sense of Boltzmann and Poincare, Boltzmann extremals, and the Hamilton-Jacobi method in non-Hamiltonian context. Contemp. Math. Fundam. Dir. 64(1), 37–59 (2018) 14. Vedenyapin, V.V., Negmatov, M.A., Fimin, N.N.: Vlasov and Liouville-type equations and its microscopic and hydrodynamic consequences. Izv. Math. 81(3), 505–541 (2017) 15. Vedenyapin, V.V., Fimin, N.N., Chechetkin, V.M.: On the Vlasov–Maxwell–Einstein equation and its non-relativistic and weakly relativistic theory. Prepr. Keldysh Inst. Appl. Math. 265 (2018)

Modeling the Explosive Phenomena Driven by Rapid Phase Transition S. E. Yakush

Abstract The paper discusses the physical background of explosive phenomena caused by internal energy release in substances undergoing liquid-to-vapor phase transition. Two types of “physical” explosions are discussed: the boiling liquid expanding vapor explosion arising upon catastrophic failure of high-pressure vessels and known as one of major hazards in process industries, and the steam explosion, arising upon interaction of high-temperature melt with water or some other coolant having low enough boiling point. In both cases, the dynamics of two-phase mixture containing liquid and vapor phases of the substance is described by Euler equations with complex equation of state which is different from the equations of state for the individual phases. Numerical aspects arising in the solution of the governing equations are discussed, and results of typical numerical simulations demonstrating the development and propagation of pressure waves from the respective types of explosions are presented.

1 Introduction Explosions of different kinds are for the most part caused by rapid yield of thermal or chemical energy concentrated in a relatively small area. Some part of this energy is then converted to mechanical energy of an explosion wave propagating on the environment. While the most widespread sources of explosion energy are of chemical nature (gaseous and solid combustibles), other sources are also possible (electric discharges, laser spark, etc.). This paper considers the explosions caused by rapid release of internal energy of substances undergoing a phase transitions, namely, evaporation resulting in release of vapor with significant increase in the specific volume. Such events are termed “physical explosions”; they can occur with chemically inert substances (water, carbon dioxide, etc.).

S. E. Yakush (B) Ishlinsky Institute for Problems in Mechanics RAS, Moscow, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_51

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2 Boiling Liquid Expanding Vapor Explosions This type of physical explosions (referred to as BLEVE) is mostly encountered in processing, transportation, or storage of hydrocarbons, although large-scale accidents of this type are also known at CO2 storage sites [1]. Once the high-pressure contents of a vessel become exposed to low-pressure atmosphere, the substance boils up and expands into the atmosphere due to significant increase in the specific volume of liquid–vapor mixture. The expanding mixture plays the role of a piston generating pressure waves in the atmosphere. A model for boiling liquid expansion into the atmosphere was developed in [2]. The model is based on the assumption of thermodynamic equilibrium between the liquid and vapor phases in the course of substance expansion into the atmosphere, with no velocity slip between the phases. The assumption of thermodynamic equilibrium between the phases means that the mixture follows the saturation curve, i.e., all non-equilibrium processes are assumed to be much faster than the dynamic processes of substance expansion (see [3]). In [2], all interphase processes were assumed isentropic, which simplifies the model by eliminating the necessity to solve the energy equation. However, it was shown that expansion of two-phase cloud is featured by development of internal shocks which are known from gas dynamics to cause the increase in entropy. Therefore, in the subsequent work [4], this model was revised to omit the isoentropic condition in favor of solving the energy equation. The governing equations are ∂ρi + ∇ (ρi U i ) = 0 ∂t ∂ρi U + ∇ (ρi U U) = −∇ P ∂t ∂ρi E i + ∇ ((ρi E i + P)U) = 0 ∂t

(1) (2) (3)

where the subscript i denotes the zone: i = m stands for the two-phase mixture, i = g stands for the ambient atmosphere considered as a perfect gas with the ratio of specific heats k = 1.4; ρi is density, U is velocity, P is pressure, E i = ei + |U i |2 /2 is the specific total energy in the ith zone (ei is the specific internal energy). Note that a single velocity field is used because the gas and mixture are not penetrating through the contact surface, therefore, do not coexist at any point except the interface. Equations (1)–(3) are Euler equations, having the same form in both the two-phase (liquid–vapor) and single-phase (ambient air) zones. The main difference between the zones is in the equations of state relating the specific energy ei and density ρi to pressure P. For the perfect gas (i = g), we have P=

eg γg (k − 1)

(4)

In the two-phase zone (i = m), the mixture density and specific energy are determined in terms of the mass fraction of vapor, xv (see [2, 4]):

Modeling the Explosive Phenomena Driven by Rapid Phase Transition

ρm−1 = xv /ρv0 + (1 − xv )/ρl0 ; em = xv ev0 + (1 − xv )el0

389

(5)

where subscripts l and v denote liquid and vapor, while 0 superscript denotes the value taken on the saturation line. Since all properties denoted by subscript 0 are single-valued functions of pressure P, Eqs. (5) can be resolved to obtain the pressure and mass fraction of vapor: xv = xv (ρm , em ), P = P(ρm , em ). The mixture entropy is defined as sm = xv sv0 + (1 − xv )sl0 . In [2], it was assumed that sm = sl0 (P0 ), where P0 is the initial (high) pressure in saturated liquid; this condition immediately gives the mass fraction of vapor as a function of pressure, eliminating the energy equation (3). In the current model, this assumption is not made; however, the isoentropic speed of sound in the two-phase mixture, Cs = (∂ P/∂ρm )1/2 is evaluated numerically with the use of local entropy sm . The governing equations (1)–(3) have to be solved in the corresponding subdomains (i = m, g) with different equations of states (4), (5). The contact discontinuity separating these subdomains is moving with time, its position is determined by matching the two solutions. The approach toward the numerical implementation, shown to be robust and non-oscillatory near the interface, is the Ghost Fluid Method (GFM) [5]. The method consists in extending the corresponding solutions beyond the interface, with the flow parameters in the fictitious cells calculated from the local pressure; this allows the scheme stencil to applied equally in the flow domain and near-interface cells. The position of interface is tracked by the signed distance function φ defined at each cell center as the distance to the interface, taken positive in the gas domain and negative in the two-phase zone (therefore, the interface corresponds to level set φ = 0); the transport equation for φ is ∂φ/∂t + (U∇) φ = 0. In each zone (i = m, g), the governing equations can be discretized by a variety of numerical schemes available for the solution of hyperbolic conservation laws. When choosing a particular solution method, it should be taken into account that the properties of two-phase mixture are very different from an ideal gas law; therefore, schemes must be formulated in a general form, not relying on similarity to a perfect gas and not utilizing the ratio of specific heats (although different numerical schemes can be applied to the gas and two-phase zones). Here, we use the explicit scheme [6] due to its simplicity, requiring only evaluation of the speed of sound, rather than the analysis of eigenvectors of the hyperbolic system. In Fig. 1, pressure waves are shown generated by the burst of a 1 m-diam pressureliquefied propane vessel with the initial pressure P0 = 10 bar; the position of contact discontinuity is denoted by a symbol. It can be seen that two distinct stages are present, the primary expansion of the superheated liquid during which the converging boiling front propagates through the liquid and the pressure falls below the ambient level; this stage is followed by the formation of inward-facing shock in the two-phase mixture causing implosion, reflection, and formation of a secondary pressure peak traveling behind the primary one. This sequence of events is also known to exist in compressed gas explosions; a multiple-peak pressure records are observed experimentally [7]. Comparison of the shock arrival times and peak overpressure with test data [4] shows reasonable agreement.

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Fig. 1 Pressure profiles for burst of 1 m-diam pressure-liquefied propane vessel, P0 = 10 bar

3 Steam Explosions This type of physical explosions is also driven by rapid phase transition (evaporation); however, the source of internal energy is not contained in the liquid itself. Rather, it comes from another hot liquid with the melting point higher than the boiling point of the evaporating liquid. Such events are encountered in underwater volcano explosions, smelting industry, etc.; however, the main interest toward this phenomenon has been driven by severe accidents at nuclear power plants where core meltdown and relocation to water pool can occur. This problem has been studied both experimentally and theoretically for decades (e.g., [8–11]). In particular, melt–water interaction in shallow water pool (so-called stratified steam explosion) was shown experimentally to have much higher energy conversion ratio than was believed [12]. Steam explosion still remains a challenge for numerical modeling, even by dedicated computer codes [13], due to significant uncertainties in the knowledge of coupled physical processes involved. Among these, melt fragmentation due to interaction with water, development of vapor film instabilities resulting in direct contact of water with high-temperature melt, spontaneous triggering (initiation), and pressure wave propagation in the three-phase mixture (water, vapor, melt). Multi-fluid formulation for the three-phase system requires numerous closures for the interaction between the phases. However, even a simpler model is capable of capturing the basic physics and fluid dynamics involved in the process of steam explosion. For example, in [14] a model was developed where water and steam are considered as a single phase (mixture), with the second phase comprised of melt droplets. Another simplification introduced in the model [14] is that both water and vapor are described by barotropic equation of state, which is reasonable for fast processes considered. The governing equations in [14] are

Modeling the Explosive Phenomena Driven by Rapid Phase Transition

∂ρm ϕm + ∇ (ρm ϕm U m ) = 0 ∂t ∂ρm ϕm U m + ∇ (ρm ϕm U m U m ) = −ϕm ∇ P − F ∂t   ∂ xv ρm ϕm + (U m ∇) xv =  ∂t ∂ρs ϕs + ∇ (ρs ϕs U s ) = 0  ∂t  ∂U s + (U s ∇) U s = −ϕs ∇ P + F ρs ϕs ∂t

391

(6) (7) (8) (9) (10)

where ϕi is the volume fraction of ith phase (i = m is liquid–vapor mixture, i = s is melt, ϕm + ϕs = 1),  is the evaporation rate, and F is drag. In contrast to (1)–(3), these equations contain the drag and evaporation rate on the right-hand side. The closures are described in detail in [14]. From the numerical point of view, we have a system of coupled Euler-type equations (6)–(10) comprised of two subsystems for the liquid–vapor mixture (6)–(7) and melt droplets (9)–(10). Since the melt density is by an order of magnitude larger than that of water, the melt possesses much higher inertia and reacts to the drag slower than the mixture. Therefore, the subsystems can be solved sequentially, which simplifies the numerical procedure. The liquid–vapor mixture subsystem was solved in [14] by an explicit numerical scheme AUSMPW+ [15]; WENO5 scheme was used for Eqs. (8)–(10). An example of pressure wave propagation and self-acceleration in the premixed zone containing ϕs = 5 · 10−3 of molten corium (density ρs = 8 · 103 kg/m3 ) initiated by the small zone of high pressure near the left (closed) wall is shown in Fig. 2. One can see that pressure wave intensity is increasing with the distance passed, indicating the continuing energy transfer from melt to two-phase mixture.

Fig. 2 Pressure profiles for steam explosion with melt volume fraction ϕs = 5 · 10−3

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4 Conclusions Physical explosions are complex phenomena where many coupled fluid dynamics, heat and mass transfer processes are involved. Despite this complexity, the principal mechanism of pressure wave formation and propagation can be captured and described by simplified models based on the physical evaluation of characteristic times, and assumptions of some processes running much faster than others. In the examples presented, it was possible to reduce the problem complexity by assuming velocity equilibrium between the liquid and vapor phases and also infinitely fast establishment of thermodynamic equilibrium between the phases (in the model for BLEVE). As a result, well-established numerical approaches to the solution of hyperbolic conservation laws can be applied to the modeling of physical explosions. Acknowledgements This work was funded by Russian Science Foundation Grant 18-19-00289.

References 1. Abbasi, T., Abbasi, S.A.: J. Hazard. Mater. 141, 489 (2007). https://doi.org/10.1016/j.jhazmat. 2006.09.056 2. Yakush, S.E.: Int. J. Heat Mass Transf. 103, 173 (2016). https://doi.org/10.1016/j. ijheatmasstransfer.2016.07.048 3. van den Berg, A.C., van der Voort, M.M., Weerheijm, J., Versloot, N.H.A.: J. Loss Prev. Process Ind. 17(6), 397 (2004). https://doi.org/10.1016/j.jlp.2004.07.002 4. Yakush, S.: In: Proceedings of the Ninth International Seminar on Fire and Explosion Hazards (ISFEH9), pp. 430–439. St. Petersburg Polytechnic University Press, St. Petersburg (2019). https://doi.org/10.18720/spbpu/2/k19-103 5. Fedkiw, R.P., Aslam, T., Merriman, B., Osher, S.: J. Comput. Phys. 152(2), 457 (1999). https:// doi.org/10.1006/jcph.1999.6236 6. Kurganov, A., Tadmor, E.: J. Comput. Phys. 160(1), 241 (2000). https://doi.org/10.1006/jcph. 2000.6459 7. Laboureur, D., Birk, A.M., Buchlin, J.M., Rambaud, P., Aprin, L., Heymes, F., Osmont, A.: Process Saf. Environ. Prot. 95, 159 (2015). https://doi.org/10.1016/j.psep.2015.03.004 8. Fletcher, D., Theofanous, T.: Adv. Heat Transf. 29, 129–213 (1997). https://doi.org/10.1016/ S0065-2717(08)70185-0 9. Magallon, D., Huhtiniemi, I., Hohmann, H.: Nucl. Eng. Des. 189(1), 223 (1999). https://doi. org/10.1016/S0029-5493(98)00274-X 10. Meignen, R., Picchi, S., Lamome, J., Raverdy, B., Escobar, S.C., Nicaise, G.: Nucl. Eng. Des. 280, 511 (2014). https://doi.org/10.1016/j.nucengdes.2014.08.029 11. Meignen, R., Raverdy, B., Picchi, S., Lamome, J.: Nucl. Eng. Des. 280, 528 (2014). https:// doi.org/10.1016/j.nucengdes.2014.08.028 12. Kudinov, P., Grishchenko, D., Konovalenko, A., Karbojian, A.: Nucl. Eng. Des. 314, 182 (2017). https://doi.org/10.1016/j.nucengdes.2017.01.029 13. Leskovar, M., Centrih, V., Uršiˇc, M.: Nucl. Eng. Des. 296, 19 (2016). https://doi.org/10.1016/ j.nucengdes.2015.10.026 14. Rashkovskiy, S.A., Yakush, S.E., Sysoeva, E.Y.: J. Phys.: Conf. Ser. 1268, 012081 (2019). https://doi.org/10.1088/1742-6596/1268/1/012081 15. Kim, H., Kim, H., Kim, C.: AIAA J. 56(7), 2623 (2018). https://doi.org/10.2514/1.J056497

Explicit-Iteration Scheme for Time Integration of the Navier–Stokes Equations V. T. Zhukov and O. B. Feodoritova

Abstract We develop a splitting-based numerical algorithm for the compressible heat-conducting gas dynamics equations. The approach is based on two-step operator splitting method applied to the three-dimensional unsteady Navier–Stokes system. The first substep is hyperbolic one and implemented according to the Godunov-type scheme. The second substep consists of solving the parabolic subsystem using the explicit-iterative scheme. The advantages of the proposed approach are the fulfilment of conservation laws and efficiency of parallel implementation.

1 Introduction We present a new approach demonstrated here in application to the three-dimensional unsteady Navier–Stokes equations of compressible heat-conducting gas dynamics. At each time step, the system is solved in two consecutive substeps:first, we solve the subproblem, which represents the hyperbolic part of the system, i.e. we take into account the convective fluxes. In the second parabolic substep, the diffusion fluxes are taken into account. The computational reason is that it is faster to solve the split terms separately than to solve the whole system. The additional advantage of the splitting technique is that the hyperbolic and the parabolic equations have different nature and can be solved by different numerical methods. In our approach, the hyperbolic subproblem is solved by the explicit Godunov’s type scheme [1, 2]. Therefore, the time step τconv is restricted by the stability condition, see [2, p. 198]. For diffusion-dominated process, the usage of the explicit scheme with time step τconv is practically impossible. In this case, one might write an implicit scheme, which can be solved by an iterative algorithm. Usually such algorithms require empirical settings V. T. Zhukov (B) · O. B. Feodoritova Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow 125047, Russia e-mail: [email protected] O. B. Feodoritova e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. V. Demidenko et al. (eds.), Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, https://doi.org/10.1007/978-3-030-38870-6_52

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(accuracy tolerance, preconditioner). Instead of an implicit scheme, we propose a new explicit-iterative scheme called LINS (Local Iterations for Navier–Stokes equations). It inherits basic properties of the LI-M scheme [3] and provides the fulfilment of the conservation laws for the entire system. The need for efficient numerical methods for solving Navier–Stokes equations is caused by their prevalence in different mathematical models. Increased interest in explicit-iteration algorithms is motivated by the development of parallel computations: computer codes for multiprocessor supercomputers are required. The developed approach ensures scalable simulation in a wide range of processors.

2 The LI-M Scheme Consider an initial-boundary value problem for the d-dimensional parabolic equation u t = −Lu + f (x, t), x ∈ G ⊂ R d , t ∈ [0, T ], where f is a given function and u is a sought function; L is linear elliptic selfadjoint positive semidefinite operator. A typical operator L is the diffusion operator Lu = −div( k(x, t) grad u). A boundary condition, Dirichlet for simplicity, is set on the boundary ∂G of G, and an initial condition is specified at t = 0. In G × [0, T ], a grid  h, τ = G h × τ is introduced, τ = { tn , 0 ≤ n ≤ Nτ } is a grid in time with a variable step τ = tn+1 − tn > 0, and G h = {xi ∈ G, 0 ≤ i ≤ Nh } is a spatial unstructured grid characterized by the mesh parameter h = (1/Nh )1/d . The grid function u(x, t) at time tn is denoted by u n . We define a discrete operator L h approximating the operator L with order of accuracy O(h 2 ), assuming that L h is self-adjoint and its eigenvalues λ belong to the real interval [0; λmax ]. In the LI-M scheme, there is no special effort to control convergence iterations. The order of accuracy O (τ + h 2 ) is provided by the special Chebyshev iterative algorithm. The transition from the level tn to the level tn+1 is implemented in q iterations defining the time step τ and the upper bound λmax of the operator L h . The estimate of λmax can be obtained by means of the Gershgorin circle theorem [4]. The iterative parameters are defined by zeros√of the Chebyshev polynomial  T p (x) = cos ( p arccos x) of degree p = 0.25 π τ λmax + 1 , where s denotes the least integer greater than or equal to s. These zeros βi = cos ((i − 0.5)π/ p), i = 1, 2, . . . , p are reordered for stability with preserving the first value β 1 = cos these zeros, we form the   set of the iterative parameters  ( 0.5 π/ p). Using b1 , . . . , bq ≡ a p , . . . , a2 , a p , . . . , a2 , a1 , where am = λmax (z 1 − βm ) / (1 + z 1 ), m = 1, . . . , p, q = 2 p − 1, z 1 = β 1 . The LI-M algorithm is rather simple, easy to code and efficient in parallel implementation. We take an initial guess y (0) = u n and calculate the grid function u n+1 at the level tn+1 by the explicit iterations. The algorithm can be written in the following form: y (0) = u n ,

Explicit-Iteration Scheme for Time Integration of the Navier–Stokes Equations

y (l) =

395

 (0)  1 y + τ bl y (l−1) − τ L h y (l−1) + τ f n+0.5 , l = 1, 2, . . . , q, (1) 1 + τ bl u n+1 = y (q) ,

where f n+0.5 = f (x, tn + 0.5τ ). The choice β1 = cos ( 0.5 π/ p) provides the equality a1 = 0, so the last iteration in (1) is equivalent to the explicit scheme u n+1 = u n − τ L h u n + τ f n+0.5 . The transfer from u n to u n+1 is executed in q = 2 p − 1 steps, and the error propagation operator Sq = (I − G 2p ) · (I + τ L h )−1 can be expressed through the Chebyshev polynomial of the first kind G p (λ) of degree p that deviates least from 0 on the interval [λ0 ; λmax ], where λ0 = λmax (z 1 − 1)/(z 1 + 1) ∈ [−1/τ ; 0] , with the condition G p (−1/τ ) = 1. On the interval [0; λmax ], the key conditions |G p (λ)| ≤ 1 and |Sq (λ)| ≤ 1 are fulfilled. These inequalities provide the stability of the sheme. The polynomial G p can be expressed by the standard Chebyshev polynomial T p (x) = cos ( p arccos x) as (see [5]) G p (λ) = H p (λ)/H p (−1/τ ), where



m= p

H p (λ) =

(am − λ) ≡ T p (z 1 − (z 1 + 1) · λ/λmax ).

m=1

The LI-M scheme is investigated in [3] where full details of the scheme and the bibliography are given. In this paper, we construct a new LINS scheme for solution of the unsteady three-dimensional Navier–Stokes equations based on the LI-M scheme.

3 Time Integration LINS Scheme for the Navier–Stokes Equations Consider the system of three-dimensional Navier–Stokes equations, written in Cartesian coordinates xk , (k = 1, 2, 3). This system can be written in the form of the conservation laws relative to the state vector U = (ρ, ρu 1 , ρu 2 , ρu 3 , E)T : ∂ F1 ∂ F2 ∂ F3 ∂U + + + = 0, ∂t ∂ x1 ∂ x2 ∂ x3

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where the flux vectors μ

Fk (U ) = Fkconv (U ) + Fk (U ) + Fkλ (U ) are split into convective fluxes Fkconv (U ) = u k U + (0, pδk1 , pδk2 , pδk3 ) and diffusion (viscous and heat) fluxes μ

Fk (U ) = − (0, τk1 , τk2 , τk3 , 0)T , Fkλ (U ) = − (0, 0, 0, 0, u 1 τk1 + u 2 τk2 + u 3 τk3 + qk )T caused by viscosity and thermal conductivity. Here qk are components of the heat flux vector, (u 1 , u 2 , u 3 )T are the  components of the velocity vector u, ρ is  Cartesian density, p is pressure, E = ρ e + 0.5u 2 is the total energy, related to volume unit; u 2 = u 21 + u 22 + u 23 , τi j are viscous stress tensor components (below, the standard summation convention for repeated indexes is assumed):

τi j = μ

∂u j ∂u i + ∂x j ∂ xi



2 ∂u k − δi j . 3 ∂ xk

Perfect gas model is used, p = eρ(γ − 1) = ρ RT . The Navier–Stokes equations are discretized in integral form with conserved variables. Finite-volume node-based algorithm [6] is employed for spatial discretization. Firstly dual grid is constructed by forming non-overlapping volumes, referred to as dual cells, around each node. Integrating over dual cells, the semidiscrete scheme is written as dU + F conv (U ) + F μ (U ) + F λ (U ) = 0, (2) D dt where D is diagonal matrix, the elements of which are the dual volumes of cells, F conv , F μ , F λ are grid approximation of convective, viscous and heat fluxes, respectively, at each control volume face. We assume that the convective fluxes are determined by solving the Riemann problem at dual cell interfaces (exactly or approximately), diffusion fluxes are obtained by the finite element approximations. For the time discretization of the system (2), the LINS scheme is constructed by integrating (2) over a time interval [tn ; tn+1 ], where tn+1 = tn + τconv and τconv is the time step restricted by the convective stability condition. In the LINS scheme, the solution U n+1 at the new time level tn+1 is obtained using the values U n = (ρ, ρu 1 , ρu 2 , ρu 3 , E)T at the level tn in two substeps: hyperbolic and parabolic. The hyperbolic substep is carried out using a Godunov-type scheme. The parabolic substep solves two parabolic equations, momentum and energy equations, respectively,

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397

d H μ /dt + L μ V μ = g μ ,

(3)

d H λ /dt + L λ V λ = g λ .

(4)

Here L μ , L λ are self-adjoint discrete operators defined by numerical fluxes F μ , F λ . For example, for the energy equation (4), the operator L λ is an approximation of and the elliptic operator−div(λ∇T ). The grid functions H μ , V μ are the momentum  velocity vectors, and the functions H λ , V λ are the total energy E = ρ e + 0.5u 2 and temperature T . The parabolic substep is implemented by the explicit-iterative scheme presented in Sect. 2. In order to implement this scheme, one should calculate upper bounds λμmax , λλmax of the operators (d H μ /V μ )−1 L μ , (d H λ /V λ )−1 L λ . For the parabolic subsystems, the iterative setting is defined by values τconv   and λmax = max λμmax ; λλmax . For each subsystem, the iterative cycle (m = 1, . . . , 2 p − 1) is implemented in accordance with simple modification of the formula (1). We apply the LINS scheme to the momentum equation (3) and the energy equation (4) in the absence of convection. We prove that the sum of the hyperbolic and parabolic substeps provides the conservation laws of mass, momentum and energy. The main advantages of the LINS scheme are fulfilment of conservation laws and the efficiency of parallel implementation. The scheme has nice features: the convection subproblem is solved explicitly, the diffusion subproblem is always selfadjoint and solved by the explicit-iteration algorithm. On the diffusion substeps, the scheme does not require any input parameter. Numerical experiments confirm efficiency of this approach [7]. Acknowledgements This work is supported by the Russian Science Foundation, project no. 1771-30014.

References 1. Godunov, S.K.: A difference scheme for numerical solution of discontinuous solution of hydrodynamic equations. Math. Sb. 47, 271–306 (1959); translated US Joint Publications Research Service, JPRS 7225, 29 Nov 1960 2. Godunov, S.K., Zabrodin, A.V., Ivanov, M.Ya., Kraiko, A.N., Prokopov, G.P.: Numerical Solution of Multidimensional Problems of Gas Dynamics [in Russian]. Nauka, Moscow (1976), 400 p 3. Zhukov, V.T.: On an explicit numerical integration methods for parabolic equations. Math. Models Comput. Simul. 3(3), 311–332 (2011) 4. Gantmacher, F.R.: The Theory of Matrices. AMS Chelsea Publishing; Reprinted by American Mathematical Society (2000), 660 p 5. Lokutsievskij, O.V., Lokutsievskij, V.O.: On numerical solution of boundary-value problems for equations of parabolic type (English. Russian original). Sov. Math. Dokl. 34, 512–516 (1987); translation from Dokl. Akad. Nauk SSSR 291, 540–544 (1986)

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6. Abalakin, I.V., Bakhvalov, P.A., Gorobets, A.V., Duben, A.P., Kozubskaja, T.K.: Parallel NOISEtte software complex for large-scale calculations of aerodynamics and aeroacoustics [in Russian]. Calc. Methods Program. 13(3), 110–125 (2012) 7. Zhukov, V.T., Feodoritova, O.B., Duben, A.P., Novikova, N.D.: An explicit time integration of the Navier-Stokes equations using the local iteration scheme [in Russian]. Keldysh Inst. Preprint 12 (2019), 32 p. https://doi.org/10.20948/prepr-2019-12