Exact finite-difference schemes 9783110489644, 3110489643, 9783110489729, 3110489724, 9783110491326, 311049132X, 9783110491333, 3110491338

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Sergey Lemeshevsky, Piotr Matus, Dmitriy Poliakov (Eds.) Exact Finite-Difference Schemes

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Computational Methods in Applied Mathematics Carsten Carstensen (Editor-in-Chief) ISSN 1609-4840, e-ISSN 1609-9389

Exact Finite-Difference Schemes | Edited by Sergey Lemeshevsky, Piotr Matus, Dmitriy Poliakov

Editors Sergey Lemeshevsky Institute of Mathematics, NAS of Belarus, 11 Surganov St., 220072 Minsk, Belarus, [email protected] Piotr Matus Institute of Mathematics, NAS of Belarus, 11 Surganov St., 220072 Minsk, Belarus; Institute of Mathematics and Computer Science The John Paul II Catholic University of Lublin, Al. Raclawickie 14, 20-950 Lublin, Poland, [email protected] Dmitriy Poliakov Institute of Mathematics, NAS of Belarus, 11 Surganov St., 220072 Minsk, Belarus, [email protected]

ISBN 978-3-11-048964-4 e-ISBN (PDF) 978-3-11-049132-6 e-ISBN (EPUB) 978-3-11-048972-9 Set-ISBN 978-3-11-049133-3 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2016 Walter de Gruyter GmbH, Berlin/Boston Cover image: Silvia Jerez and Saúl Díaz-Infante Typesetting: Konvertus, Haarlem, NL Printing and binding: CPI books GmbH, Leck ♾ Printed on acid-free paper Printed in Germany www.degruyter.com

Preface This book is devoted to exact numerical methods of mathematical models. A lot of such models can be formulated as initial boundary-value problems (IBVPs) for partial differential equations (PDEs), which may be multidimensional, as well as non-linear. However, it is not possible in general to determine the solution of such problems in a closed form. Therefore, the exact solution must be approximated by numerical techniques. One of commonly used computational methods is the finite-difference method (FDM). Considerable progress has been made in developing the theory of finite-difference schemes (FDSs), established by the famous Russian mathematician Aleksandr Samarskii (see ref. [1]). One of the main problems in constructing FDSs for equations of mathematical physics is the accuracy, i.e., the convergence of the approximate solution y of a discrete problem to the exact solution of a differential problem u in some norm. It is natural to desire the maximum order of convergence rate for a minimum number of grid nodes participating in the notation of the FDS. It turns out that for some PDEs it is possible to construct exact numerical algorithms, i.e., the exact solution u of PDE coincides with the approximate solution y and approximation error equals zero at all grid nodes. Godunov in 1959 [2] pointed out the possibility of constructing an exact-difference scheme (EDS) for linear one-dimensional transport equation. In his famous monograph [1], A. Samarskii constructed an EDSs for the Dirichlet problem for stationary one-dimensional convection-diffusion equation. EDS in this book are developed on the basis of general theory of FDS with the use of special Steklov averaging and weighting of difference derivatives by spacial variables. Main features of EDSs are: – on the basis of the EDSs one can construct simple computational algorithms of a high order of accuracy; – for complicated problems of mathematical physics, it is expedient to construct such computational methods that would be exact in the case of degeneration of the equation into equations of a more simpler structure in some domains. The aim of the authors is to describe the state-of-the-art in this field of numerical analysis. The EDSs can be used as a starting point for the construction of FDS suitable for implementation on a computer. It is shown that the combination of the EDS with the modern quadrature formulas results in numerical algorithms that are highly efficient. Questions of construction FDSs of an arbitrary order of accuracy for boundary value problems for ordinary differential equations are considered in the monograph of Gavrilyuk et al., [3]. For PDSs the pioneering works are the monographs of American mathematician Mikkens [4, 5], which use an approach of the so-called non-standard FDSs. Let us also notice the approaches to constructing FDSs of increased order given in the monographs of Tolstykh [6], Ashyralyev and Sobolevskii [7].

VI | Preface

The book is addressed to graduate students of mathematics and physics, as well as to working scientists and engineers as a self-study tool and reference. We now outline the contents of the book. The introductionary chapter is designed to acquaint the reader with EDS for ordinary differential equations and Euler–MacLauren quadrature formula. Chapter 2 deals with IBVPs for hyperbolic equations and systems. In particular, it concerns linear and non-linear transport equations and systems of the first order and the equation of vibrations of a string with Dirichlet, Neumann or Robin boundary conditions. It is shown that algorithms for semilinear transport equation can be used for exact numerical simulation of shock waves with a variable amplitude without smearing and non-physical oscillations. Chapter 3 is devoted to EDS for parabolic equations on the solution of the type of the travelling waves and with separated variables. The algorithms are proposed also for the multidimensional case. The last section of this chapter is devoted to development of the L1 -conservative EDSs. The following chapters of the book contain the various applications of EDSs. They are prepared by several groups of leading scientists working in this field. Chapter 4 was written by the outstanding mathematicians, the pioneer of non-standard finite difference (NSFD) methodology, Ronald E. Mickens and Talitha M. Washington. The main purpose of this chapter is to provide a brief summary of some of the results coming from the application of the NSFD methodology to the discretization of both ordinary and partial differential equations. Chapter 5 is prepared by a well-known group of Ukrainian scientists Ivan Gavrilyuk, Myroslav Kutniv and Volodymyr Makarov and is devoted to construction of EDS and associated truncated difference schemes (TDS) for arbitrary ODEs. Chapter 6 by Silvia Jerez and Saúl Díaz-Infante presents a new way to construct explicit numerical methods for stochastic differential equations (SDEs) via the Steklov mean. Chapter 6 prepared by Ryszard Kozera, Agnieszka Paradzinska and Denis Schadinskii proposes algorithms that are used for determinations of the blow-up effects in numerical methods approximating IBVP for quasilinear parabolic equations. The leading authors S. Lemeshevsky, P. Matus and D. Poliakov want to express gratitude to the above-mentioned mathematicians for submitted surveys. Finally, we wish to thank our many colleagues and P. Matus, students from Institute of Mathematics, NAS of Belarus (U.A. Irkhin, A.A. Kirshtein) and from Department of Mathematics of Catholic University of Lublin, Poland (A. Kolodynska, M. Lapinska-Chrzczonowicz) for their contribution to EDSs and applications.

Bibliography [1] A. A. Samarskii. The theory of difference schemes. Marcel Dekker Inc., New York - Basel, 2001. [2] S. K. Godunov. A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat. Sb. (N.S.), 47(89)(3):271–306, 1959. (in Russian).

Preface

|

VII

[3] I. P. Gavrilyuk, M. Hermann, V. L. Makarov, and M. V. Kutniv. Exact and truncated difference schemes for boundary value ODEs, volume 159 of International series of numerical mathematics. Birkhäuser / Springer Basel AG, Berlin, 2010. [4] R. E. Mickens. Applications of nonstandard finite difference schemes. World Scientific, Singapore, 2000. [5] R. E. Mickens. Advances in the applications of nonstandard finite difference schemes. World Scientific, Singapore, 2005. [6] A. I. Tolstykh. High accuracy non-centered compact difference schemes for fluid dynamics applications, volume 21. World Scientific, Singapore, 1994. [7] A. Ashyralyev and P. E. Sobolevskii. New difference schemes for partial differential equations, volume 148. Birkhäuser, 2012.

Contents Basic notation | XIII Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov Preliminary results | 1 1 Exact finite-difference schemes for ordinary differential equations | 1 2 Effective computations of integrals | 4 Bibliography | 6 Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov Hyperbolic equations | 7 1 Linear and semi-linear transport equation | 7 1.1 One-dimensional transport equation | 7 1.2 Two-dimensional transport equation | 14 1.3 Three-dimensional transport equation | 15 2 Exact numerical simulation of shock waves with a variable amplitude | 24 2.1 Two-dimensional advection-reaction equation | 27 3 Linear system of two equations | 28 3.1 Cauchy problem | 28 3.2 IBVP problem in Riemann invariants | 33 4 Riemann problem for non-linear transport equation | 44 4.1 Numerical experiment | 47 5 Non-homogeneous quasilinear transport equation | 49 5.1 Statement of the problem | 49 5.2 Finite-difference scheme | 50 5.3 Approximation error | 50 5.4 Stability of the finite-difference scheme | 52 5.5 Convergence | 56 5.6 Numerical experiment | 57 6 Linear equation of vibrations of a string | 62 6.1 Dirichlet IBVP | 62 6.2 Robin IBVP | 67 6.3 Numerical experiment for discontinuous input data | 72 7 Non-linear equation of vibrations of a string | 80 7.1 Statement of the problem | 80 7.2 Finite-difference schemes on characteristic grids | 81 7.3 Numerical experiment | 84

X | Contents

8

Non-linear gas dynamic system | 86

8.1

Statement of the problem | 87

8.2

Finite-difference scheme | 89

8.3

Numerical experiment | 91

Bibliography | 93 Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov Parabolic equations | 95 1

Travelling wave type solutions | 95

1.1

One-dimensional parabolic equation | 95

1.2

Two-dimensional convection-diffusion-reaction equation | 102

2

Solutions of the separated variables | 115

2.1

Introduction | 116

2.2

The Cauchy problem for parabolic equations | 116

2.3

Arbitrary-order finite-difference schemes for the BVP for parabolic equations | 122

2.4

The BVP for parabolic equations with small parameter | 126

2.5

The Cauchy problem for multidimensional parabolic equations | 129

3

L1 -conservative FDS for the Neumann problem | 133

3.1

Introduction | 133

3.2

Exact L1 -conservative finite-difference scheme | 135

3.3

Exact conservative iteration process | 137

3.4

Flow finite-difference scheme | 138

3.5

Exact L1 -conservative flow algorithm | 139

3.6

Multidimensional generalization | 140

Bibliography | 142 Ronald E. Mickens and Talitha M. Washington Use of exact difference schemes to construct NSFD discretizations of differential equations | 144 1

Introduction | 144

2

Exact finite-difference schemes | 145

3

Logistic equation | 146

4

Second-order ODE having constant coefficients | 146

5

Jacobi differential equations | 148

6

A non-linear reaction-advection PDE | 149

7

Comment | 150

8

NSFD methodology | 150

9

Discrete derivatives | 151

Contents

| XI

10 Non-locality of functional terms | 152 11 Other properties | 152 12 Dynamic consistency | 153 13 Subequations | 153 14 NSFD applications | 154 15 Conservative oscillators | 154 16 Time-independent Schrödinger equation | 155 17 Linear, advective-diffusive PDE | 156 18 Linear advection, non-linear reaction PDE | 158 19 Combustion model | 159 20 Coupled interacting population | 160 21 Black–Scholes equation | 162 22 Summary | 163 Bibliography | 163 Ivan Gavrilyuk, Myroslav Kutniv, and Volodymyr Makarov Exact and truncated difference schemes for boundary-value problem | 165 1 Introduction | 165 2 Three-point difference schemes of high-order accuracy for linear boundary-value problems | 168 3 Exact and truncated difference schemes for non-linear boundary-value problems | 174 4 Two-point difference schemes for systems of first-order ODEs | 175 5 Three-point difference schemes for non-linear second-order ODEs | 178 6 Three-point difference schemes for non-linear BVPs on the half-axis | 184 7 Three-point difference schemes for singular non-linear BVPs | 191 8 Exact- difference schemes for PDEs | 196 Bibliography | 200 Silvia Jerez and Saúl Díaz-Infante Exact difference schemes for stochastic differential equations | 204 1 Introduction | 204 2 General settings | 205 3 Steklov method | 206 4 Convergence and Stability | 209 5 Linear Steklov method | 210 6 Almost sure stability | 213 7 Numerical simulations | 215 Bibliography | 218

XII | Contents

Ryszard Kozera, Agnieszka Paradzinska, and Denis Schadinskii Numerical blow-up time | 220 1 Introduction | 220 2 The Cauchy problem for ODE | 221 3 Statement of the problem and FDS | 221 4 Solvability of the implicit FDS for ODE | 225 5 Numerical experiment | 226 6 The Neumann problem for a parabolic equation with a non-linear source of power form | 228 Bibliography | 231 List of contributors | 233

Basic notation ¯ h = {x | x = x i = ih, ω

i = 0, 1, . . . , N,

Nh = l}

ω h = {x | x = x i = ih,

i = 1, 2, . . . , N − 1,

Nh = l}

uniform rectangular grid on [0, l]

internal nodes of the uniform rectangular grid on [0, l]

u x¯ ≡

h d2 u u i − u i−1 du (x ) + O(h2 ) = (x i ) − h dx 2 dx2 i

left difference derivative

ux ≡

h d2 u u i+1 − u i du (x ) + O(h2 ) = (x i ) + h dx 2 dx2 i

right difference derivative

u x˚ ≡

u i+1 − u i−1 du h2 d3 u (x ) + O(h3 ) = (x ) + 2h dx i 3 dx3 i

central difference derivative

u x¯ x =

u x − u x¯ u i+1 − 2u i + u i−1 = h h2

second difference derivative

    ω τ = t n = nτ, n = 0, N T , τN T = T = ω τ ∪ t N T = T

uniform time grid

ω = ωh × ωτ

uniform rectangular space-time grid

(y, w) ≡



y(x)w(x)h, (y, w)± ≡

N−1 

hg h,i f h,i ,



N−1   gh , fh = hu g,i v f ,i ,

i=0

[ g h | 2 =

N−1 

y(x)w(x)h

x∈ω±

x∈ω

[g h , g h ) =



i=1

2 hg 2h,i , |g h | =

i=0

N−1  i=1

g (·)C = g C = max g (x ) x∈[0, l]

hg 2h,i

scalar products

Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov

Preliminary results 1 Exact finite-difference schemes for ordinary differential equations One of the main problems in constructing FDSs for equations of mathematical physics is the accuracy, i.e., the convergence of the approximate solution y of a discrete problem to the exact solution u of a differential problem in some norm. It is natural to desire the maximum order of convergence rate for a minimum number of grid nodes participating in the notation of the FDS (the stencil of a FDS). Definition 1.1. Finite-difference scheme (FDS) is called exact-difference scheme (EDS) if the approximation error is equal to zero or y = u at all grid nodes. Example 1.1 (Samarskii [1]). For a boundary value problem for the second-order ODE

k(x)u (x)



= 0,

0 < x < l,

u(0) = μ 1 ,

 u l = μ2 ,

(1)

FDS y −y y −y  1

a i+1 i+1 i − a i i i−1 = 0, i = 1, N − 1, y0 = μ1 , y N = μ2 , h h h ⎞−1 ⎛ x i 1 dx ⎟ ⎜ ai = ⎝ ⎠ , y = y i = y(x i ), x i ∈ ω h , x i − x i−1 k(x)

(2)

(3)

x i−1

defined on the uniform grid: ¯ h = {x | x = x i = ih, ω

i = 0, 1, . . . , N,

Nh = l},

  in the segment 0, l with a constant step h = x i − x i−1 is EDS in the class of piece-wise continuous coefficient k(x) with a finite number of discontinuity points of first kind. In ref. [1], the EDS is constructed for a more difficult problem

  k(x)u (x) − q(x)u(x) = −f (x), 0 < x < l, u(0) = μ1 , u l = μ2 . However, the coefficients of the appropriate FDS are defined via the integral relationship of the functions k(x), q(x), f (x), and their practical use in the general case is difficult. For example, the stencil functional (3) cannot be calculated exactly for an

2 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov

arbitrary function k(x). Nevertheless, using the simplest quadrature formulas, we can   construct numerical methods of any order of accuracy O h m , m ≥ 2. So we can construct simple numerical algorithms of high order of accuracy on the basis of EDS. We can also construct FDSs that will be exact just on some types of input data. Let us note some results on EDSs. In ref. [2], the main focus is the theory of numerical methods for boundary-value problems (BVPs) that are based on the EDSs as well as on the well-studied and sophisticated solvers for initial value problems (IVPs). It is shown that the combination of the EDS with the modern IVP-solvers results in numerical algorithms that are highly efficient. The theory of the EDS permits the construction of a posteriori error estimators, which in turn are the basis for adaptive algorithms. In this section, using the special Steklov averaging ⎛  ⎜ f u(t) ≈ ⎝ 

u 

n+1

1 u n+1 − u n

⎞−1 du ⎟ ⎠ , f (u)

t n ≤ t ≤ t n+1 ,

u n = u(t n ),

t n = nτ,

un

the EDSs for some types of non-linear ODEs are constructed [3]. The mathematical apparatus given here will be used for developing the theory of EDSs for partial differential equations (PDEs). Consider the Cauchy problem for a non-linear equation of the form du = f (u, t), dt

f (u, t) = f 1 (u)f2 (t),

t > 0,

u(0) = u0 .

(4)

Introduce the uniform grid ω τ = {t n = nτ, n = 0, 1, . . . } with a constant step τ. We will denote by y = y n = y(t n ) the value of the grid function at the node t n . Since the variables in equation (4) are decoupled, the problem of constructing the EDS is trivial: 

y n+1 − y n = φ y n , y n+1 , t n , n = 0, 1, . . . , y0 = u0 , τ 



φ y n , y n+1 , t n = φ1 y n , y n+1 φ2 (t n ), ⎛ ⎞−1 y n+1  t n+1

 ⎜ ⎟ 1 1 du n n+1 ⎟ , φ2 (t n ) = =⎜ f2 (t)dt. φ1 y , y ⎝ y n+1 − y n f1 (u) ⎠ τ yn

(5)

(6)

tn

In this case, the function f1 (u) is replaced by the functional φ1 (y) by means of the specific Steklov averaging with respect to the solution u or to the non-linearity. We should note the surprising connection between this approximation and

Preliminary results | 3

approximation (3), which was used for constructing the EDS (2) for a differential equation of an absolutely different nature (1). We show that FDS (5), (6) is EDS for the problem (4). The residual or the approximation error ψ = ψ(t n ) can be presented in the form 

u n+1 − u n − φ u n , u n+1 , t n τ ⎞ ⎛ ⎛ ⎞ t n+1 u n+1 un    ⎟ u n+1 − u n ⎜ du du ⎟ 1 ⎜1⎜ = n+1 f2 (t)dt⎟ − ⎠− ⎝ ⎠. ⎝ τ f1 (u) f1 (u) τ u  u0 u0 t n+1 du f1 (u)

ψn =

(7)

un

Using equation (4), we can obtain the presentation ⎞ ⎛ u 1 du d⎜ du ⎟ = f2 (t), ⎠ = f2 (t), ⎝ f1 (u) dt dt f1 (u) u0

u u0

du = f1 (u)

t f2 (t)dt,

⎛

1⎜ ⎝ τ

u 

n+1

u0

0

du − f1 (u)

u u0

n

⎞ du ⎟ 1 ⎠= f1 (u) τ

t n+1 f2 (t)dt. tn

Substituting the later expression into (7), we obtain ψ n = 0 for arbitrary n = 0, 1, . . . , i.e., the FDS (5), (6) is EDS. Note that the integrals on the right-hand sides of formula (6) can be calculated directly in many cases. Therefore, in numerical practice it is convenient to use the EDS (5), (6). Example 1.2. Consider the Cauchy problem (4) with the right-hand side f (u, t) = −λu + c; λ, c = const: du = −λu + c, dt

t > 0,

u(0) = u0 .

(8)

Averaging the right-hand side of equation (8) according to rule (6), we obtain the following EDS: ⎛ y

n+1

n



n+1

⎞−1

−y 1 +c ⎠ 1 −λy , = ⎝− n+1 n ln τ −λy n + c y −y λ

y0 = u0 .

(9)

It is easy to see that from formula (9), we can find an explicit presentation of the approximate solution

c c + , y(t n+1 ) = e−λt n+1 u0 − λ λ

4 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov

which coincides with the exact solution at the nodes t = t n+1 :

c c + , t ≥ 0. u(t) = e−λt u0 − λ λ Example 1.3. Consider a more difficult example du = −λu2 , dt

t > 0,

u(0) = u0 .

Averaging the function u2 (t) by rule (6): ⎛ ⎜ u2 (t) ≈ ⎝

u 

⎞−1

n+1

1 u n+1 − u n

du ⎟ ⎠ u2

= u n u n+1 ,

t n ≤ t ≤ t n+1 ,

un

we obtain a linearized EDS of the form y n+1 − y n = −λy n y n+1 , τ

y0 = u0 .

Example 1.4 (Mickens [4]). Let us note an EDS for (8) with c = 0 is constructed in the form y n+1 − ψ(τ)y n = −λy n , φ(τ)

y0 = u0 ,

with ψ(τ) = 1,

φ(τ) =

1 − e−λτ . λ

Example 1.5. It is interesting to note the EDS t2 + t n t n+1 + t2n+1 y n+1 − y n = y n y n+1 n , τ 3

y0 = u0 ,

approximates the Cauchy problem of the form du = u2 t2 , dt

t > 0,

u(0) = u0 .

2 Effective computations of integrals In this section, the simple method to calculating the quadrature coefficints of the Euler–MacLaurin formula of high orders for numerical integration is given. The whole material is taken from the article [5].

Preliminary results | 5

Let u(x) be sufficiently smooth, i.e., it has bounded continuous derivatives as long   as necessary. Consider uniform grid ω h = x0 + ih, 0 ≤ i ≤ K with constant step h, where Kh = l = x K − x0 . The simplest quadrature rules on the grid are the trapezoidal and midpoint rules. They have an accuracy O(h2 ). It is well known, that minor adjustments of Euler–MacLaurin to the trapezoidal rule ⎛ ⎞ x K K−1  h4

  u u h2   + ··· u(x)dx = h⎝ 0 + ui + K ⎠ − u K − u0 + u K − u0 2 2 12 720 x0

i=1

significantly increase its order of accuracy. Similar adjustments can be constructed for the midpoint rule. The general form of the Euler–Maclaurin formula can be easily found. But methods for deriving numerical coefficients of this formula, described in the literature, are too bulky. Here, the simple method for deriving them is given and sufficiently many coefficients to meet the needs of the practice are found. The general Euler–Maclaurin formula based on trapezoidal rule can be written only for uniform grid. The formula with M terms is the following: ⎛ ⎞ x K K−1 M     u u m 2m (2m−1) u(2m−1) . (10) u(x)dx ≈ h⎝ 0 + ui + K ⎠ + − u (−1) a m h 0 K 2 2 x0

m=1

i=1

If u(2M+2) (x) is continuous, then the approximation error is O(h2M+2 ). Let us give a simple method for the computation of coefficients a m . The values of these coefficients do not depend on u(x), grid step h, or number of intervals K of the grid. Take u(x) = x 2M , N = 1, h = 1, x0 = 0, x N ≡ x1 = 1; substitution of these values to (10) gives 1  1 (2M )! m am . = + (−1) 2M + 1 2 (2M − 2m + 1)! M

(11)

m=1

Setting M = 1, we have only one unknown coefficient a1 in (11), which can be found. Then setting M = 2, with a1 already found, we compute a2 . So, increasing M by 1 we can find any number of coefficients. The six first coefficients were computed and five of them are given in Table 1, but 6th is not because it is too big. For the first five coefficients, the quantity 1/a m is an integer, but for 6th it is a fraction. We can significantly simplify the computations by introducing new coefficients c m , which are related to the old ones by the ratio a m = c m /(2m + 2)!.

6 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov

Tab. 1. Coefficients of Euler–Maclaurin formula. m

1

2

3

4

5

6

1/a m cm dm

12 2 1

720 1 1

30240 4/3 1

1209600 3 1

47900160 10 25/24

... 691/15 691/600

Then formula (11) can be rewritten as M 

m 2M (−1) C2m−1

m=1

cm 2M − 1 = −2 . m(m + 1)(2m + 1) 2M + 1

(12)

Here, C2M 2m−1 are binomial coefficients. Setting M = 1, 2, . . . in (12), one gets c m given in Table 1; they look simple enough. It is interesting to notice another simplification. Introduce the coefficients d m with the relation cm =

2 (m − 1)!d m . m

These coefficients are also given in Table 1. First four of them are exactly equal to 1, and further ones start lightly deviating from unity. The formulas described above allow us to numerically integrate smooth enough functions with high accuracy. Coefficients given in Table 1 reduce the error to O(h14 ), that is enough to meet most needs of practice.

Bibliography [1] A. A. Samarskii. The theory of difference schemes. Marcel Dekker Inc., New York - Basel, 2001. [2] I. P. Gavrilyuk, M. Hermann, V. L. Makarov, and M. V. Kutniv. Exact and truncated difference schemes for boundary value ODEs, volume 159 of International series of numerical mathematics. Birkhäuser / Springer Basel AG, Berlin, 2010. [3] P. Matus, U. Irkhin, and M. Lapinska-Chrzczonowicz. Exact difference schemes for time-dependent problems. Comput. Meth. Appl. Math., 5(4):422–448, 2005. [4] R. E. Mickens. Advances in the applications of nonstandard finite difference schemes. World Scientific, Singapore, 2005. [5] N. N. Kalitkin. The Euler–Mcloren formulae of high orders. Matem. Mod., 16(10):64–66, 2004. (in russian).

Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov

Hyperbolic equations Hyperbolic equations and systems include various linear and non-linear problems of mathematical physics. For many of them, it is possible to construct EDS and FDS of high order of approximation. In this chapter, we start with linear and semi-linear transport equations. Although they may seem to be rather simple, we are going to show general concepts of our technique (special Steklov averaging, non-standard weighing by spatial variables) to obtain EDSs. Let us note that constructed algorithms for semilinear equation are used for exact numerical simulation of shock waves with a variable amplitude. Obtained results are generalized to linear acoustics system, quasi-linear transport equations, second-order hyperbolic equations with boundary conditions of Dirichlet, Neumann or Robin types, and gas dynamics system.

1 Linear and semi-linear transport equation In this section, the FDSs for linear and semi-linear transport equations are described and analyzed. Transport PDEs model the reactive transport of solutes in subsurface flows, fluid dynamics, and many other important applications. These equations admit solutions with moving steep fronts, which need to be resolved accurately in applications and often cause severe numerical difficulties. Standard finite-difference or finite-element methods of second order of accuracy construct numerical solutions with severe non-physical oscillations. While upstream weighing methods of the first order of accuracy can eliminate these oscillations, they introduce excessive numerical dispersion [1, 2]. One of the approaches for constructing effective numerical algorithms for such problems is the concepts of EDS. The idea for constructing such algorithms is using the special Steklov averaging of spacial variables and of the reaction term, which lead to difference equations with integral terms.

1.1 One-dimensional transport equation 1.1.1 Basic features of transport equation   In the domain Q¯ T = 0, l × [0, T ], let us consider the IBVP for linear transport equation ∂u ∂u +a = f (x, t), 0 < x ≤ l, 0 < t ≤ T, a > 0, ∂t ∂x u(x, 0) = u0 (x), 0 ≤ x ≤ l, u(0, t) = μ1 (t), 0 < t ≤ T.

(1) (2)

8 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov

A weak solution of the problem (1)–(2) is piecewise continuously differentiable function u(x, t) that satisfies the initial and boundary conditions (2) with compatibility condition u0 (0) = μ(0), and an integral conservation law [3, p. 505]   udx − audt + f (x, t)dxdt = 0, Q

∂Q

∀ Q ⊂ Q T . (Here ∂Q is the boundary of the domain Q .)

Since the solution of problem (1)–(2) along the characteristics equation [3, p. 25] du = f (x, t), dt dx =a

dx dt

= a satisfies the

dt

so

⎧    t

⎪ ⎪ ⎪ ⎨u0 (x − at) + f x − a t − ξ , ξ dξ , u(x, t) =

x  0x  x−ξ ⎪ ⎪ ⎪ dξ , μ ⎩ 1 t − a + f ξ, t − a 0

0 < at ≤ x < l, t ≤ T, (3) 0 < x < at ≤ aT, x < l.

From this formula, triangle inequality and integral properties, the following estimate follows !

" t #  #   # # u (t )C ≤ max u 0 C , max μ 1 ξ + #f ·, ξ # dξ , C

ξ ∈[0, t]

∀t > 0,

0

where ·C is uniform norm in space, defined by g (·)C = g C = max g (x ). x∈[0, l]

1.1.2 Weighted finite-difference scheme In the domain Q T , we introduce uniform rectangular grid ω = ωh × ωτ ,     ω h = x i = ih, i = 0, N, hN = l = ω h ∪ x0 = 0, x N = l ;

Hyperbolic equations

| 9

    ω τ = t n = nτ, n = 0, N0 , τN0 = T = ω τ ∪ t N0 = T , ω = ωh × ωτ . For approximation of problem (1)–(2), consider a weighted FDS [4] on the grid ω = φ, y(α)t + ay(σ) x x ∈ ωh ,

y(x, 0) = u0 (x),

(x, t) ∈ ω, y(0, t) = μ1 (t),

(4) t ∈ ωτ ,

(5)

where y = y ni = y(x i , t n ),

y ni+1 − y ni y n+1 − y ni , y nt,i = i , h τ y(σ),n = σy n+1 + (1 − σ)y ni , α, σ ∈ [0, 1], i i

y nx,i =

n = αy ni + (1 − α)y ni−1 , y(α),i

(σ) and φ - some stencil functional of f , i.e., one can set φ = f(α) . Rewrite FDS (4)–(5) in the canonical form: n+1

(α + γσ)y i

= (α + γσ − γ)y ni + (α + γσ − 1)y n+1 i−1   + 1 − α + γ(1 − σ) y ni−1 + τφ ni , i = 1, N,

n = 0, N T − 1.

(6)

Here, γ is a Courant number γ=

aτ . h

Let us consider FDS in an abstract form [5]. We introduce the families of linear spaces ¯ with vector parameter h provided with and B(2) given on grid ω of grid functions B(1) h h (1) (2) norm |h| > 0. Suppose that B h , B h are finite-dimensional spaces whose dimension depends on h and can tend to infinity as |h| → 0. We say that L h y = φ,

y ∈ B(1) h ,

φ h ∈ B(2) h ,

(7)

is the FDS in abstract form. Here, L h y is the non-linear operator from D(L h ) = B(1) to h . The operator L defines a structure of the concrete FDS y the solution of R(L h ) ⊆ B(2) h h the FDS, and φ the input data of the problem. For example, for the time-dependent problems we have ⎧ ⎪ ⎪ is the right-hand side at the inner grid-points, ⎪ ⎨f h φ = μ h are the boundary conditions at the boundary grid-points, ⎪ ⎪ ⎪ ⎩u is the initial condition at t = 0. 0h

˜ ∈ B(2h ) in (7), we get the equation with respect to the Perturbing the input data φ perturbed solution y˜ : ˜ L h y˜ = φ.

(8)

10 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov

˜ − φ ≥ 0(≤ 0) implies Definition 1.1. [6, 7]. FDS (7) is called monotone if the condition φ the inequality y˜ − y ≥ 0(≤ 0), where y˜ is the solution of the perturbed problem (8). In the case of a linear operator L, Definition 1.1 is equivalent to the following definition. Definition 1.2. [6, 7]. FDS (7) with linear operator L is called monotone if the condition φ ≥ 0(≤ 0) implies the inequality y ≥ 0(≤ 0). For the monotonicity, it is necessary to require the positivity of all the coefficients in equation (6). It results in the following conditions on weights and grid steps ⎧ ⎪ ⎪ γ ≤ 1, σ = 0, α = 1, ⎪ ⎨ 1−α α ≤γ≤ , 0 < σ < 1, α + σ ≥ 1, ⎪ σ 1−σ ⎪ ⎪ ⎩γ ≥ 1 − α, σ = 1. which are equivalent to the following system of inequalities α + γσ ≥ max(γ, 1).

(9)

It is easy to show that under (9), the solution of FDS (4)–(5) is stable in the uniform norm and the estimate ! " n # # # #  # k# # n+1 # t ≤ max max | μ | ,  u  + τ y # #φ # # 1( k ) 0 C C

k=1,n+1

k=0

C

holds, where # n# #y # = max y ni . C i=0,N

When α = 1, σ = 0, the FDS (4)–(5) is transformed into the well-known upstream FDS y t + ay x = φ, with the stability criterion γ ≤ 1.

1.1.3 Theorem about EDS for one-dimensional linear transport equation Theorem 1.1. [8] FDS (4)–(5) is an EDS with φ=

1 τ

t n+1  h f x i−1 + (t − t n ), t dt, τ

(10)

tn

α + σ = γ = 1.

(11)

Proof. Condition γ = 1 means that nodes of the grid (x i , t n+1 ) and (x i−1 , t n ) are located on a characteristic curve dx dt = a.

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Now we have to show that the solution of the FDS changes to the same value as exact solution does. Substituting relations (10)–(11) to the canonical form of FDS (6), we get − y ni−1 = y n+1 i

t n+1  h f x i−1 + (t − t n ), t dt, τ tn

which coincides with the change of the exact solution (3) between points (x i , t n+1 ) and (x i−1 , t n ). Thus, the FDS is an EDS. Let us notice that the constructed EDS is monotone and stable, as condition (9) is fulfilled in the form of an equality.

1.1.4 One-dimensional semi-linear transport equation   In the domain Q T = 0, l × [0, T ] let us consider the IBVP for the semi-linear transport equation ∂u ∂u +a = f (u), 0 < x ≤ l, 0 < t ≤ T, a > 0, ∂t ∂x u(x, 0) = u0 (x), 0 ≤ x ≤ l, u(0, t) = μ1 (t), 0 < t ≤ T.

(12) (13)

On the uniform rectangular grid ω we approximate the differential problem (12)–(13) by FDS (4)–(5), but the stencil functional φ has a form similar to (3): ⎛  ⎜ 1 =⎜ φ = φ y ni−1 , y n+1 i ⎝ y n+1 − y n

i

i−1

⎞−1

n+1

y i

du ⎟ ⎟ . f (u) ⎠

(14)

y ni−1

Same as in the linear case, if the Courant number γ = 1, then the nodes of the uniform grid (x i−1 , t n ), (x i , t n+1 ) on a common characteristic curve dx dt = a. We show that the FDS (4)–(5), (14) is EDS under condition (11). To do this, we consider the expression for the approximation error 

n n+1 . − φ u , u Ψ = u (α)t + au(σ) i−1 i x Applying (11), we obtain ⎛ Ψ=

n+1

ui 

⎞

⎟ u n+1 − u ni−1 ⎜ du i ⎜ − τ⎟ n+1 ⎠. ⎝ ui f (u)  du n u i−1 τ u n f (u) i−1

(15)

12 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov

Using the differential equation (12), we obtain a sequence of relationships ∂u ∂u dx ∂u ∂u du = + = +a , dt dx =a ∂t ∂x dt ∂t ∂x

1 du = 1, f (u) dt dx =a

dt

u dξ f (ξ ) dx u0 dt

dt

⎞ ⎛ u dξ ⎟ d⎜ ⎠ ⎝ dt f (ξ ) u0 dx dt

u n+1 i

 = t,

u ni−1

dξ − f (ξ )

u0

=a



= 1, =a

u n+1 i

dξ = τ, f (ξ )



dξ = τ. f (ξ )

u ni−1

u0

Substituting the last formula into (15), we obtain Ψ = 0. Thus, the FDS (12)–(13) is an EDS under conditions (11) and (14).

1.1.5 One-dimensional transport equation with separated variables In the case of variable coefficients, one cannot build EDS on rectangular grids. So we use moving grids.

1.1.5.1 Non-divergent equation Consider the following problem ∂u ∂u + c(x, t) = f (x, t, u), 0 < x ≤ l, ∂t ∂x u(0, t) = μ(t), u(x, 0) = u0 (x), c(x, t) = c1 (x)c2 (t) > 0,

t > 0,

(16) (17)

f (x, t, u) = f1 (x, t)f2 (u).

To develop appropriate numerical methods, we use the idea of the method of characteristics. We assume a moving non-stationary grid, and we consider ⎛ − x ni x n+1 i τ

n+1

⎜ 1 =⎜ ⎝ x n+1 − x n i

i

x i x ni

⎞−1 dx ⎟ ⎟ c1 (x) ⎠

1 τ

t n+1 c2 (t)dt,

i = 0, N,

n = 0, N T − 1,

(18)

tn

where x0i = x i0 is the initial partition of the segment [0, l], x0n+1 ∈ (0, T], n = 0, N T − 1. On the basis of the foregoing, we state that the FDS (18) is an EDS on the characteristic meshes dx i (t) = c(x i (t), t), dt

x i (0) = x i0 .

(19)

Hyperbolic equations

Equation (16) along the characteristic x i (t) is written in the form du = f (x, t, u), u(0) = u0 . dt dx =c(x,t)

| 13

(20)

dt

The EDS for previous equation (20) was given earlier ⎛

⎞−1

n+1

− y ni ⎜ y n+1 1 i =⎜ ⎝ y n+1 − y n τ i i

y i y ni

du ⎟ ⎟ f2 (u) ⎠

1 τ

t n+1 f1 (x i (t), t)dt. tn

Suppose the right-hand side f (x, t, u) = 0, then problem (16)–(17) is linear, while the appropriate problem (19)–(20) is non-linear.

1.1.5.2 Divergent equation Consider the following IBVP ∂u ∂ + (c(x, t)u) = 0, 0 < x ≤ l, ∂t ∂x u(0, t) = μ(t), u(x, 0) = u0 (x),

t > 0,

(21)

c(x, t) = c1 (x)c2 (t) > 0. If coefficient c(x, t) is negative for all 0 ≤ x < l, t > 0, then the boundary condition has to be posed on the right boundary {x = l, t ≥ 0}. We transform equation (21) to the considered above case: ∂u ∂u + c(x, t) = c3 (x, t)u, ∂t ∂x

c3 (x, t) = −c1 (x)c2 (t).

In turn, linear equation (22) is equivalent to the system of non-linear ODEs dx du = c3 (x, t)u. = c(x, t), dt dt dx =c(x,t) dt

Using the above approach, we construct the EDS on a moving grid ⎞ ⎛ x n+1 i  t n+1 x n+1 − x ni ⎜ dx ⎟ 1 i ⎟1 c2 (t)dt, =⎜ ⎝ x n+1 − x n τ c1 (x) ⎠ τ i i x ni

tn

⎛

⎞−1 u n+1 t n+1 i   ⎜ u n+1 − u ni 1 1 du ⎟ i ⎟ . c3 (x i (t), t)dt⎜ = n+1 n ⎝ τ τ u ⎠ ui − ui tn

u ni

(22)

14 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov

The last equation can be rewritten in the following form: ⎧ ⎫ ⎪ ⎪ t n+1 ⎨ ⎬ 1 n u n+1 = u exp − c (x (t), t)dt , i = 0, N, 3 i i i ⎪ ⎪ ⎩ τ ⎭

n = 0, N T − 1.

tn

1.2 Two-dimensional transport equation   ¯ × [0, T], Ω ¯ = x = (x1 , x2 ) : 0  x k  l k , k = 1, 2 = Ω ∪ ∂Ω, In the parallelepiped Q¯ = Ω   ¯ : x k = 0, k = 1, 2 , let us consider the IBVP ∂Ω = x ∈ Ω ∂u ∂u ∂u + b2 = 0, x ∈ Ω, t ∈ (0, T], + b1 ∂t ∂x1 ∂x2 u x∈∂Ω = g(x, t), (x, t) ∈ ∂Ω × (0, T), u(x, 0) = u0 (x),

x ∈ Ω,

where b k = const > 0, k = 1, 2. Let us introduce the following uniform grids   ¯h=ω ¯ h1 × ω ¯ h2 , ω ¯ h α = x(iα α ) = i α h α , i α = 0, N α , h α N α = l α , α = 1, 2, ω ¯ h ∩ ∂Ω, ¯ h α = ω h α ∪ {0} , α = 1, 2, ∂ω h = ω ω   ¯ τ = t n = nτ, n = 0, N T , τN T = T = ω τ ∪ {0}. ω Let us consider y(μ1 ) = μ1 y i1 ,i2 + (1 − μ1 )y i1 −1,i2 ,

y(μ2 ) = μ2 y i1 ,i2 + (1 − μ2 )y i1 ,i2 −1 ,

0  μ1 , μ2  1.

In ref. [9] for this problem, the non-standard EDS is considered, which is a particular case (μ1 = μ2 = 0.5) of the following FDS, discussed in ref. [10], y t + b1 y(μ2 )x¯ 1 + b2 y(μ1 )x¯ 2 = 0, x ∈ ω h , t ∈ ω τ , y x∈∂ω = g(x, t), (x, t) ∈ ∂ω h × (0, T),

(23)

h

y(x, 0) = u0 (x),

x ∈ ωh .

(24)

It is easy to verify that FDS (23)–(24) is an EDS under conditions μ1 + μ2 = 1,

γ1 = γ2 = 1,

γk =

bk τ , k = 1, 2. hk

FDS (23)–(24) is monotone [10] if 0  μ 1 , μ2  1,

max {γ1 , γ2 }  γ1 μ2 + γ2 μ1  1,

(25)

Hyperbolic equations

| 15

and due to the grid maximum principle (Lemma 2.2 in ref. [7]), we obtain the following estimate for the solution of FDS (23)–(24): ( ' # # # # # k# # n+1 #  max max ,  u  g y # # # # 0 C(ω ¯ h) , ¯ h) C(ω

# n# #v #

0kn+1

C(S)

C(∂ω h )

= max v n (x) , x∈S

n = 0, N T − 1,

that expresses the stability of the FDS in the uniform norm with respect to small perturbations of the initial and boundary conditions. Thus, EDS (23)–(25) is monotone and stable. Let us note that the approach in ref. [9] was not extended by the authors for the three-dimensional problem.

1.3 Three-dimensional transport equation 1.3.1 Transport equation with constant velocity   ¯ × [0, T], Ω ¯ = x = (x1 , x2 , x3 ) : 0  x k  l k , k = 1, 2, 3 = In the parallelepiped Q¯ T = Ω ! " 3 ) ¯ x k = 0 , let us consider the IBVP Ω ∪ ∂Ω, ∂Ω = x ∈ Ω : k=1

∂u + (b · ∇)u = f (x, t), ∂t b = (b1 , b2 , b3 )T ,

x ∈ Ω,

(b · ∇)u =

t ∈ (0, T],

3  i=1

u

x∈∂Ω

= g(x, t),

bi

∂u , ∂x i

(26)

(x, t) ∈ ∂Ω × (0, T],

(27)

¯ x ∈ Ω,

(28)

u(x, 0) = u0 (x),

where b k = const > 0, k = 1, 2, 3. In Q¯ T , we introduce uniform grids 

¯h=ω ¯ h1 × ω ¯ h2 × ω ¯ h3 , ω



¯ h α = x(iα α ) = i α h α , i α = 0, N α , h α N α = l α , ω

(29) (30)

¯ h α = ω h α ∪ { 0} , ω

α = 1, 2, 3,   ¯ τ = t n = nτ, n = 0, N T , τN T = T = ω τ ∪ {0}, ω

(31)

¯ h ∩ ∂Ω. ∂ω h = ω ¯ τ , the problem (26)–(28) is approximated by the following ¯ =ω ¯h ×ω On the grid ω two-layer explicit FDS with a non-standard procedure of weighing of the convective

16 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov

terms by spatial variables and special averaging of the right-hand side )y = φ, x ∈ ω h , t ∈ ω τ , y t + (b · ∇−μ h

 −μ −μ −μ −μ ∇h y = ∇h 1 y, ∇h 2 y, ∇h 3 y , ±μ ∇h i y

y

= y(±μ i,j ,±μ i,k )x¯ i ,

x∈∂ω h

= g(x, t),

j, k =  i,

1 ≤ j < k ≤ 3,

(x, t) ∈ ∂ω h × [0, T],

y(x, 0) = u0 (x), φ = φ ni1 i2 i3

(32)

(33)

x ∈ ω¯h ,

(34)

t n+1 1 dx = f (x(ξ ), ξ )dξ , where = b. τ dt tn

Here and below, we use the following notation for the weighting in space y(±μ i,j ,±μ i,k ) = μ i,j y±μ i,k + (1 − μ i,j )y(±1j ) , y(±μ i,k ) = μ i,k y(x) + (1 − μ i,k )y(x(±1k ) ), 0 ≤ μ i,j , μ i,k ≤ 1,

i=  j,

1 ≤ i, j, k ≤ 3,

i=  k.

Let us obtain the conditions of monotonicity and stability of FDSs (32)–(34). To do this, we rewrite (32) in the following canonical form (see, e.g. ref. [11]) yˆ = τφ + αy +

3 

α i y−1i +

i=1

+α123 y

−11 ,−12 ,−13



α ij y−1i −1j

1≤i x , u20 (x), x > x* , where

lim v10 (x) =  lim v20 (x),

x→x*−

lim u10 (x) =  lim u 20 (x). In this case solution

x→x*+

x→x*−

x→x*+

of (66), (78) exists only in general sense. Definition 3.1. For generalized solution of problem (66), (78), we define integrable functions v(x, t), u(x, t), which fulfill initial conditions (78) and the following integral relations    vdx + audt = 0, (79) C





 udx + avdt = 0

(80)

C

for arbitrary domain G¯ ⊂ Q¯ T bounded with closed contour C. Note that if first derivatives of functions v, u are continuous, then using Green formula we can easily obtain acoustic system (66) from (79), (80), i.e. in this case functions v, u satisfy (66), (67), as well as (79), (80), (67). Let us find velocity for the front of the shock wave (discontinuous solution) [19]. Discontinuity curve x(t) is the curve that moves in the space with some velocity D(t). We shall use integral equations (79), (80) to describe its moving. Let us introduce

32 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov contour AA B B within the contour C. Line PP  showed on Fig. 3 is the piece of the discontinuity curve, so along PP we have the following relation dx = D(t), dt

x(0) = x* .

Let us denote values of functions v and u on the left and the right side of discontinuity by v1 , u1 and v2 , u2 respectively. Our goal is to find relations between these values. To this end we use integral equations (79), (80) for contour AA B B. We are going to bring closer left and right side of curvilinear quadrilateral AA B B, so as they stay on different sides of discontinuity curve. Then integrals for bases of quadrilateral AA and BB converge to zero, so integral equations for contour AA B B can be written in the form P





v1 dx + au1 dt +

P

P P

P





 v2 dx + au2 dt = 0,

P





P

u1 dx + av1 dt +





 u2 dx + av2 dt = 0.

P

Hence, we easily obtain the following relations for discontinuity curve of functions u and v D(t) = ±a. Since D(t) is parallel to characteristic lines of problem (66), (78) on which lie the ¯ for γ = 1, the FDS (70), (71) can be used to find values nodes of rectangular grid ω of discontinues functions when γ = 1. Remark 3.1. For approximating differential problem (66), (67), we can also use FDS  n  n v n+1 u n − u nh i−1 h i − 0, 5 v h i−1 + v h i+1 = a h i+1 , τ 2h   n n u n+1 vn − vn h i − 0, 5 u h i−1 + u h i+1 = a h i+1 h i−1 , for n = 0, τ 2h n−1 n − v − u nh i−1 v n+1 u hi hi = a h i+1 , 2τ 2h n+1 n−1 n n −v uh i − uh i v = h i+1 h i−1 , for n = 1, 2, . . . , 2τ 2h v0h i = v0 (x i ), u0h i = u0 (x i ), i = 0, ±1, ±2, . . . which is EDS when condition for mesh steps γ = 1 is fulfilled.

Hyperbolic equations

| 33

2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4

Fig. 4. Functions v and v h , t = 0.24, γ = 1.

3.1.4 Numerical experiment Let us consider functions ! v(x, t) =

− sin(x − 2t), sin(x − 2t) + 1,

! sin(x − 2t) + 1.5, u(x, t) = − sin(x − 2t) + 0.5,

x < 2t + x* , x > 2t + x* , x < 2t + x* , x > 2t + x* ,

that are generalized solution of problem (66), (78) with a = 2. Here, x* is not a node of the grid. In the experiment we take x* = 1. On the Fig. 4, 5 and 6, 7, we can see the exact solution of the differential problem and solution approximated by FDS (70), (71) for t = 0.24 and t = 1.04, respectively, when γ = 1.

3.2 IBVP problem in Riemann invariants In this section, we introduce the exact solution of the IBVP for acoustic equations system with the boundary conditions of the arbitrary form. The mathematical nature of arbitrary discontinuity decay is analysed. We propose a new class of weighted FDSs, which are EDS under γ = 1. It describes the behavior of discontinuous solution well with the Courant number considerably different from one. The comparative analysis of the FDS proposed with the well-known Lax and Lax–Wendroff [18] FDSs is presented.

34 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov

2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4

Fig. 5. Functions u and u h , t = 0.24, γ = 1.

2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4

Fig. 6. Functions v and v h , t = 1.04, γ = 1.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4

Fig. 7. Functions u and u h , t = 1.04, γ = 1.

2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

Hyperbolic equations

| 35

In the rectangle Q T , we consider IBVP for the firstorder linear system of two equations ∂u ∂u ∂v ∂v =a , = a , a > 0, ∂t ∂x ∂t ∂x u(x, 0) = u 0 (x), v(x, 0) = v0 (x), 0 ≤ x ≤ l, α0 v(0, t) + β0 u(0, t) = μ1 (t),     α l v l, t + β l u l, t = μ2 (t),    α0 − β0 α l + β l = 0.

0 < t ≤ T, 0 < t ≤ T,

(81)

(82)

Using Riemann invariants s = v − u, r = v + u, it is not hard to find exact solution of this problem [4]. This can be done by solving transport equations for each of the invariants. Notice, that in transport equations for Riemann invariants, fluxes are directed to different boundaries. If the discontinuity in initial data is such that in the same point initial conditions for both of the Riemann invariants are discontinuous at the same time, then the discontinuity in the solution of problem (81)–(82) will inevitably decay into two different discontinuities moving to opposite direction in every function. Thereby, the behavior of the solution is fully determined by the transport of Riemann invariants, where there is no decay of discontinuity [20, p. 80]. It can be shown that in this case, continuity of the initial data for one of the invariants is equivalent to fulfilment of Hugoniot conditions. Hugoniot conditions can be defined as follows [3, p.507] ⎧ ⎨D(v − v ) = −a(u − u ), + − + − (83) ⎩D(u+ − u− ) = −a(v+ − v− ), where f+ = limf (x), f− = limf (x), D − − velocity of the a shock wave. x→+0

x→−0

Multiplying one of the equations in (83) by D, and another by a, we get the equivalent system: ⎧

 ⎪ ⎪ D2 − a2 (v+ − v− ) = 0, ⎪ ⎨

 D2 − a2 (u+ − u− ) = 0, ⎪ ⎪ ⎪ ⎩D(v+ − v− ) = −a(u+ − u− ).



Assuming that the discontinuity exists (v+ − v− )2 + (u+ − u− )2 = 0 , we get ⎧ ⎨|D| = a, ⎩D(v+ − v− ) = −a(u+ − u− ),

36 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0 0

0.5

1 V(x,0)

1.5

2

0

0.5

1 U(x,0)

1.5

2

Fig. 8. Distribution of initial data with t = 0.

that gives two different solutions / v+ + u+ = v− + u− , v+ − u+ = v− − u− ,

D = a, D = −a,

or /

r+ = r− , s+ = s− ,

D = a, D = −a.

(84)

Hence, one of the invariants is continuous and the discontinuity is moving to the flux direction of another invariant. Since all of the conversions were equivalent, so (83) are equal to the fulfilment of one of the conditions from (84). The nature of interaction of two discontinuities moving to the opposite directions can be described by simple addition. Discontinuities moving to the same direction cannot interact, because they have the same speed a. On Fig. 8–10, the exact solution of the Riemann problem is presented. The initial data for this problem doesn’t meet Hugoniot conditions.

3.2.1 Stability with respect to right-hand side For simplicity we consider the following version of problem (81)–(82) ∂v ∂u = + f (x, t), ∂t ∂x 1

∂u ∂v = + f (x, t), ∂t ∂x 2

a > 0,

(85)

Hyperbolic equations

| 37

1

1 0.8

0.5 0.6 0.4 0 0.2 0 0

−0.5 0.5

1 V(x,0.15)

1.5

2

0

0.5

1 U(x,0.15)

1.5

2

0.5

1 U(x,0.5)

1.5

2

Fig. 9. The solution at the moment t = 0.15.

1

1 0.8

0.5 0.6 0.4 0 0.2 0 0

−0.5 0.5

1 V(x,0.5)

1.5

2

0

Fig. 10. The solution at the moment t = 0.5.

u(x, 0) = u0 (x), v(x, 0) = v0 (x),   v(0, t) = v l, t = 0,

0 ≤ x ≤ l, 0 < t ≤ T.

Let us take L2 ([0, l]) inner product of the second equation in (85) with 2v. Integrating it by parts, one gets    d ∂v 2 + 2 f2 , v . v = −2 u, dt ∂x

(86)

38 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov

Substituting the expression from (85) ∂v ∂u = −f ∂x ∂t 1 results in      d

 u 2 +  v 2 = 2 f 1 , u + 2 f 2 , v . dt

  Using Cauchy–Schwarz inequality first in L2 0, l and then in R2 gives

 12 d

 12 # # # # 2  u 2 +  v  2  u 2 +  v 2 ≤ 2u#f1 # + 2v#f2 # dt  12 # # # #  12

# f 1 #2 + # f 2 #2 . ≤ 2  u 2 +  v 2 Hence,  2 # # # #  2 d

2 2  u 2 +  v 2 ≤ #f1 # + #f2 # . dt 1

Integrating

t 0



1

gives the final estimate 2

u + v

2

 12



2

≤ u0  + v0 

2

 12

t

+

# #2 # #2  12 #f1 # + #f2 # dt.

(87)

0

3.2.2 Scheme with half-integer nodes For the problem (85)–(86) on the grids,  1 ¯ h = {x i = ih, x i+ 1 = i + h, i = 0, 1, . . . , N − 1, ω 2 2     ω τ = t n = nτ, n = 0, N T , τN T = T = ω τ ∪ t N T = T ,

x N = Nh = l, }. ω = ωh × ωτ

consider the following FDS: 1) 2) + φ1 , v h,t = a u¯ (σ + φ2 , u¯ h,t = av(σ h,x h, x¯  h n+1 , v0h,i = v0 (x i ), v n+1 u¯ 0h,i = u nh,i+ 1 = u0 x i + h,0 = v h,K = 0, 2 2

(88) (89)

where 0 ≤ σ1 , σ2 ≤ 1, φ1 , φ2 some grid functionals, and grid function u¯ h,i = u h,i+ 1 2 takes values on the half-integer nodes in space.

Hyperbolic equations

| 39

We introduce the grid scalar products and norms in the grid functions space: [g h , g h ) =

N−1 

hg h,i f h,i ,



N−1   gh , fh = hu g,i v f ,i ,

i=0

[g h | 2 =

N−1 

i=1

hg 2h,i ,

| g h | 2 =

i=0

N−1 

hg 2h,i .

i=1

Let us prove the following statement. Theorem 3.1. Suppose the following condition is true σ1 = σ2 =

1 . 2

Then the FDS (88)–(89) is stable with respect to initial data and right-hand side, and for its solution, the following a priori estimates hold    1   2 2  12  1 n 0 0  k 2 k 2 2  n+1 2 n+1 2 2 u¯ h + v h ≤ u¯ h + v h + τ φ1 + φ2 . k=0

Proof. Under homogenious boundary conditions on v h , it is easy to establish the following “summation by parts” formula: 

   u¯ h , v h,x = − u¯ h, x¯ , v h .

Now taking the inner product of the first equation from (88) with 2τ u¯ (0.5) and using h this formula, we obtain    n+1 2  n 2 u¯ h − u¯ h = 2τ u¯ (0.5) , u¯ h,t h     n (0.5) (0.5) ¯ = 2τa u¯ (0.5) + 2τ u , v , φ 1 h h,x h   

(0.5) (0.5) (0.5) + 2τ u¯ h , φ1n . = −2τ a u¯ h, x¯ , v h Substituting the second equation from (88) and using the fact that 

n+1 2 n 2 v h − v h = 2τ v(0.5) , v h,t h one gets  n+1 2  n 2 n+1 2 n 2 u¯ h − u¯ h + v h − v h 

  = 2τ u¯ (0.5) , φ1n + 2τ v(0.5) , φ2n h h 

      n n+1 n ¯ nh , φ1n + τ v nh , φ2n . = τ u¯ n+1 h , φ1 + τ v h , φ2 + τ u

40 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov ¯) Twice using Cauchy–Schwarz inequality, first in the space of discrete functions L2h (ω and then in R2 , gives  n n   n n   n  n n n u¯ h , φ1 + v h , φ2 ≤ u¯ h φ1 + v h φ2  1  1  2 2 2  n 2 n 2 2 + , ≤ u¯ nh + v nh φ φ 1 2 which results in  n+1 2 n+1 2  n 2 n 2 u¯ h + v h − u¯ h − v h ⎛  1 2 2 2   2 2  ⎜ n+1 n+1 ≤ τ⎝ u¯ h + v h + u¯ nh + v nh

⎞ 1 2

 ⎟  n 2 n 2 ⎠ φ1 + φ2

⎛

 1 2 2 2   2 2  ⎜ n+1 n+1 Dividing by ⎝ u¯ h + v h + u¯ nh + v nh   1   n+1 2 n+1 2 2  2 2 u¯ h + v h ≤ u¯ nh + v nh

1 2

1 2

.

⎞ 1 2

⎟ ⎠ one gets

  2 2 + τ φ1n + φ2n

1 2

,

which by induction ends the proof. Remark 3.2. This estimate gives grid analog of the inequality (87).

3.2.3 Conditions of the EDS for Lax and Lax–Wendroff FDS Lax finite-difference scheme In case of Cauchy problem, Lax FDS is the most well known [18] 

n n v n+1 − u nh,i−1 un h,i − 0.5 v h,i+1 + v h,i−1 = a h,i+1 , 2h 

τ n n u n+1 h,i − 0.5 u h,i+1 + u h,i−1

v nh,i+1 − v nh,i−1 , τ 2h 0 0 v h,i = v0 (x i ), u h,i = u0 (x i ), i ∈ Z, =a

which is EDS under γ = 1 and γ ≤ 1. Disadvantage of this FDS is its 

stable under 2 conditional approximation O h2 + τ + hτ .

Hyperbolic equations

| 41

Lax–Wendroff finite-difference scheme For numerical solution of Cauchy problem, Lax–Wendroff FDS is often used as well [18]: v h,t = au u0h,i

◦

h, x

+

a2 τ a2 τ , v h,xx , u h,t = av ◦ + u h, x 2 2 h,xx

(90)

= u0 (x i ), v0h,i = u0 (x i ), i ∈ Z,

which is a FDS of second order of approximation O(h2 + τ2 ), and is a monotone EDS only under γ = 1. Disadvantage of the FDS (90) is absence of the monotonicity under γ = 1. Same as most of the explicit FDSs, it is stable under the fulfilment of the Courant criterion γ ≤ 1.

Explicit finite difference with viscosity Bellow we introduce new classes of FDSs for IBVP for the system of acoustic equations, which are EDSs under γ = 1 and save the property of stability and monotonicity under γ = 1. Consider the FDS for the problem (81)–(82) on the grid ω: v h,t = au

◦

h, x

+

ah ah , u h,t = av ◦ + , v u h, x 2 h,xx 2 h,xx

(91)

i = 1, N − 1, n = 0, N0 − 1, u0h,i = u0 (x i ), v0h,i = u0 (x i ), i = 0, N, n+1 α0 v n+1 h,0 + β 0 u h,0 = μ 1 ( t n+1 ) , ( v h + u h )0,t − a ( v h + u h )0,x = 0, n+1 α l v n+1 h,N + β l u h,N = μ 2 ( t n+1 ) , ( v h − u h )N,t + a ( v h − u h )N,x = 0.

(92)

Second and fourth of the equations in (92) were obtained by taking half-sum and half-difference of the equations (91) correspondingly. Let us show that the FDS (91) for the problem (81)–(82) is EDS under γ = 1. Rewrite first equality from (91) in the index form: n v n+1 h,i − v h,i =

 aτ  n − u nh,i−1 + v nh,i+1 + v nh,i−1 − 2v nh,i . u 2h h,i+1

Under the condition γ = 1, it gives v n+1 h,i =

 1 n − u nh,i−1 + v nh,i+1 + v nh,i−1 . u 2 h,i+1

42 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov Since for the exact solution r ni+1 = r n+1 , s ni−1 = s n+1 , so i i + u n+1 + v n+1 − u n+1 = 2v n+1 , u ni+1 − u ni−1 + v ni+1 + v ni−1 = v n+1 i i i i i which means that the first equation of the FDS (91) is EDS. The exactness of the second equation from (91), second and fourth equations from (91) can be shown analogously. Approximation, stability and monotonicity of the FDS are important properties of the FDS. It is not hard to show that this FDS approximates the corresponding problem    1. Also from the maximum principle [21], it with the first order O h + τ when γ = follows that the FDS is uniformly stable and monotone under γ ≤ 1.

Scheme, weighted along the time and space Defining α

Λu ni =

 1 2αu ni + (1 − α)u ni−1 + (1 − α)u ni+1 , 2

consider the FDS α

Λv h,t + (1 − α)hu

◦

= au(σ)◦ +

ah (σ) , v 2 h,xx

◦

= av(σ)◦ +

ah (σ) , u 2 h,xx

h,t x

α

Λu h,t + (1 − α)hv

h,t x

h, x

h, x

(93)

i = 1, N − 1, n = 0, N0 − 1, u0h,i = u0 (x i ), v0h,i = u0 (x i ), i = 0, N, (σ) n+1 α0 v n+1 h,0 + β 0 u h,0 = μ 1 ( t n+1 ) , ( v h + u h )1,(1−α)t − a ( v h + u h )0,x = 0, (σ) n+1 α l v n+1 h,N + β l u h,N = μ 2 ( t n+1 ) , ( v h − u h )N,(α)t + a ( v h − u h )N,x = 0.

(94)

To find the numerical solution, the matrix elimination method can be used [22, p. 106]. It is nothard to show that the FDS approximates the corresponding problem with 



the order O α − 12 h + σ − 12 τ + h2 + τ2 . Also it can be shown that the FDS is EDS under (11): α + σ = γ = 1. Remark 3.3. A comparison of the above mentioned FDSs in the numerical simulations can be found in the paper by Matus and Colleagues [4].

Hyperbolic equations

| 43

3.2.4 Stability of weighted finite-difference scheme For simplicity, we consider IBVP (81)–(82) in the case, when the Riemann invariants   are given on the bound α0 = −β0 = α l = β l = 1 . Theorem 3.2. Let the following condition be true α + σγ ≥ max(1, γ).

(95)

Then the FDS (93)–(94) is uniformly stable with respect to initial data and boundary conditions and for its solution a priori estimate # # # # # # n+1 # # max #v n+1 h # , #u h # C C ⎛ ⎞  # # # # 1 # # # # 0 0 ≤ max⎝#v h # + #u h # , max |μ1 (t k )| + max |μ2 (t k )| ⎠ C C 2 k=1,n+1 k=1,n+1 holds. Proof. First, we estimate the norm of the solution by the norm of Riemann invariants from the same time layer:



# # # #   max #v nh #C , #u nh #C = max v nh,i , u nh,i  n n = max v h,¯i , u h,¯i ¯   i∈{i ,i }  = 0.5 v nh,¯i + u nh,¯i + v nh,¯i − u nh,¯i

# # # #  ≤ 0.5 #v nh + u nh #C + #v nh − u nh #C . For the Riemann invariants, we have the following expression: ⎧ ⎪ n n+1 ⎪ ⎪(α + γσ − γ)r i + (α + γσ − 1)r i+1 + ⎨ i = 0, N − 1,   = (α + γσ)r n+1 + 1 − α + γ(1 − σ) r ni+1 , i ⎪ ⎪ ⎪ ⎩(α + γσ)μ2 (t n+1 ), i = N, ⎧ ⎪ n n+1 ⎪ ⎪ ⎨(α +γσ − γ)s i + (α + γσ − 1)s i−1 + = (α + γσ)s n+1 + 1 − α + γ(1 − σ) s ni−1 , i ⎪ ⎪ ⎪ ⎩(α + γσ)μ1 (t n+1 ),

i = 1, N, i = 0.

44 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov

Hence, under the condition (95), the estimate follows ! " # # # # # 0 # n+1 n+1 # 0# #v h + u h # ≤ max max μ2 (t k ), #v h + u h # , C C !k=1,n+1 " # # # # # n+1 # 0 n+1 # 0# #v h − u h # ≤ max max μ1 (t k ), #v h − u h # . C

C

k=1,n+1

It is obvious that # # # # 1 # 0 # 0 0# 0# #v h − u h # + #v h + u h # 2 C C

# # # # # # # # ≤ #v0h # + #v0h # . C

C

This completes the proof of the theorem. Remark 3.4. Under α = 1, σ = 0, the FDS (93)–(94) transformes into the FDS (91)–(92), and monotonicity and stability condition (95)—into the Courant criterion γ ≤ 1.

4 Riemann problem for non-linear transport equation In this section, we consider the Riemann problem for non-linear transport equation. We show that the simple explicit conservative FDS, approximating the Riemann problem, is stable [23] EDS. Consider the Riemann problem ∂u ∂F(u) + = 0, ∂t ∂x

0 < t  T,

u(0, x) = u0 (x) =

0 < x  l,

F  (u) > 0,

⎧ ⎨u L

for 0 < x < ξ ,

⎩u R

for ξ < x  l,

u(t, 0) = u L ,

0 < t  T,

F  (u) > 0,

(96)

u L > u R  0. (97)

where u L and u R are constants. The weak solution of this problem is defined in a class of piecewise continuous functions satisfying (for any closed piecewise-smooth contour ∂V) the following integral equation [24]  udx − F(u)dt = 0 ∂V

and conditions (4) and (97).

Hyperbolic equations

| 45

Let the solution u(t, x) be discontinuous along a line x(t) = ξ + X(t). In the case of the considered Riemann problem (96)–(97), velocity D(t) of the discontinuity movement is determined by the Rankine–Hugoniot condition [6]: D=

F(u R ) − F(u L ) dX = . dt uR − uL

(98)

If D ≡ const, then D = dx/dt = h/τ. Thus, for the constant velocity D it follows from (98) that F(u R ) − F(u L ) h = . τ uR − uL Hence, assuming τD = 1, h

(99)

u R − u L F(u R ) − F(u L ) − = 0. τ h

(100)

we obtain the identity

For u R , from (100) we have τ u R = u L + (F(u R ) − F(u L )). h ¯ h as follows: the point ξ (where To approximate problem (96)–(97), we take the grid ω the initial function u0 is discontinuous) is not a grid-point. Let ξ ∈ (x i0 −1 , x i0 ) (see ¯ consider the FDS Fig. 11). On the grid ω, − u ni F(u ni ) − F(u ni−1 ) u n+1 i + = 0, t ∈ ω τ , x ∈ ω h , τ h ¯ h , u(t, 0) = u L , t ∈ ω τ . u(0, x) = u0 (x), x ∈ ω

u0(x) uL

x i0− 1

Fig. 11

+

uR ξ

x i0

x

(101) (102)

46 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov

From equation (107), we have τ = u ni − (F(u ni ) − F(u ni−1 )). u n+1 i h Thus, since u ni0 −1 = u L , u ni0 = u R near the discontinuity, it follows that τ R R L u n+1 i0 = u − (F(u ) − F(u )). h Combining this with (100), we get the identity L n u n+1 i0 = u = u i0 −1 ,

and, consequently, FDS (107) approximates the Rankine–Hugoniot conditions (100) exactly near the discontinuity. It follows that, on the shock wave under assumptions (99), FDS (107) is EDS for the problem (96)–(97). Consider the problem with perturbed input data ∂ u˜ ∂F(u˜ ) + = 0, ∂t ∂x

0 < t  T, 0 < x  l, F  (u˜ ) > 0, F  (u˜ ) > 0, ⎧ ⎨u˜ L for 0 < x < ξ , u˜ (0, x) = u˜ 0 (x) = u˜ L > u˜ R  0. ⎩u˜ R for ξ < x  l, u˜ (t, 0) = u˜ L ,

(103) (104)

0 < t  T,

¯ we approximate this problem by the where u˜ L and u˜ R are constants. On the grid ω, following FDS: − u˜ ni F(u˜ ni ) − F(u˜ ni−1 ) u˜ n+1 i + = 0, t ∈ ω τ , x ∈ ω h , τ h ¯ h , u˜ (t, 0) = u˜ L , t ∈ ω τ . u˜ (0, x) = u˜ 0 (x), x ∈ ω ¯ with FDS (105), (106) is EDS for the problem (103), (104) on the same grid ω F(u˜ R ) − F(u˜ L ) h = = D. D˜ = τ u˜ R − u˜ L Under conditions (99), (4), we have = u ni−1 , u n+1 i

i = 1, 2, . . . , N x ,

n = 0, 1, . . . , N − 1,

u˜ n+1 i

i = 1, 2, . . . , N x ,

n = 0, 1, . . . , N − 1.

=

u˜ ni−1 ,

Therefore, = δu ni−1 , δu n+1 i

i = 1, 2, . . . , N x ,

n = 0, 1, . . . , N − 1,

Consequently, δu n+1 C(ω¯ h )  δu 0 C(ω¯ h ) .

(105) (106)

Hyperbolic equations

| 47

Thus, we have proved that the simple explicit conservative FDS (107), (108), approximating the Riemann problem (96)–(97), is stable EDS for τ = hD > τ K and D˜ = D. We stress that this value of τ is greater than value τ K admissible by the Courant–Friedrichs–Lewy condition.

4.1 Numerical experiment ¯ h as follows: the point ξ (where To approximate problem (96)–(97), we take the grid ω ¯ consider the FDS the initial function u0 is discontinuous) is a grid-point. On the grid ω, − y ni F(y ni ) − F(y ni−1 ) y n+1 i + = 0, t ∈ ω τ , τ h ¯ h , y(t, 0) = u L , y(0, x) = u0 (x), x ∈ ω

x ∈ ωh ,

(107)

t ∈ ωτ .

(108)

It follows that on the shock wave under assumptions (99), FDS (107) is EDS for the problem (96)–(97). In numerical simulation, we used the following input data (see Fig. 12). u L = 2,

u R = 1,

l = 1,

ξ = 0.5,

F(u) =

u2 . 2

3 2.8 2.6 2.4 2.2 2 1.8 u0(x)

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

0.1

0.2

0.3

0.4

Fig. 12. Initial condition for FDS (107).

0.5 x

0.6

0.7

0.8

0.9

1

48 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov

We stress that in this case D=

uL + uR , 2

τ = τD =

h > τC , D

τC =

h . uL

As the initial data have a discontinuity, we propose to fulfil computations twice: for first time to set y01,h (ξ ) = lim u0 (x), x→ξ −0

and for the second one, we set y02,h (ξ ) = lim u0 (x). x→ξ +0

In this way, we obtain two difference solutions y1,h and y2,h and plot them simultaneously. It allows us to know exactly the position of the shock wave and plot it without smearing. Computations were carried out with following parameters (see Fig. 13): T = 0.3,

h = 0.01,

τ = 0.015.

3 2.8 2.6 2.4 2.2 2

yh(x,0.3)

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

0.1

0.2

0.3

Fig. 13. Solution of FDS (107).

0.4

0.5 x

0.6

0.7

0.8

0.9

1

Hyperbolic equations

| 49

5 Non-homogeneous quasilinear transport equation In this section, FDS of any order of accuracy O(τ M ), M ≥ 1, τ = t n+1 − t n — time step, for IBVP for quasilinear transfer equation with non-homogeneous and non-linear right-hand side on the moving characteristic grid and minimal stencil are constructed. All the theoretical aspects of FDSs such as approximation, stability, and convergence in non-linear case are also considered.

5.1 Statement of the problem In the domain (see Fig. 14)   Q T = (x, t) : 0 ≤ x ≤ x l (t), 0 ≤ t ≤ T , IBVP is considered for the transport equation ∂u ∂u +u = λu2 , ∂t ∂x

0 < x ≤ x l (t), 0 < t ≤ T, λ < 0,

(109)

u(0, t) = μ(t), u(x, 0) = u0 (x), 0 < u0 ≤ u(x, t) ≤ u1 ,



u0 (x) ≥ 0

Equation (109) along the characteristic [17]

dx dt

dx = u, dt

u(x, t) ∈ C11 (Q T ).

= u can be written in the following form:

du | dx = λu 2 , 0 ≤ x ≤ x l (t), λ < 0, dt dt =u   dx l 0 ≤ x ≤ x l (t), 0 ≤ t ≤ T, = u x l (t), t , x l (0) = l. dt

1

t

xl(t)

– QT

0

l

x

Fig. 14. Domain of the problem definition (109–110).

(110)

(111)

50 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov

Selection of the domain Q T of the problem with a moving right boundary is explained by using characteristic method. In this case, some nodes of the moving grid will be located on the boundary of the domain.

5.2 Finite-difference scheme Let us introduce the uniform grid on the segment [0, l] ( ' l , ω0h = x0i = ih, i = 0, N, hN = l, h = N with a constant step h > 0 where x0i is the initial segment partition. The following moving grid is constructed in Q T   ω hτ = (x nhi , t n ), i = −n, N, x nh(−n) = 0, t n = nτ, n = 0, N T , τN T = T . Here, the following notation is used:   x ni = x i (t n ), u ni = u x ni , t n ,

  u nhi = u x nhi , t n .

Applying the specific Steklov averaging, the differential problem is approximated by the FDS [25]   M−1 n  λτ k  n k+1 x n+1 hi − x hi , M ≥ 1, i = −n, N, n = 0, N T − 1; = u τ k + 1 hi

(112)

k=0

x0hi = x0i ,

x nh(−n) = 0,

i = 0, N, n = 0, N T ;

n u n+1 n hi − u hi = λu n+1 hi u hi , i = −n, N, n = 0, N T − 1; τ

 n = 0, N T . u0hi = u0 x0i , u nh(−n) = μ(t n ), i = 0, N,

5.3 Approximation error The problem for the method error can be written in the following form: δx nt,i

−

M−1  k=0



k k   λτ  k + 1 !  n k+1−j  n j n   , δu i u i = ψ0i k+1 k + 1 − j !j! j=0

i = −n, N, n = 0, N T − 1, n δx0i = 0, δx(−n) = 0,

i = 0, N, n = 0, N T ,

(113)

Hyperbolic equations



n = ψ1i δu ni + δu ni δu n+1 , δu nt,i − λ u ni δu ni + u n+1 i i n = 0, δu0i = 0, δu(−n)

| 51

i = −n, N, n = 0, N T − 1,

i = 0, N, n = 0, N T ,

where δx ni = x nhi − x ni , δu ni = u nhi − u ni , δv nt,i =

− δv ni δv n+1 i . τ

The approximation error is expressed by the equality n n + ψ1i , ψ ni = ψ0i

where n ψ0i

= −x nt,i

+

M−1  k=0



k λτ  n k+1 , u k+1 i

and n = −u nt,i + λu n+1 u ni , ψ1i i

i = −n, N,

n = 0, N T − 1.

Due to (111), the following equalities hold: dx = u, dt

d2 x du = = λu2 , dt2 dt

d3 x du = 2λu = 2λ2 u3 , . . . dt dt3

Taking into account the above equalities and substituting them into the Taylor formula, we obtain − x ni x n+1 dx n τ d2 x ni τ M−1 d M x ni λ M τ M  n M+1 i 0 u + · · · + + = i + τ dt 2! dt2 M! dt M M+1 i     M M−1  λτ k  n k+1 λτ  n M+1 0 u = + . ui k+1 M+1 i

x nt,i =

k=0

From here the equality follows n ψ0i =−

 λ M τ M  n M+1 0 ui = O τM , M+1

where   0 u ni = x i t0n , t0n , t0n = t n + Θ n τ, 0 < Θ n < 1.

(114)

52 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov

In equation 1, it was shown that for the Cauchy problem for a non-linear ordinary differential equation du = f1 (u)f2 (t), dt

f1 (u) =  0,

t > 0,

u(0) = u0 ,

the FDS with the specific Steklov averaging ⎛

⎞−1

n+1

y

1 y n+1 − y n ⎜ =⎜ ⎝ y n+1 − y n τ

yn

du ⎟ ⎟ f1 (u) ⎠

1 τ

t n+1 f2 (t)dt,

y0 = u0 ,

(115)

tn

is EDS. In our case, f1 (u) = λu2 ,

f2 (t) = 1.

Consequently, from (115), we obtain n = −u nti + λu n+1 u ni = 0. ψ1i i

(116)

Taking into account (116), the expression for ψ ni can be written in the form

 n n + ψ1i = O τM . ψ ni = ψ0i From here it follows that the initial differential problem is approximated by the FDS (112), (113) with any order of M ≥ 1.

5.4 Stability of the finite-difference scheme We introduce the following notation for the grid norms: # n# #u h #

Cn

= max u nhi . −n+1≤i≤N

Perturbing the initial and boundary conditions in (112), (113), we arrive at the problem 0 x nht,i

=

M−1  k=0



k λτ  n k+1 0 u , k+1 i

i = −n, N, 0 x0hi = x0i ,

0 x nh(−n) = 0,

u nhi ≤ 0 u1 , 0 0, M2 > 0 independent of τ and n exist, such that for any n, the inequality # # # # # n+1 # #0n+1 n+1 # + #0 u hi − u n+1 (118) #x h,i − x h,i # hi # C n+1 C n+1 ( ' # # # # # 0 0# # # 0 , ≤ M1 #0 x h − x h # + M2 max max 0 μ(t k ) − μ(t k ) , #0 u h − u0h # C0

1≤k≤n+1

C0

is true. Let x nhi − x nhi , ∆x ni = 0

∆u ni = 0 u nhi − u nhi ,

i = −n, N, n = 0, N T .

Subtracting from (117) the respective equation (112),(113), we arrive at the perturbation problem ∆x nt,i

M−1 

∆x0i

k k  n k+1−j  n j λτ  0 u hi u hi , k+1

= =

n λu n+1 hi ∆u i

k=0

∆u nt,i



∆u ni

= 0,

i = 0, N,

j=0

n ∆x(−n)

+ λ0 u nhi ∆u n+1 , i = 0,

∆u0i

i = −n, N, n = 0, N T − 1,



 n =0 u i x0i − u i x0i , ∆u(−n) =0 μ(t n ) − μ(t n ),

n = 0, N T .

To get corresponding estimates expressing the stability of the FDS, it is necessary to u nhi are bounded. show that u nhi and 0 Lemma 5.1. Let the conditions 0 < u0 ≤ u(x, t) ≤ u1 ,

0 0,

α = 1, 2,

(12)

of the equation (7), the following equality is satisfied 2 

Λ(σ) α u i1 i2 = −

α=1

2



 n+1 n σc α u(μ . + (1 − σ)c α u(μ 3−α )+ x α ,i 1 i 2 3−α )x α ,i 1 i 2

α=1

Proof. First, let us notice that for a travelling-wave solution of the form (12) follows  2  ∂φ α (u) ∂ u ∂x α ∂x α α=1

=−

2  α=1

cα

∂u . ∂x α

By integrating above equality on the closed rectangle [x1 , x1 + h1 ] × [x2 , x2 + h2 ] ⊂ Ω T and by performing simple transformations, we obtain ⎡ ⎤ x3−α +h3−α  2  ∂φ α (u) ⎢ 1 ⎥ u + c α ⎦ = 0. ⎣ h3−α ∂x α α=1

x3−α

xα

Parabolic equations

| 109

Since x1 , x2 , h1 , h2 are chosen arbitrary, the following conditions are satisfied  ∂  φ α (u) = −c α , α = 1, 2. ∂x α   Let α = 1. Integrating the equation ∂x∂ 1 φ1 (u) = −c1 for t = t n+1 , x2 = x2i2 on the closed n+1 n+1   intervals [x1i1 −1 , x1i1 ] and [x1i1 , x1i1 +1 ], we have φ1 (u) x ,i i = −c1 , φ1 (u) x ,i +1,i = 1 1 2 1 1 2 −c1 . As a consequence, we obtain

 n+1  u(μ2 )+ φ1 (u) x 1

x1 ,i1 i2

n+1 = −c1 u(μ , 2 )+ x 1 ,i 1 i 2

and similarly we get

 n  u(μ2 ) φ1 (u) x 1

x1 ,i1 i2

n = −c1 u(μ . 2 )x 1 ,i 1 i 2

For α = 2, an analogous calculation leads to n+1 n Λ(σ) 2 u i1 i2 = −σc 2 u (μ1 )+ x2 ,i1 i2 − (1 − σ)c 2 u (μ1 )x2 ,i1 i2 .

After adding obtained equalities, we get 2 

Λ(σ) α u i1 i2 = −

2



α=1

 n+1 n σc α u(μ . + (1 − σ)c α u(μ 3−α )+ x α ,i 1 i 2 3−α )x α ,i 1 i 2

α=1

Now we are ready to prove the main theorem. Theorem 1.4. Let γ1 = γ2 = 1, where γ α = (a α + c α )τ/h α , α = 1, 2, and one of the following conditions be satisfied σ = 0, μ1 + μ2 = 1,

(13)

or c1 = c2 , μ1 + μ2 = 1, h1 = h2 , or c1 =  c2 , μ 1 = μ 2 = 0.5.

(14)

Then the FDS (9)–(10) is EDS for the travelling-wave solution of the form (12). Proof. The error of approximation of the FDS (9)–(10) is given by the formula (11). In view of the Lemma 1.2, for a travelling-wave solution, it reduces to ψ ni1 i2 = −u t,i1 i2 −

2

 α=1

 n+1 n σc α u(μ + (a + (1 − σ)c )u α α (μ3−α )x α ,i1 i2 . 3−α )+ x α ,i 1 i 2

110 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov

Since γ1 = γ2 = 1, we have u n+1 i1 +1,i2 +1 = U(x 1i1 +1 − (a 1 + c 1 )t n+1 , x 2i2 +1 − (a 2 + c 3 )t n+1 ) = U(x1i1 − (a1 + c1 )t n + h1 − (a1 + c1 )τ, x2i2 − (a2 + c2 )t n + h2 − (a2 + c2 )τ) = U(x1i1 − (a1 + c1 )t n , x2i2 − (a1 + c1 )t n ) = u ni1 i2 . and after some tedious manipulations, the error of approximation can be expressed as 

   n n n , ψ ni1 i2 = −d/τ u n+1 i1 i2 + u i1 i2 − u i1 −1,i2 + u i1 ,i2 −1 where d = μ1 + μ2 − 1 + σ(1 − 2μ2 )c1 τ/h1 + σ(1 − 2μ1 )c2 τ/h2 , It is obvious, that if one of the conditions (13)–(14) is satisfied, then the error ψ ni1 i2 equals 0.

1.2.5 Numerical experiments for linear equation In this section, we present numerical experiments to illustrate our theoretical results. Here, the method of matrix elimination, presented in ref. [4], was used to solve the system of vector equations that are the final results of the difference approximation of the IBVP for a convection-diffusion equation. First, let us consider the linear equation (7) with k α (u) ≡ b α , b α = const > 0, α = 1, 2 together with the exact solution u(x1 , x2 , t) = e−(x1 −(a1 +b1 /2)t)/2 e−(x2 −(a2 +b2 /2)t)/2 . In this case, we have c α = γ1 =

bα 2 .

The FDS (9)–(10) is EDS for

(a1 + b1 /2)τ (a + b2 /2)τ = 1, γ2 = 2 =1 h1 h2

under one of the conditions (13)–(14). It is worth to notice here, that the explicit FDS, i.e., for σ = 0, is EDS but unstable. The results presented in Tables 5–7 illustrate, that the FDS (9)–(10) is EDS. Remark 1.7. In practice, the velocity of wave propagation cannot be evaluated directly from d = (d1 , d2 ), d α = a α −

∂φ α (u) , α = 1, 2, ∂x α

Parabolic equations

| 111

Tab. 5. σ = 0.5, μ1 = 0.3, μ2 = 0.7, γ1 = γ2 = 1, a1 = a2 = b1 = b2 = 1, l = T = 1, ϵ = 1.0 · 10−17 . h1 = h2

τ

0.5 0.25 0.125 0.05

0.3(3) 0.16(6) 0.083(3) 0.03(3)

max y n − u n C

Number of iterations

5.42 · 10−20 2.60 · 10−18 3.66 · 10−17 9.30 · 10−16

1 1 4 7

t n ∈ω τ

Tab. 6. σ = 1.0, μ1 = 0.3, μ2 = 0.7, γ1 = γ2 = 1, a1 = a2 = b1 = b2 = 1, l = T = 1, ϵ = 1.0 · 10−17 . h1 = h2

τ

0.5 0.25 0.125 0.05

0.3(3) 0.16(6) 0.083(3) 0.03(3)

max y n − u n C

Number of iterations

2.17 · 10−19 1.84 · 10−18 2.39 · 10−18 4.77 · 10−18

1 2 4 7

t n ∈ω τ

Tab. 7. σ = 1.0, μ1 = μ2 = 0.5, γ1 = γ2 = 1, a1 = a2 = 1, b1 = 2, b2 = 1, l = T = 1, ϵ = 1.0 · 10−17 . h1 0.3(3) 0.16(6) 0.083(3) 0.03(3)

h2 0.25 0.125 0.0625 0.025

τ

max y n − u n C

Number of iterations

2.60 · 10−18 2.39 · 10−18 3.47 · 10−18 6.29 · 10−18

2 3 4 6

t n ∈ω τ

0.16(6) 0.083(3) 0.0416(6) 0.016(6)

since the value of the derivative is unknown. Nevertheless, one can find it approximately by the formula d˜ nαi1 i2 = a α − k α (y ni1 i2 )y ◦n

x α i1 i2

/y ni1 i2 ≈ d α , α = 1, 2,

where y is the solution of the FDS [4] yt +

2  α=1

a α y xα =

2  

r α (y)y x α

(σ) xα

, x ∈ ωh , t ∈ ωτ ,

α=1

y0i1 i2 = u0 (x), x ∈ ω h , y|x∈∂Ω = g(x, t), t ∈ ω τ . Here, r1 (y i1 i2 ) = 0.5(k1 (y i1 −1,i2 ) + k1 (y i1 i2 )), r2 (y i1 i2 ) = 0.5(k2 (y i1 ,i2 −1 ) + k2 (y i1 i2 )).

(15)

112 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov

Tab. 8. σ = 0.5, a1 + b1 /2 = 1.5, a2 + b2 /2 = 2, l = T = 1. h1 = h2 = τ 0.5 0.2 0.1 0.05 0.02 0.01

  ˜ ˜ n max D 1 − d1 

˜1 D

t n ∈ω τ

1.5098 1.5016 1.5004 1.5001 1.5000 1.5000

  ˜ ˜ n max D 2 − d2 

˜2 D

C

0.0012 0.0161 0.0120 0.0073 0.0035 0.0018

t n ∈ω τ

2.0196 2.0032 2.0008 2.0002 2.0000 2.0000

C

0.0024 0.0325 0.0225 0.0131 0.0058 0.0030

Table 8 presents the average values of the velocity of wave propagation ˜ = (D˜ 1 , D˜ 2 ), D

D˜ α = avg(d˜ nαi1 i2 )

computed by formula (15) for the considered problem. From this one can conclude that the velocity of the travelling wave is constant.

1.2.6 Numerical experiments for non-linear equation Now, let us consider in the domain [0, 2]×[0, 2]×[0, 1] the non-linear equation (7) with k α (u) = χ0 u β , χ0 , β = const > 0, α = 1, 2 together with the exact solution [4] 1/β b  (2D + a1 + a2 )t − x1 − x2 + 4 D1/β 8 β where b = const > 0 and D = χ0βb . In this case, we have c α = D. The FDS (9)–(10) is EDS for u(x1 , x2 , t) =

γ1 =

(a1 + D)τ (a + D)τ = 1, γ2 = 2 =1 h1 h2

under one of the conditions (13)–(14). The results presented in Tables 9 and 10 illustrate, that the FDS (9)–(10) is EDS.

Tab. 9. σ = μ1 = μ2 = 0.5, γ1 = γ2 = 1, a1 = a2 = b = 1, β = χ0 = 8, ϵ = 1.0 · 10−17 . h1 = h2 = τ 0.5 0.25 0.125 0.05

max y n − u n C

Number of iterations

2.17 · 10−19 3.25 · 10−19 9.76 · 10−19 3.36 · 10−18

1 1 1 2

t n ∈ω τ

Parabolic equations

|

113

Tab. 10. σ = 1.0, μ1 = 0.3, μ2 = 0.7, γ1 = γ2 = 1, a1 = a2 = b = 1, β = χ0 = 2, ϵ = 1.0 · 10−17 . h1 = h2 = τ 0.5 0.25 0.125 0.05

max y n − u n C

Number of iterations

2.17 · 10−19 4.34 · 10−19 1.30 · 10−18 3.69 · 10−18

1 2 2 2

t n ∈ω τ

Numerical experiments illustrate that the FDS (9)–(10) is not EDS in the case of non-linear first derivative of u.

1.2.7 Finite-difference scheme for a convection-diffusion-reaction equation In the present section, results obtained for equation (7), which were discussed in the previous section, will be generalized to the convection-diffusion equation with reaction term λu. In the domain Q T = Ω × [0, T], let us consider the IBVP for a convection-diffusionreaction equation  2 2 ∂u ∂u  ∂ ∂u  k α (u) aα = + ∂t ∂x α ∂x α ∂x α α=1

+ λu, (x, t) ∈ Ω × (0, T],

(16)

α=1

together with the input conditions (8). Following [6], we introduce a new function v(x1 , x2 , t) u(x1 , x2 , t) = e λt v(x1 , x2 , t), for which we obtain the problem ∂v  ∂ ∂v  aα = + ∂t ∂x α ∂x α 2

2

α=1

α=1

v(x, 0) = u0 (x), x ∈ Ω,



 ∂φ α (e λt v) v , ∂x α

v|x∈∂Ω = e−λt g(x, t), (x, t) ∈ ∂Ω × (0, T].

The problem (17)–(18) is approximated by the FDS with weights   2 2     n λt n+1 n+1 a α v h(μ3−α )x α ,i1 i2 = vh ) σ v n+1 v hti1 i2 + h(μ3−α )+ φ α (e α=1



α=1



(17)

+ (1 − σ) v h(μ3−α ) φ α (e λt n v h )



 xα

xα

, x ∈ ωh , t ∈ ωτ , x α ,i1 i2

(18)

x α ,i1 i2

(19)

114 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov v0hi1 i2 = u0 (x), x ∈ ω h ,

v h |x∈∂Ω = e−λt g(x, t), t ∈ ω τ .

(20)

Here, μ1 , μ2 , σ ∈ R and 0 ≤ μ1 , μ2 , σ ≤ 1. Remark 1.8. If the conditions of Theorem 1.4 are satisfied, then the FDS (19)– (20) is EDS for the travelling-wave solution v(x, t) of the form (12). Following [6], we apply the relation v nhi1 i2 = e−λt n y ni1 i2 to the problem (19)–(20). Then we obtain the following FDS /  2 2 n     e−λτ y n+1 i1 i2 − y i1 i2 n n+1 n+1 φ a α y(μ3−α )x α ,i1 i2 = (y ) σe−λτ y(μ + α 3−α )+ τ xα α=1 α=1 2 

  , x ∈ ωh , t ∈ ωτ , + (1 − σ) y(μ3−α ) φ α (y) x α

y0i1 i2

= u0 (x), x ∈ ω h ,

x α ,i1 i2

x α ,i1 i2

y|x∈∂Ω = g(x, t), t ∈ ω τ ,

which approximates the problem (16), (8).

1.2.8 Conclusions The additional weights in space were introduced in the FDS to improve its property of EDS. The problem (7)–(8) can be also approximated by the FDS with two weights y t,i1 i2 +

2



2   n a α μy n+1 Λ(σ) x α ,i1 i2 + a α (1 − μ)y x α ,i1 i2 = α* y i1 i2 , x ∈ ω h , t ∈ ω τ ,

α=1

(21)

α=1

y0i1 i2 = u0 (x), x ∈ ω h , y|x∈∂Ω = g(x, t), t ∈ ω τ ,

(22)

n+1 − n − where Λ(σ) α* y i1 i2 = σΛ α* y i1 i2 + (1 − σ)Λ α* y i1 i2 and the operators Λ α* , Λ α* are given by formulas



     , Λ−α* y i1 i2 = y φ α (y) x . Λ α* y i1 i2 = y φ α (y) x α

x α ,i1 i2

α

x α ,i1 i2

The FDS (21)–(22) is EDS for a travelling-wave solution u(x1 , x2 , t) = U(x1 − (a1 + c1 )t, x2 − (a2 + c2 )t), a α + c α > 0, α = 1, 2, if μ = σ = 0.5, γ1 = γ2 = 1,

Parabolic equations

| 115

or a1 = a2 = a, c1 = c2 = c, μ = 0, σ =

(a + c) , γ1 = γ2 = 1. 2c

For other cases, this FDS has a second order of approximation, if σ = μ = 0.5, i.e., ψ ni1 i2 = O((σ −0.5)(h1 + h2 + τ)+(μ −0.5)(h1 + h2 + τ)+ h21 + h22 + τ2 ). The FDS is monotone and stable in the linear approximation y t,i1 i2 +

2



2   n a α μy n+1 y(σ) x α ,i1 i2 + a α (1 − μ)y x α ,i1 i2 = x α x α ,i1 i2 , b 1 , b 2 = const > 0,

α=1

α=1

for 0 ≤ σ, μ ≤ 1,

τ≤

h21 h22 /



a1 h1 /b1 ≤ σ/μ, a2 h2 /b2 ≤ σ/μ,

 (2b1 h22 + 2b2 h21 )(1 − σ) + (a1 h1 h22 + a2 h21 h2 )(1 − μ) .

The last condition in case h1 = h2 = h takes the form   τ ≤ h2 / (2b 1 + 2b2 )(1 − σ) + h(a1 + a2 )(1 − μ) . The main disadvantage of this FDS, in comparison with the FDS (9)–(10), is a second condition, which imposes very small grid steps for a wide classes of problems, e.g., problems with small parameters.

2 Solutions of the separated variables The Cauchy problem for the parabolic equation  ∂u ∂u ∂ k(x, t) + f (u, x, t), x ∈ R, t > 0, = ∂t ∂x ∂x u(x, 0) = u0 (x), x ∈ R, is considered [7]. Under conditions u(x, t) = X(x)T1 (t) + T2 (t),

∂u =  0, ∂x

k(x, t) = k1 (x)k2 (t),

f (u, x, t) = f1 (x, t)f2 (u),

it is shown that the above problem is equivalent to a system of two first-order ordinary differential equations for which EDSs with special Steklov averaging and FDS with any order of approximation are constructed on the moving mesh. On the basis of this approach, the EDS are constructed also for BVP and multidimensional problems. Presented numerical experiments confirm the theoretical results.

116 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov

2.1 Introduction Consider the Cauchy problem for the one-dimensional parabolic equation  ∂u ∂u ∂ k(x, t) + f (u, x, t), x ∈ R, t > 0, = ∂t ∂x ∂x u(x, 0) = u0 (x), x ∈ R.

(23) (24)

Under conditions u(x, t) = X(x)T1 (t) + T2 (t),

∂u =  0, ∂x

k(x, t) = k1 (x)k2 (t),

f (u, x, t) = f1 (x, t)f2 (u),

we show that problem (23)–(24) is equivalent to the following system of two ordinary differential equations: dx = c1 (x)k2 (t), dt

du = f1 (x(t), t)f2 (u), u(x(0), 0) = u0 (x(0)), dx dt dt =c1 (x)k2 (t) where c1 (x) = −

(k1 (x)u0 (x)) . u0 (x)

(25) (26)

From (25), we find the curve x = x(t), along which we get

from (26) the solution u(x, t) = u(x(t), t) of problem (23)–(24). Here, x(0) = x0 ∈ R is the initial state of the curve x = x(t). Special Steklov averaging ⎡ ⎢ c(x(t)) ≈ ⎣

x

n+1

1 x n+1 − x n

⎤−1 dx ⎥ n ⎦ , t n ≤ t ≤ t n+1 , x = x(t n ), t n = nτ, c(x)

xn

is used to construct EDS only on the moving mesh. On the basis of this approach, the EDSs are constructed also for BVP and for multidimensional problems. A FDS of arbitrary order of approximation is proposed in the case when the integral in the Steklov averaging cannot be evaluated exactly.

2.2 The Cauchy problem for parabolic equations In this section, using the special Steklov averaging, EDS for the Cauchy problem for parabolic equations is constructed. Let us consider in the domain Q T = R × [0, ∞) the Cauchy problem for the one-dimensional parabolic equation:  ∂u ∂u ∂ k(x, t) + f (x, t, u), x ∈ R, t > 0, (27) = ∂t ∂x ∂x u(x, 0) = u0 (x),

x ∈ R.

(28)

Parabolic equations

| 117

Assume that problem (27)–(28) has an unique solution u(x, t) ∈ C21 (Q T ),

u(x, t) = X(x)T1 (t) + T2 (t),

∂u =  0, ∂x

and that the input data have the following form k(x, t) = k1 (x)k2 (t),

f (x, t, u) = f1 (x, t)f2 (u).

The coefficient k is bounded from above and below, i.e., 0 < k1 ≤ k(x, t) ≤ k2 ,

(x, t) ∈ R × (0, ∞),

k1 , k2 = const,

k(x, t) ∈ C11 (Q T ).

Rewriting equation (27) as ∂u − ∂t

∂ ∂u ∂x (k(x, t) ∂x ) ∂u ∂x

∂u = f (x, t, u) ∂x

and using the notation ∂

∂u

(k(x, t) ) dx = − ∂x ∂u ∂x dt ∂x

yields the following form: ∂u dx ∂u + = f (x, t, u). ∂t dt ∂x For

dx the following holds: dt ∂ ∂ (k(x, t) ∂u ) (k(x, t)X  (x)T1 (t)) dx = − ∂x ∂u ∂x = − ∂x dt X  (x)T1 (t)

=− =−

∂x ∂ (k(x, t)X  (x)) ∂x  X (x)

∂

= − ∂x

∂ (k(x, t)X  (x)T1 (0)) (k1 (x)k2 (t)u0 (x)) = − ∂x X(x)1 T(0) u0 (x)

k2 (t)(k1 (x)u0 (x)) = c1 (x)k2 (t), u0 (x)

where c1 (x) = − we have

(k1 (x)u0 (x)) . It follows that, instead of the differential problem (27)–(28), u0 (x)

dx = c1 (x)k2 (t), dt

du = f1 (x(t), t)f2 (u), dx dt dt =c1 (x)k2 (t)

u(x(0), 0) = u0 (x(0)),

(29) (30)

118 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov where x(0) = x0 ∈ R is the initial state of the curve x = x(t). Solving this problem, we obtain the following integral equations   dx = k2 (t)dt, c (x)  1  du = f1 (x(t), t)dt. f2 (u) Let ω0h = {x0i = ih0i , i = 0, ±1, ±2, . . .}, ω0hL = {x0i = −L + ih0i , h0i =

2L , i = 0, N } N

be uniform grids in space at t = 0 and ω τ = {t n = nτ, n = 0, 1, 2, . . . }, ( ' T ω τT = t n = nτ, n = 0, N0 , τ = N0 be uniform grids in time. Here, h ni = x ni+1 − x ni is the space step at time t = t n . Let us approximate Problem (29)–(30) by FDS ⎛

⎞−1

n+1

− x ni ⎜ x n+1 1 i =⎜ ⎝ x n+1 − x n τ i i

x i

dx ⎟ ⎟ c1 (x) ⎠

x ni

t n+1 k2 (t)dt,

1 τ

tn

x0i ∈ ω0h , i = 0, ±1, ±2, . . . , n = 0, 1, . . . , ⎞−1 ⎛ y n+1 i  t n+1 − y ni ⎜ y n+1 1 du ⎟ 1 i ⎟ ⎜ f1 (x(t), t)dt, = ⎝ n+1 n τ f2 (u) ⎠ τ yi − yi y ni

(31)

(32)

tn

y0i = u0 (x0i ), i = 0, ±1, ±2, . . . , n = 0, 1, . . . . Here, equation (31) represents the space-grid as a moving mesh, where x0i ∈ ω0h is the initial partitioning. Then, the following theorem holds: Theorem 2.1. FDS (31)–(32) is EDS. Proof. We show that the FDS (31) is the EDS for (29). The truncation error ψ = ψ ni is ⎛ ψ ni

=

− x ni x n+1 i τ

n+1

⎜ 1 −⎜ ⎝ x n+1 − x n i

i

x i x ni

⎞−1 dx ⎟ ⎟ c1 (x) ⎠

1 τ

t n+1 k2 (t)dt, tn

Parabolic equations

⎛ ⎜ − x ni ⎜ x n+1 ⎜ i = n+1 ⎜ xi  dx ⎜ ⎝ x ni

xn+1 i x0

⎞

n

dx c1 (x)

−

x i x0

| 119

dx c1 (x)

τ

1 − τ

c1 (x)

⎟ t n+1 ⎟ ⎟ k2 (t)dt⎟. ⎟ ⎠ tn

(33)

On the basis of (29), we obtain ⎞ ⎛ x x t d⎜ dx ⎟ dx dx = k2 (t)dt, = c1 (x)k2 (t), ⎝ ⎠ = k2 (t), dt dt c1 (x) c1 (x) 0

x0

0

x0

0

⎞ ⎛ x x t dx d⎜ dx ⎟ dx = k2 (t)dt, = c1 (x)k2 (t), ⎝ ⎠ = k2 (t), dt dt c1 (x) c1 (x) ⎛ 1⎜ ⎜ τ⎝

0

n+1

x i

x0

n

dx − c1 (x)

x i x0

⎞ dx ⎟ ⎟= 1 c1 (x) ⎠ τ

t n+1 k2 (t)dt.

(34)

tn

Substituting (34) into (33), we obtain ψ ni = 0, for n = 0, 1, . . . and i = 0, ±1, ±2, . . .. Similarly, we can show that the FDS (32) is EDS for the differential problem (30). Hence, FDS (31)–(32) is EDS. Example 2.1. Let us consider the following Cauchy problem: ∂u ∂2 u − An1 (n1 − 1)x n1 −2 + Bn2 t n2 −1 , t > 0, u(x, 0) = Ax n1 , x ∈ R, = ∂t ∂x2

(35)

where n1 , n2 > 2. The solution of this problem exists and equals u(x, t) = Ax n1 + Bt n2 , where A, B =const. Using the above technique, equation (35) is replaced by the problem u (x) dx n −1 =− 1 = − 0 , dt x u0 (x)

∂u dx ∂u du + = f (x(t), t), u(x(0), 0) = u0 (x(0)) = A(x0 )n1 , n1 −1 = dt dx ∂t dt ∂x dt =− x

(36) (37)

where x(t) is the solution of equation (36) and f (x, t) = −An1 (n1 − 1)x n1 −2 + Bn2 t n2 −1 . Solving problem (36)–(37) analytically, we obtain !7 −2(n1 − 1)t + (x0 )2 , if x0 ≥ 0, 7 x(t) = − −2(n1 − 1)t + (x0 )2 , if x0 < 0, ! n1 Bt n2 + A((x0 )2 − 2(n1 − 1)t) 2 if x0 ≥ 0, u(x(t), t) = n1 n2 n1 −2 0 2 A((x ) − 2(n1 − 1)t) 2 if x0 < 0, Bt + (−1)

120 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov (x0 )2 . Substituting (x0 )2 = x2 + 2(n1 − 1)t in the above 2(n1 − 1) equation, we find the solution of the Cauchy problem (35) in the explicit form: ! if x ≥ 0, Bt n2 + A|x|n1 , u(x(t), t) = Bt n2 + (−1)n1 −2 A|x|n1 if x < 0, where x0 ∈ R and 0 < t
0, n = 0, N T − 1, t0 = 0, t N T = T }, ω   ¯ τ \ t N T = T , ω h = {x i = ih, h = l/N, i = 1, N − 1}, ωτ = ω   ¯ h = ω h ∪ 0, l , ω

136 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov

and Steklov averaging operators [13]: 1 S x p(x) = h

x+h p(ξ )dξ ,

S nt q(t) =

1 τn

t+τ n  q(η)dη,

x

t

S ntx s(x, t) = S nt S x s(x, t),

x ∈ ωh ,

t ∈ ωτ .

For approximation of the problem (62)–(64), consider the next FDS with Steklov averaging of the source term and boundary conditions and special approximation of the initial data n n y n+1 y nt,i = (a n,(σ) x¯ ,i )x + (S tx f )i , i

(x, t) ∈ ω,

ω = ωh × ωτ ,

(68)

¯ h. x∈ω

(69)

l

u0 (ξ )dξ 0 N−1 5

y(x, 0) = u0 (x)

h 2 u 0,0 +

− a1n,(σ) y n+1 x,0 + y n+1 a n,(σ) x¯ ,N + N

i=1

,

hu0,i + 2h u0,N

h y = (S nt φ)(t n ), 2 t,0

h y = (S nt ψ)(t n ) 2 t,N

t ∈ ωτ , (70)

t ∈ ωτ .

Here and below, we use standard notation of the theory of FDS [13] a ni = a(x i , t n , y(x i , t n )) = 0.5(k(x i−1 , t n , y(x i−1 , t n )) + k(x i , t n , y(x i , t n ))), = σa n+1 + (1 − σ)a ni , a n,(σ) i i

σ ∈ [0, 1].

It is easy to check that on smooth solutions, this FDS has the order of accuracy O(h2 + τ* ), τ* = max τ n [4]. Let us introduce the grid quantity of heat 0≤n≤N T −1

h n  n h n hy i + y N , y + 2 0 2 N−1

Q nh =

i=1

which approximates the quantity of heat Q(t n ) with the order of O(h2 ). Theorem 3.1. FDS (68)–(70) is exact L1 -conservative, i.e., Q h (t n ) ≡ Q(t n ),

n = 0, N T .

Proof. According to initial condition (69), we immediately obtain Q h (0) = Q(0).

Parabolic equations

|

137

Multiplying the difference equation (68) by h and summing the result over the internal nodes ω h , we obtain n (y t,n , 1) = ((a n,(σ) y n+1 x¯ )x , 1) + (S tx f , 1).

Using difference Green formula [4] and approximation of boundary conditions (70), we get ˆ x¯ ,N ) − (a(σ) ˆ x,0 ) = S nt ψ + S nt φ − ((a(σ) yˆ x¯ )x , 1) = (a(σ) 1 y N y

h h y − y . 2 t,N 2 t,0

Thus, for all n ⎛ ⎜ Q nh,t = S nt ⎝ψ(t) + φ(t) +

l

⎞ ⎟ f (x, t)dx⎠.

0

The last formula is the EDS for (67) (see Section 1). Thus we obtain that which expresses exact L1 -conservativeness.

3.3 Exact conservative iteration process In the case σ = 0, FDS (68)–(70) is called linearized and its solution is found by explicit formulas (Thomas algorithm [4]). Otherwise, we need to use iteration processes. Our aim is to implement the iteration process for the solution of the difference problem that also exactly satisfies the law of heat conservation for the differential problem on each iteration. For convenience, consider the case σ = 1 and the next iteration process [4]: (s+1)

(s) (s+1)

y n+1 − y ni n n i = (a n+1 y n+1 i i x¯ )x + (S tx f )i , τn

(0)

y n+1 = y ni , i

(x, t) ∈ ω,

(71)

with boundary conditions (s) (s+1) − a1n+1 y0n+1 x (s) (s+1) n+1 a n+1 N y N x¯

+

(s+1)

h y0n+1 − y0n + = (S nt φ)n , 2 τn h 2

(s+1) n y n+1 N − yN

τn

= (S nt ψ)n ,

t ∈ ωτ (72) t ∈ ωτ .

The important property of this iteration process is the consistent approximation of boundary conditions and flux in the iteration process (71)–(72). Define iteration

138 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov

quantity of heat as (s)

h n+1  n+1 h n+1 y + hyi + yN . 2 0 2 (s)

Q n+1 = h

N−1

(s)

(s)

i=1

Repeating the proof of the Theorem 3.1, one can prove Theorem 3.2. The iteration process (71)–(72) exactly satisfies the law of heat conservation for the differential problem (68)–(70) on each iteration, i.e., (s)

Q nh ≡ Q(t n ),

s = 0, 1, . . . ,

n = 0, N T .

3.4 Flow finite-difference scheme The proposed FDS (68)–(70) has the order of accuracy O(h2 + τ* ), τ* =

max τ n ,

0≤n≤N T −1

because we use the trapezoid quadrature formula l

 h h hu0,i + u0,N u + 2 0,0 2 N−1

u0 (ξ )dξ ≈

i=1

0

of second order of approximation and also the approximation of boundary conditions (70) has the order of accuracy O(τ* + h2 ). Moreover, as we have shown the exact solution of the FDS exactly satisfies the L1 -conservation law. Unfortunately, as for the approximate solution of the linear FDS (especially in multi-dimensional problems) is found by iteration method the conservation law may not be valid. In refs. [12, 14], it is shown that for constant coefficient k ≡ 1, the following estimate for the dis-balance for the quantity of heat is obtained for Seidel iteration method (s)

Q nh − Q nh  ≤

ετ(n − 1) . h2

So one should choose ε from the condition above, which can be very rather strong. Alternatively in the paper mentioned above, how to choose iteration process without dis-balance is presented. Unfortunately, these conditions are valid only for the constant coefficient k ≡ 1. That is why Shashkov [14] proposed so-called flow FDS for heat equation. Let us first rewrite (62) in the form of system: ∂u ∂w = + f (x, t), ∂t ∂x

w=k

∂u , ∂x

(x, t) ∈ Ω T ,

(73)

Parabolic equations

where w is the flux of heat. Boundary conditions (64) take the form −w = φ(t), w = ψ(t), t ∈ (0, T]. x=0

x=l

|

139

(74)

Shashkov’s flow FDS approximating (73), (74)and (63) for the case f (x, t) ≡ 0, k ≡ 1, is u¯ nht,i = (w h )n+1 x,i ,

i = 0, N − 1,

−w n+1 h,0 = φ(t n+1 ),

¯ n+1 w n+1 h,i = u h x¯ ,i ,

w n+1 h,N = ψ(t n+1 ),

 u¯ 0h,i = u0 x i+ 1 , 2

i = 1, N − 1,

(75)

n = 0, N T − 1.

i = 0, N − 1.

(76)

Here, the temperature u¯ h refers to the centres of the cells, and the flow w h refer to the surrounding nodes. FDS (75)–(76) is also of order O(τ* + h2 ). Moreover,it has no dis-balances and for difference quantity of heat defined by Q hn,* =

N−1 

h u¯ nhi ,

i=0

it satisfies [14] the following difference analogue of (67) for f (x, t) ≡ 0: n,* = ψ(t n ) + φ(t n ). Q h,t

So flux FDS is L1 -conservative.

3.5 Exact L1 -conservative flow algorithm As the relation n,* = ψ(t n ) + φ(t n ). Q h,t

is the explicit Euler method for differential equation dQ = ψ + φ, dt it means that Shashkov’s flow FDS (75)–(76) satisfies the differential conservation law with the first order. Our aim is to modify this FDS to make it exact L1 -conservative. Consider system (73)–(74) and appropriate FDS with Steklov averaging of the source term and boundary conditions and special approximation of initial data. n n u¯ nht,i = (w h )n+1 x,i + (S tx f )i ,

w n+1 h,i

=

a ni u¯ n+1 h x¯ ,i ,

i = 0, N − 1,

i = 1, N − 1,

(77)

140 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov

n n −w n+1 h,0 = (S t φ) ,

n n w n+1 h,N = (S t ψ) ,

l

n = 0, N T − 1,

u0 (ξ )dξ

u¯ 0h,i = u¯ 0,i 0

N−1 5

,

i = 0, N − 1.

(78)

h u¯ 0,i

i=0

Let us note that Q hn,* approximates Q(t) with the second order as it is midpoint quadrature formula. Analogously to Theorems 3.1 and 3.2, we can prove Theorem 3.3. FDS (77)–(78) is flow exact L1 -conservative, i.e., Q hn,* ≡ Q(t n ),

n = 0, N T .

3.6 Multidimensional generalization In this section, the exact L1 -conservative flow FDSs are constructed for the Neumann problem for multi-dimensional heat equation in an isotropic medium. We describe the heat state of a solid body, which has volume Ω m = Ω1 × Ω2 × . . . Ω m ,

Ω α = {x α : 0 ≤ x α ≤ l α },

α = 1, m,

beginning with the initial time t = 0 upto a final time t = T, T > 0. Let Ω m T = {( x, t) | m m x ∈ Ω , 0 < t ≤ T } and let Γ = {(x, t) | x ∈ ∂Ω , 0 < t ≤ T } be the lateral surface of Ω. The propagation of heat in an isotropic medium is described by parabolic equation [15]  m ∂u ∂u  ∂ k α (x, t, u) = ∂t ∂x α ∂x α

+ f (x, t),

(x, t) ∈ Ω T ,

(79)

α=1

where we assume that 0 < k1 ≤ k α (x, t, u) ≤ k2 ,

k1 , k2 = const,

α = 1, m.

Let n denote the external with respect to the domain Ω m , normal to the boundary ∂Ω m and cos(n, x α ), α = 1, m, be the direction cosines of the external normal. Then, the flow is specified by equation ∂u  ∂u kα cos(n, x α ). = ∂ν ∂x α m

α=1

Parabolic equations

|

141

Equation (79) is complemented by initial and Neumann boundary conditions: u(x, 0) = u0 (x), ∂u = φ(x, t), ∂ν

x ∈ Ωm , (80)

(x, t) ∈ Γ.

The quantity of heat Q contained in the system at time t is determined by relation  m Q (t) = u(x, t)dΩ m . (81) Ωm

Let Ω m,α = Ω m /Ω α , α = 1, m, φ α,0 = φ(x, t) x α =0 , φ α,l = φ(x, t) x α =l α ,

x ∈ Ω m,α .

Integrating (79) by x in Ω m from equation (81) and boundary conditions (80), we obtain ODE dQ m  = dt p



(φ α,0 + φ α,l )dΩ m,α +

α=1 Ω m,α



f (x, t)dΩ m ,

(82)

Ωm

which expresses the differential law of conservation of heat. Finally integrating (82) by t in the interval [0, t], we get ⎛ ⎞ t   m  ⎜ ⎟ (φ α,0 + φ α,l )dΩ m,α + f (x, t)dΩ m ⎠dt, Q m (t) = Q m (0) + ⎝ 0

α=1 Ω m,α

Ωm

which expresses the integral law of conservation of heat. Let us introduce the next grids α = 1, m, ωm α,h = { x α,i α = i α h α , h α = l α /N α , i = 1, N α − 1} ,   m m −m m ¯ α,h = ω α,h ∪ 0, l α , ω α,h = ω α,h ∪ {0}, α = 1, m, ω −m m −m −m ω−m,α = ω−m 1,h × . . . × ω α−1,h × ω α,h × ω α+1,h × . . . × ω m,h , h

¯m ω h m

=

¯ = ω

m m ¯ 1,h ¯ 2,h ω ×ω ¯m ¯ τ, ω h ×ω

¯m ×...×ω ω−m m,h , h ω−m = ω−m × ω , τ h

=

ω−m 1,h m

× ω−m 2,h m

α = 1, m,

× . . . × ω−m m,h ,

¯ ∩ Γ. ∂ω = ω

Introduce the following notation:

S x α p(x) =

1 hα

x α +h α

p(x)dx α ,

α = 1, m,

(x, t) ∈ ω−m ,

xα

S m,n tx q(x, t) =

S nt S x1 S x2

. . . S x m q(x, t),

(x, t) ∈ ω−m .

142 | Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov ¯ m , consider the FDS On the grid ω u¯ nh,t = w n+1 h,α

=

m 

α=1 a α u¯ n+1 h, x¯ α ,

m,n (w n+1 h,α )x α + S tx f ,

(x, t) ∈ ω

−m,α

=

(x, t) ∈ ω−m ,

ω−m,α h

a α = 0.5(k α,i α −1 + k α,i α ),

× ωτ ,

(83)

α = 1, m,

α = 1, m.

Here, the temperature u¯ h refers to the centres of the cells, and the flows w h,α , α = 1, m, refer to the surrounding nodes. For convenience, we use the following notation for the scalar products: (w, n) =

m 

 w α cos(n, x α ),

α=1

(u, 1)l1m =

⎛



⎝u h

0≤i α ≤N α −1 α=1,m

m *

(u, 1)L1m = ⎞

udΩ m ,

Ωm

h α ⎠.

α=1

Initial and Neumann boundary conditions (80) are approximated by: (u0 , 1)L1m , x ∈ ω−m h , (u¯ 0 , 1)l1m (w, n) (x,t)∈∂ω m = S m,n tx φ.

u¯ 0h = u¯ 0

(84)

Let us define the difference quantity of heat by Q hm,n,* = (u h , 1)l1m . This relation approximates (81) with the order O((h* )2 ), h* = max h α . Analogously to 1≤α≤m

Theorems 3.1, 3.2 and 3.3, we can prove Theorem 3.4. FDS (83)–(84) is flow exact L1 -conservative, i.e., Q hm,n,* ≡ Q m (t n ),

n = 0, N T .

Bibliography [1]

A. N. Kolmogorov, I. G. Petrovskii, and N. S. Piskunov. A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem. Bjul. Moskovskogo Gos. Univ, 1(7):1–26, 1937.

Parabolic equations

[2] [3] [4] [5]

[6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

|

143

A. I. Volpert, V. A. Volpert, and V. A. Volpert. Travelling wave solutions of parabolic systems, volume 140. American Mathematical Soc., 1994. B. H. Gilding and R. Kersner. Travelling waves in nonlinear diffusion-convection reaction, volume 60. Birkhäuser, 2012. A. A. Samarskii. The theory of difference schemes. Marcel Dekker Inc., New York - Basel, 2001. M. Lapinska-Chrzczonowicz and P. Matus. Exact difference schemes and schemes of higher order of approximation for a convection-diffusion equation. I. Ann. UMCS, Inform., 13(1):37–51, 2013. A. N. Tikhonov and A. A. Samarskii. Equations of mathematical physics, volume 39. Courier Corporation, 1990. M. Lapinska-Chrzczonowicz and P. Matus. Exact difference schemes for parabolic equations. Int. J. Numer. Anal. Model., 5(2):303–319, 2008. P. Farrell, A. Hegarty, J. M. Miller, E. O’Riordan, and G. I. Shishkin. Robust computational techniques for boundary layers. CRC Press, 2000. A. Samarskii, V. Galaktionov, S. Kurdyumov, and A. Mikhailov. Blow-up in quasilinear parabolic equations. De Gruyter, Berlin, 1995. A. N. Tikhonov and A. A. Samarskii. Convergence of difference schemes in the class of discontinuous coefficients. Dokl. Akad. Nauk SSSR, 124(3):529–532, 1959. Yu. P. Popov and A. A. Samarskii. Completely conservative difference schemes. USSR Comput. Math. Math. Phys., 9(4):296–305, 1969. M. Yu. Shashkov. Violation of conservation laws when solving difference equations by iteration methods. USSR Comput. Math. Math. Phys., 22(5):131–139, 1982. A. A. Samarskii, P. P. Matus, and P. N. Vabishchevich. Difference schemes with operator factors. Springer, 2002. M. Yu. Shashkov. Conservative finite-difference methods on general grids, volume 6. CRC Press, 1995. A. Samarskii and P. Vabishchevich. Computational heat transfer, volume 1 of Mathematical modelling. John Wiley & Sons, Chichester, 1995.

Ronald E. Mickens and Talitha M. Washington

Use of exact difference schemes to construct NSFD discretizations of differential equations 1 Introduction The main purpose of this chapter is to provide a brief summary of some of the results coming from the application of the NSFD methodology to the discretization of both ordinary and partial differential equations. A fuller and deeper examination and discussion of various issues related to the NSFD methodology is given in various publications of Ronald E. Mickens and his collaborators [1–9]. The need for discretizations of differential equations follows directly from the fact that few differential equations have known exact solutions expressible in terms of a finite number of the elementary functions [1, 10]. Further, even in situations where such exact solutions exist, they may not be of practical value for applications [11, 12]. A way to resolve this issue is to calculate numerical solutions. Within the context of finite-difference procedures, the standard methods are based mathematically on the extensive use of the concepts of convergence, stability and consistency. However, the NSFD methodology of Mickens is focused on ensuring that the important physical properties of the system, modelled by the differential equations, are incorporated both into the construction of the discretizations and into the calculated numerical solutions [1, 2, 7]. Thus, standard methods for constructing finite-difference models of differential equations and the NSFD methodology differ not only in their mathematical foundations, but also in what goals are to be achieved. In summary, the two procedures have different philosophical bases as to what constitutes valid discretization models for differential equations. The NSFD methodology has been extensively applied to many of the elementary and complex differential equations appearing in the mathematical models of systems in the biomedical, engineering and natural sciences. Particular applications include the following topics: – simulation of robotic system – computational electromagnetics – non-linear heat transport – non-smooth, vibro-impact, mechanical system – interacting populations – singular perturbation mathematical problems – robust numerical methods for chaotic systems

Use of EDS to Construct NSFD Discretizations

– –

| 145

advection-diffusion-reactioning systems mathematical issues relating to the NSFD methodology

An examination of references [2, 3, 5] and [6] provide the names of the major researchers working on these issues. The remainder of this chapter has the following format. Section 2 gives a brief introduction to the definition of exact schemes and illustrates its use by explicitly constructing such schemes for four differential equations. Section 8 outlines the general NSFD methodology for constructing discretizations of differential equations. The critical issues related to the actual construction of NSFD schemes are illustrated in Section 14 by explicitly carrying out such tasks for a number of differential equations which model important phenomena occurring in a broad range of the sciences. Finally, in Section 22, we give a concise summary of our general results and state several research problems of great interest for future study.

2 Exact finite-difference schemes Given a system of ODEs such that a solution exists for the initial-value problem, i.e., du = f (u, t, λ), u(t0 ) = u0 , dt u(t) = ϕ(u0 , t0 , t, λ), where λ = (λ1 , λ2 , . . . , λ N ) is the set of parameters characterizing the system. Then, an exact finite-difference scheme is [1, 2, 6] u k+1 = ϕ(u k , t k , t k+1 , λ), where t k = (∆t)k,

u k = u(t k ),

k = (0, 1, 2, . . . ).

For the case of PDEs, it is not possible to give a general characterization or definition of what constitutes and exact finite-difference representation. However, when such occurs, the solutions to the discrete scheme are exactly equal to the PDEs at the nodes of the (computational) nodes. In the following four subsections, it will be demonstrated how exact schemes can be determined for several specific differential equations.

146 | Ronald E. Mickens and Talitha M. Washington

3 Logistic equation The logistic differential equation is du = λ1 u − λ2 u2 , dt

u(t0 ) = u0 ,

(1)

where λ1 and λ2 are parameters. The general solution for (1) in ref. [1] is u(t) ≡ ϕ(u0 , t0 , t, λ1 , λ2 ) =

λ1 u0 . (λ1 − u0 λ2 ) exp[−λ1 (t − t0 )] + λ2 u0

(2)

Making the replacements t0 → t k ,

t → t k+1 ,

u0 → u k ,

u(t) → u k+1

in (2) and rearranging the resulting expression, gives the following result for the exact finite-difference scheme of the logistic equation u − uk  k+1  = λ1 u k − λ2 u k+1 u k . e λ1 h − 1 λ1 The expression e λ1 h − 1 = D(λ1 , h) λ1 is the denominator function [1], and it has the property D(λ1 , hy) = h + O(λ1 h2 ). Also, note that the non-linear, quadratic in u factor, is modeled at two difference grid notes, “k” and “k + 1”, i.e., u2 → u k+1 u k .

4 Second-order ODE having constant coefficients A broad range of physical phenomena in the fields of vibration, acoustics, and seismology can be modelled by a second-order, linear ODE having constant coefficients [13]. a

dy(x) d2 y(x) +b + cy(x) = 0 dx dx2

(3)

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The exact finite-difference scheme for (3) is [13] 2 1 2 1 y − ψy k−1 y − 2y k + y k−1 +b k a k+1 D1 D2 / 6 2 2 2(1 − ψ)y k + (ϕ + ψ − 1)y k−1 +c = 0, D3 where h = ∆t, t k = hk, u k = u(t k ), and D1 = D1 (a, b, c, h) = D2 = D2 (a, b, c, h) =

(4)

a

[ϕ(a, b, c, h)]2 ,

c

a

ϕ(a, b, c, h), c D3 = D3 (a, b, c, h) = ϕ(a, b, c, h),

and /

6 / 6  β

7 e−β 1 − ϵ2 · ϕ(a, b, c, h) = √ · sin , ϵ 1 − ϵ2 / 6  β

7 ϵe−β −β + e cos 1 − ϵ2 · ψ(a, b, c, h) = √ , ϵ 1 − ϵ2 with ϵ2 =

b2 , 4ac

β=

bh . 2a

Careful inspection of (4) and the definitions of (D1 , D2 , D3 ) gives the results: (i) The functions D1 and D3 have the properties D1 (a, b, c, h) = h2 + O(h4 ), D2 (1, b, c, h) + h + O(h2 ). (ii) In the limits h → 0, k → ∞, hk = t = fixed, / 6 2(1 − ψ)y k + (ϕ2 + ψ2 − 1)y k−1 lim = y(t). D3 (iii) Taking the appropriate limits for the parameters (a, b, c), the exact-difference scheme (4) reduces to the exact schemes for the following differential equations a

dy d2 y +b = 0, dt dt2 d2 y a 2 + cy = 0, dt dy b + cy = 0. dt

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5 Jacobi differential equations The Jacobi elliptic functions, cn(t, k) and sn(t, k), satisfy the following differential equations, with associated initial conditions [14] x = cn(t, k) :

x = sn(t, k) :

d2 x + (1 − 2k2 )x + 2k2 x3 = 0, dt2 dx(0) x(0) = 1, = 0; dt 2 d x + (1 + k2 )x − 2k2 x3 = 0, dt2 dx(0) x(0) = 0, = 1, dt

(5)

(6)

where the k-dependency in x has been suppressed, and, in general, 0 ≤ k2 ≤ 1. The Duffing differential equation is used to model a broad range of phenomena in the theory and application of single-degree-of-freedom non-linear, oscillatory systems [15, 16]. The Duffing equation takes the form d2 y + ay + by3 = 0. dt2

(7)

Rescaling, i.e., y(t) = Ax(t) and

t = T t¯ ,

gives d2 x + (aT 2 )x + (bT 2 A2 )x3 = 0, dt2

(8)

where the over-bar in t¯ has been dropped. If x(0) = 1 and dx(0)/dt = 0, then comparison of (5) and (8) gives T2 =

1 , a + bA2

k2 =

bA2 . 2(a + bA2 )

If a > 0 and b ≥ 0, then 0 ≤ k2 < 1/2 and A is the initial amplitude, i.e., y(0) = A. Consequently, the solution to (7) is y(t) = A cn(t, k2 ). Starting with the addition theorem [14] for the Jacobi cosine function [17] cn(u + v) =

(cn u)(cn v) − (sn u)(dn u)(sn v)(dn v) , 1 − k2 (sn u)2 (sn v)2

t m = hm,

h = ∆t,

and defining x m = cn(t m ),

m = integer,

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the following expression can be derived [17] !1 " 2   1 − cn h x m+1 − 2x m + x m−1 2 x m+1 + x m−1 xm − k +2 2 (sn h)2 (sn h)2

x  +x + 2k2 m+1 m−1 x2m = 0. 2

(9)

(Note that there are three Jacobi elliptic functions [14]. The third is denoted dn(t, k2 ).) Taking the limits h → 0,

m → ∞,

hm = t = fixed,

gives the differential equation d2 x + (1 − 2k2 )x + 2k2 x3 = 0, dt2 with (9) representing its exact-difference scheme [17]. In a similar way, the exact-difference scheme for sn(t), see (6), is [17] 2 1

 x m+1 − 2x m + x m−1 1 − (cn h)(dn h) 2 x m+1 + x m−1 x + 2 − 2k x2m = 0. m 2 (sn h)2 (sn h)2

6 A non-linear reaction-advection PDE The first-order, non-linear PDE ∂w ∂w + v0 = λ1 w − λ2 w2 , ∂t ∂x where w = w(x, t), and (v0 , λ1 , λ2 ) are positive parameters, can be rescaled using t = T t¯ ,

x = X x¯ ,

w = w* u

to the form ∂u ∂u + = u − u2 , ∂t ∂x

(10)

where the over-bars have been dropped from (t¯ , x¯ ), and T=

1 , λ1

X=

v0 , λ1

u* =

λ1 . λ2

With the initial value u(x, 0) = f (x), where f (x) is bounded and has a continuous first derivative, the solution to (10) is [1] u(x, t) =

f (x − t) . e−t + (1 − e−t )f (x − t)

(11)

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By means of a direct series of essentially algebraic manipulations, it can be shown that given (11), the exact finite- difference scheme for (10) is [1] k u k − u km−1 u k+1 m − um + m = u km−1 (1 − u k+1 m ), ϕ(∆t) ϕ(∆x)

(12)

where ϕ(h) = e h − 1,

∆t = ∆x = h,

and t → t k = (∆t)k,

x → x m = (∆x)m,

u km = u(x m , t k ).

It is important to note that there is a relationship between the time and space step-sizes. In addition, since (12) is linear in u k+1 m , it can be rewritten to the expression u k+1 m =

[1 + ϕ(h)]u km−1 . 1 + ϕ(h)u km−1

Consequently, on the discrete space-time grid, (x m , t k ), knowledge of only u km−1 gives u k+1 m .

7 Comment The above examples of exact schemes illustrate the fact that such constructions are essentially impossible to derive from application of the standard rules generally used in the numerical analysis of differential equation community [11, 18, 19]. Therefore, it is clear that a radically different methodology is required if new and (in some sense) improved finite- difference structures are to be constructed for the discretization of differential equations. The next section is devoted to beginning this task.

8 NSFD methodology The NSFD basic philosophy for constructing finite-difference models of differential equations is to use discretization procedures that directly incorporate important features of the differential equations. Which features are selected as essential are to be determined a priori [1, 2, 6] to the actual construction of the scheme. One consequence of this way of proceeding is that the inclusion of different properties can give rise to different discretizations of the same differential equations. This “physical” approach may produce vastly different mathematical structures for the

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finite-different mathematical structures in comparison to those constructed using conventional methods. This section gives a brief overview of the basic NSFD methodology. A much more extensive and deeper discussion of many of the associated issues and topics are to be found in the references [1, 4, 5] and [7, 8].

9 Discrete derivatives Within the NSFD methodology, the discrete representation of the first-order derivative is [1, 20] ⎧ ⎪ x k+1 − ψ(h, λ)x k ⎪ ⎪ , ⎪ ⎨ ϕ1 (h, λ) dx → dt ⎪ x k − ψ(h, λ)x k−1 ⎪ ⎪ ⎪ , ⎩ ϕ (h, λ) 1

where the first item is a generalized forward-Euler, and the second item is a generalized backward-Euler representations. The functions ψ(h, λ) and ϕ1 (h, λ) have the properties (for fixed λ) ψ(h, λ) = 1 + O(h2 ),

ϕ1 (h, λ) = h + O(h2 ),

where λ can be one or more of the parameters that occur in the differential equation. An implication of this type of representation is that both ψ(h, λ) and ϕ1 (h, λ) may not be determinable until later on in the discretization construction. The ψ and ϕ functions are called, respectively, the “numerator” and “denominator” functions. For the standard methods of constructing discrete first-derivatives, ψ(h, λ) = 1,

ϕ1 (h, λ) = h.

Within the context of the NSFD methodology, it is always possible to select ψ(h, λ) = 1, while ϕ1 (h, λ) will generally have a more complex structure. Second-order derivatives may always be represented as [1, 7, 20] x − 2x k + x k−1 d2 x → k+1 , dx2 ϕ2 (h, λ) where ϕ2 (h, λ) = h2 + O(h4 ).

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A detailed examination of the exact schemes, given in the previous section, show that in general it is not expected that the standard representations ϕ1 (h, λ) = h

or

ϕ2 (h, λ) = h2

will apply.

10 Non-locality of functional terms The NSFD representation of polynomial terms is generally based on non-local (on the computational grid) discrete models of these functions [1, 20]. Furthermore, particular representation depends on the order of the differential equation. For example: ⎧ ⎪ x k+1 x k , first-order ODE; ⎪ ⎪ ⎪ ⎨ x k+1 + x k + x k−1 x2 → x k , second-order ODE; ⎪ 3 ⎪ ⎪ ⎪ ⎩2x2 − x x ; first-order ODE; k+1 k k and

⎧ ⎪ ⎪ x2k x k+1 , ⎪ ⎪ ⎪ ⎪ ⎨ 2 2 2x k x k+1 x3 → , ⎪ x ⎪ k+1 + x k ⎪

 ⎪ ⎪ ⎪ x k+1 + x k−1 x2 , ⎩ k 2

first-order ODE; first-order ODE; second-order ODE.

Non-polynomial functions of x generally requires both thought and ingenuity for their discretizations [20, 21].

11 Other properties There are several other properties of the NSFD methodology, which must be incorporated into the construction of the associated finite-difference scheme. – The orders of the discrete derivatives must be exactly the orders of the corresponding derivatives in the differential equations. If the order of the discrete derivatives are greater, then ’spurious’ solutions will appear [1, 11]. Such solutions do not equate to any solution of the original differential equations. – Special solutions and conditions of physical importance should also be solutions and conditions for the NSFD scheme. Particular examples are fixed-points (constant solutions) and their (in most cases, linear) stability properties, and the requirement of positivity when applicable.

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It must be realized that the NSFD methodology may not give a unique discretization for a given set of differential equations. Further, the discretizations obtained are, in general, not exact schemes. However, because of the nature of their construction, they should be expected to not possess the usual numerical instabilities associated with the standard procedures for constructing finite-difference discretizations of differential equations [1, 2, 6, 20].

12 Dynamic consistency An important concept that places severe restrictions on the construction of NSFD schemes is the principle of dynamic consistency (DC) [7]. The basic idea is to first specify which properties and features are desired to be incorporated into the discretization of the differential equations and then formulate the construction of the NSFD scheme such that their properties and features are also possessed by the discretization. Particular examples of such items are boundedness, monotonicity, and the existence of special types of solutions such as periodic travelling waves, rational, and solitons [1, 6, 7, 20].

13 Subequations One of the most powerful techniques available to aid in the construction of NSFD schemes is the method of subequations [1, 7, 8, 20]. The procedure will be illustrated by means of an explicit example. Consider the linear, advection-diffusion PDE ∂u ∂u ∂2 u + =D 2, ∂t ∂x ∂x

(13)

where D is a positive parameter. This PDE has the following three, non-trivial sub-equations ∂u ∂u + = 0, ∂t ∂x

du d2 u =D 2, dx dx

∂u ∂2 u =D 2. ∂t ∂x

The first two equations have known exact-difference schemes, while no such schemes exist for the third equation. To formulate a NSFD scheme for (13), one starts with the two exact schemes and match or fit them together such that the ∂u/∂x and du/dx terms agree. This technique can be easily generalized to differential equations containing more than three terms.

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14 NSFD applications This section lists and provides brief discussions on the application of the NSFD methodology to the construction of difference equation schemes for seven ordinary and partial differential equations. There equations model interesting and important phenomena in the natural, engineering, and economic sciences.

15 Conservative oscillators The general undamped, unforced Duffing oscillator equation is [15] d2 x + ω2 x + ax2 + bx3 = 0. dt2 Its first-integral or energy function is given by the expression by        2 1 dx ω2 a 3 b 4 dx 2 = x + x = constant. + x + E x, dt 2 dt 2 3 4

(14)

The corresponding NSFD discrete energy function, E(x k , x k−1 ) is [2, 22, 23] 22  2  1  x k − x k−1 1 ω E(x k , x k−1 ) = + x k x k−1 2 2 ϕ(h)     x2k x k−1 + x k x2k−1 a b 2 2 x x , + + 3 2 4 k k−1 where E(x k , x k−1 ) = constant. Note that this expression is invariant under the interchange of k and (k − 1). This is but the discrete formulation of the property that (14) is invariant under the transformation t → (−t) [2]. Also, observe that the functions, (x2 , x3 , x4 ) have non-local representations. To determine the second-order equation of motion, ∆E = 0 must be calculated, and doing this gives ∆E(x k , x k−1 ) = E(x k+1 , x k ) − E(x k , x k−1 ) = 0, and



x +x  x k+1 + x k + x k−1 x k+1 − 2x k + x k−1 2 x k + b k+1 k−1 x2k = 0. + ω x + a k 2 3 2 [ϕ(h)]

(15)

Inspection of (15) shows that it is linear in x k+1 , i.e., there exists a function F such that x k+2 = F(x k+1 , x k ), where F(x k , x k−1 ) can be obtained from solving (15) for x k+1 .

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If x(0) and dx(0)/dt are specified, then x0 and x1 may be determined, and x k can then be calculated for k ≥ 2. The denominator function, ϕ(h) is not determined at this level of the analysis. An elementary choice is   ωh 2 sin . ϕ= ω 2 Finally, note that the NSFD scheme (15) differs radically in mathematical structure from what is obtained using a basic application of standard finite-difference techniques [11] x k+1 − 2x k + x k−1 + ω2 x k + ax2k + bx3k = 0. h2

16 Time-independent Schrödinger equation The generalized time-independent Schrödinger equation is d2 u + f (x)u = 0. dx2

(16)

based on the exact-difference scheme for the harmonic oscillator differential equation [1], d2 u + λu = 0 dx2

→



u k+1 − 2u k + u k−1 ⎡  ⎤ + λu k = 0, √  2 h λ ⎦ 4 ⎣ sin λ 2

[24] derived the following NSFD scheme 

u k+1 − 2u k + u k−1 ⎡  7 ⎤2 + f k u k = 0, h fk ⎦ 4 ⎣ sin fk 2

where x → x k = hk,

h = ∆x;

u(x) → u k ,

f k = f (x k ).

Using the identity 2 sin2 θ = 1 − cos θ, the latter equation becomes 1 7 2 u k+1 + u k−1 = 2 cos h f k u k .

(17)

156 | Ronald E. Mickens and Talitha M. Washington

This is the Mickens–Ramadhani scheme (MRS). However, based on (17), [25] extended this scheme to what they named the combined Numerov–Mickens finite-difference scheme (CNMFDS). / 6 6 6 / 1 7 2/ h2 f k−1 h2 f k+1 h2 f k u k+1 + 1 + u k−1 = 2 cos h f k uk . 1+ 1+ 12 12 12 The Numerov scheme is a widely used standard procedure for obtaining numerical solutions to differential equations having the form given in (16). [25] reached the following conclusions: – The MRS and CNMFDS are exact-difference schemes if f (x) is constant. This is not the case for the Numerov method. – The MRS is (formally) of O(h2 ), while the Numerov scheme is O(h4 ). However, the MRS outperforms, i.e., gives more accurate numerical results than the Numerov schemes for large values of h. – The CNMFDS is O(h4 ), same as the Numerov method, but gives more accurate solutions for large h than the Numerov method. In summary, the NSFD scheme CNMFDS provides an enhanced numerical accuracy in comparison to the standard Numerov scheme and the MRS.

17 Linear, advective-diffusive PDE The linear, advective, diffusive, reactive PDE provides a good approximate model for a broad range of phenomena in acoustics and fluid dynamics [12, 26]. This equation takes the form ∂u ∂u ∂2 u + =D 2 ∂t ∂x ∂x

(18)

where D is positive and u = u(x, t). It is assumed that the solutions satisfy a positivity condition, i.e., u(x, t) ≥ 0 and t ≥ 0, where,e.g., the variable u denotes a density of some quantity. In the following discussion, the following notation is used x → x m = (∆x)m,

t → t k = (∆t)k,

u km = u(x m , t k ).

Examination of (18) shows that it has the following three subequations [27] ∂u ∂u + = 0, ∂t ∂x

du d2 u =D 2, dt dx

∂u ∂2 u =D 2, ∂t ∂x

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The first and second equations have known exact-difference schemes [1], and they are k u k − u km−1 u k+1 m − um + m = 0, ∆t = ∆x ∆t ∆x − 2u m + u m−1 u u m − u m−1 . = D m+1 ∆x/D ∆x D(e − 1)∆x

Matching the discrete ∂u/∂x and du/dx terms gives the NSFD scheme [27] 6 / k u km − u km−1 u km+1 − 2u km + u km−1 u k+1 m − um . + =D ∆t ∆x D(e ∆x/D − 1)∆x This equation is linear in u k+1 m and solving for it gives k k k u k+1 m = Au m+1 + (1 − A − 2B)u m + (A + B)u m−1 ,

(19)

where A=

∆t , ∆x

B=

A . e ∆x/D − 1

(20)

For the discretization equation, the positivity condition can be enforced if 1 − A − 2B ≥ 0. (Note, this is not the only way this requirement can be fulfilled.) Substituting A and B into (19) gives 6 / e ∆x/D − 1 ∆x. (21) ∆t ≤ ∆x/D e +1 However, for D = 0 and ∆t = ∆x, the equal sign is required, i.e., 6 / e ∆x/D − 1 ∆x. ∆t = ∆x/D e +1 In summary, a NSFD scheme for the liner, advective-diffusive PDE (18) is given by (19), where A and B are the expressions in (20). Further, the relation between the space and time step-sizes is (21). Finally, the positivity condition u km ≥ 0 ⇒ u k+1 m ≥ 0, holds for this scheme.

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18 Linear advection, non-linear reaction PDE This non-linear PDE is ∂u ∂u + = f (u), ∂t ∂x

(22)

where it is assumed that u(x, t) is non-negative. For the case where f (u) is a polynomial, write it as follows. f (u) = f (+) (u) − f (−) (u) where f (+) (u) = a0 u +

I1 

a i (u)i = a0 u + g (+) (u)u

i=1

f (−) (u) = b0 u +

I2 

b j (u)j = b0 u + g (−) (u)u

j=1

with ai ≥ 0 :

i = 0, 1, 2, . . . , I1 ,

bj ≥ 0 :

j = 0, 1, 2, . . . , I2 ,

for some positive integers I1 and I2 . With these results, (22) becomes ∂u ∂u + = (a0 − b0 )u + g (+) (u) − g (−) (u)u. ∂t ∂x The subequation ∂u ∂u + = (a0 − b0 )u ∂t ∂t has the exact-difference scheme, see Section 6 and [1], k u k − u km−1 u k+1 m − um + m = (a0 − b0 )u km−1 ϕ(∆t) ϕ(∆t)

where ∆t = ∆x = h,

ϕ(h) =

e(a0 −b0 )h − 1 . (a0 − b0 )

(23)

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Therefore, a NSFD scheme for (23), which preserves the positivity condition, is the expression k u k − u km−1 u k+1 m − um + m = (a0 − b0 )u km−1 + g (+) (u km−1 )u km−1 − g (−) (u km−1 )u k+1 m . ϕ(h) ϕ(h)

Since this equation is linear in u k+1 m , it can be solved to obtain the result / 6 e(a0 −b0 )h + ϕ(h)g (+) (u km−1 ) k k+1 u m−1 . um = 1 + ϕ(h)g (−) (u km−1 ) Note that the solution at grid-node (x m , t k+1 ) is completely determined by the value at (x m−1 , t k ). The above results have been extended by [28] and [29] to N-dimensional productive-destructive systems.

19 Combustion model Combustion is a one-space dimension can be modelled by the following non-linearreaction-diffusion PDE [2, 30] ∂u ∂2 u + u2 (1 − u), = ∂t ∂x2

(24)

where the solutions satisfy the conditions 0 ≤ u(x, 0) ≤ 1

⇒

0 ≤ u(x, t) ≤ 1, t > 0.

Consider the NSFD discretization  k u k − 2u km + u km−1  k 2 u k+1 k 2 m − um + (u ) + (u ) = m+1 m+1 m−1 ∆t (∆x)2 / 6 6 / k (u km+1 )2 + (u km−1 )2 k+1 u m+1 + u km−1 k+1 um − um . − 2 2 This scheme was obtained by making the following replacements in (24)   u km+1 + u km−1 2 2 2 k 2 k 2 u k+1 u = 2u − u → (u m+1 ) + (u m−1 ) − m , 2 / 6 (u km+1 + (u km−1 )2 k+1 3 u → um . 2

(25)

160 | Ronald E. Mickens and Talitha M. Washington

Observe that the discrete space indices in these expressions have the symmetry that they all are invariant under the interchange (m + 1) ↔ (m − 1). This is the discrete correspondence of the fact that (24) is invariant under x → (−x) [3, 20, 22]. Solving (25) for u k+1 m gives u k+1 m =

R(u km+1 + u km−1 ) + ∆t[(u km+1 )2 + (u km−1 )2 ] + (1 − 2R)u km

 , k k k 2 + uk 2] [u 1 + ∆t + (u ) + (u ) m+1 m+1 m−1 m−1 2

where R=

∆t . (∆x)2

Now if u km ≥ 0, then u k+1 m ≥ 0, and certainly if 1 − 2R ≥ 0. For 1 − 2R = 0, it follows that  1 (u km+1 + u km−1 ) + ∆t[(u km+1 )2 + (u km−1 )2 ] 2 k+1  . (26) um = ∆t [u km+1 + (u km+1 )2 + u km−1 + (u km−1 )2 ] 1+ 2 Using the fact that for w real 0≤w≤1

⇒

w2 ≤

w + w2 , 2

it can be shown that 0 ≤ u km ≤ 1 for all m and fixed k, which implies that 0 ≤ u k+1 m ≤ 1. Thus, the NSFD scheme (26) for (24) satisfies the appropriate positivity and boundedness requirements, which hold for the PDE.

20 Coupled interacting population A SEIR model for the spread of disease is given by the following set of coupled differential equations [31] ⎧  ⎪ dS I ⎪ ⎪ S, = B − mS − β ⎪ ⎪ dt N ⎪ ⎪ ⎪  ⎪ ⎪ dE I ⎪ ⎪ ⎨ S − gE − mE, =β dt N (27) ⎪ dI ⎪ ⎪ = gE − mI − γI, ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ dR ⎪ ⎪ = γI − mR, ⎩ dt

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where (B, m, β, g, γ) are positive parameters, and the total population is P(t) = S(t) + E(t) + I(t) + R(t). Note that [S(0) ≥ 0, E(0) ≥ 0, I(0) ≥ 0, R(0) ≥ 0] ⇒ [S(t) ≥ 0, E(t) ≥ 0, I(t) ≥ 0, R(t) ≥ 0]. Adding the four equations in (27) gives dP(t) = B − mP(t), dt and this differential equation has the exact-difference scheme [1] P k+1 − P k = B − mP k , ϕ

ϕ(h, m) =

1 − e−mh . m

(28)

A NSFD scheme that preserves the positivity restriction and also satisfies (28) is [9] ⎧  ⎪ ⎪ ⎪ S k+1 − S k = B − mS k − β I k S k+1 , ⎪ ⎪ ϕ Nk ⎪ ⎪ ⎪  ⎪ ⎪ E − E I ⎪ k ⎪ = β k S k+1 − gE k+1 − mE k , ⎨ k+1 ϕ Nk (29) ⎪ − I I ⎪ k+1 k ⎪ = gE k+1 − mI k − γI k+1 , ⎪ ⎪ ϕ ⎪ ⎪ ⎪ ⎪ ⎪ R k+1 − R k ⎪ ⎪ = γI k+1 − mR k . ⎩ ϕ In the above relations, t → t k = hk, h = ∆t and for each dependent variable, v(t) → v k . Solving (29) for (S k+1 , E k+1 , I k+1 , R k+1 ) gives ⎧ ⎪ ϕB + (e−mh )S k ⎪ ⎪  S k+1 = , ⎪ ⎪ Ik ⎪ ⎪ 1 + (βϕ) ⎪ ⎪ Nk  ⎪ ⎪ ⎪ ⎪ I ⎪ −mh ⎪ (e )E k + (βϕ) k S k+1 ⎪ ⎪ N ⎪ k ⎨E , k+1 = 1 + gϕ (30) ⎪ ⎪ ⎪ −mh ⎪ (e )I k + (gϕ)E k+1 ⎪ ⎪ I k+1 = , ⎪ ⎪ 1 + γϕ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ R k+1 = (γϕ)I k+1 + (e−mh )R k , ⎪ ⎪ ⎪ ⎪ −mh ⎩P )P k + ϕB. k+1 = (e Given (S k , E k , I k , R k ), then (S k+1 , E k+1 , I k+1 , R k+1 ) is calculated by the evaluation of the equations in (30) in the order presented in this set of expressions.

162 | Ronald E. Mickens and Talitha M. Washington

21 Black–Scholes equation The Black–Scholes equation is one of the most important mathematical expressions appearing in the modern theory and applications of quantitative finance [32]. Its mathematical structure can take one or another of the forms [32, 33] ∂u ∂2 u ∂u = Dx2 2 + rx − ru ∂t ∂x ∂x ∂u ∂2 u ∂u = D 2 + (r − D) − ru, ∂t ∂x ∂x

(31) (32)

where u = u(x, t), and (D, r) are positive parameters. The construction of the NSFD schemes proceeds by first obtaining the exactdifference schemes for the subequations given by the right-hand sides of (31) and (32). Next the exact-difference is determined for the subequation ∂u = −ru, ∂t where x is taken as fixed in value. The final result is obtained by matching the two “ru” terms in the exact-difference for the two subequations. Carrying out these steps, the following NSFD scheme is obtained for the Black–Scholes equation as formulated in (32) 2 1 1 2 k −G1 r −r 2 k u k+1 m − um ∆ um + ∆u km − ru k+1 (33) = m , G2 G2 ϕ(∆t, r) where e r∆t − 1 , r h −β G1 = 2 − (e + e ), ϕ(∆t, r) =

rh , D k k k ∆u m = u m+1 − u m , β=

G2 = 1 + e(h−β) − (e h − e−β ) h = ∆x, ∆2 u km = u km+2 − 2u km+1 + u km .

Since (33) is linear in u k+1 m , it can be simplified to the expression / 6 1 − e−r∆t k k+1 −r∆t k )u m − u m+2 − (2 + G1 )u km+1 + (1 + G1 )u km . u m = (e G2 The NSFD discretization for the form of the Black–Scholes equation given in (31) is derived and listed in ref. [33].

Use of EDS to Construct NSFD Discretizations

| 163

22 Summary This chapter has presented a concise summary of how to construct (at least some) exact-difference schemes for both ordinary and partial differential equations, the general background features of the NSFD methodology, and the application of their procedures to the determination of NSFD schemes for a range of phenomena modelled by differential equations. However, a great effort is still required to fully understand the mathematical basis of the NSFD way of constructing discretizations. Further, there are a number of unresolved issues remaining, and these include the construction of denominator functions for oscillating systems, the difficulties of cross-diffusion for coupled PDEs, and how to extend these methods to multi-space-dimensional systems. Finally, one point is quite clear: better or enhanced finite-difference schemes, such as provided by the NSFD methodology, must be based on considering simultaneously all the terms in a differential equation. The standard procedure of discretizing the individual terms and then adding them together to construct a finite- difference representation is generally not valid [1, 7, 8].

Bibliography [1]

R. E. Mickens. Nonstandard finite-difference models of differential equations. World Scientific, Singapore, 1994. [2] R. E. Mickens. Applications of nonstandard finite-difference schemes. World Scientific, Singapore, 2000. [3] R. E. Mickens. Nsfd schemes for reaction-diffusion equations. Numer. Methods Partial Differ. Equ., 15:201–204, 1999. [4] R. E. Mickens. The role of positivity in the construction of nsfd schemes for pde’s. in d. schultz, et al., (editors). Proc. Int. Conf. Sci. Comput. Math. Model., May 24–27:294–307, 2000. [5] R. Anguelov and J. M.-S. Lubuma. Contributions to the mathematics of the nonstandard finite-difference method and applications. Numer. Methods Partial Differ. Equ., 17:518–543, 2001. [6] R. E. Mickens. Advances in the application of nonstandard finite-difference schemes. World Scientific, Singapore, 2005. [7] R. E. Mickens. Dynamic consistencey: A fundamental principle for constructing nsfd schemes for differential equations. J. Differ. Equ. Appl., 11:645–653, 2005. [8] R. E. Mickens. Calculation of denominator functions for nonstandard finite- difference schemes for differential equations satisfying a positivity condition. Numer. Methods Partial Differ. Equ., 23:672–691, 2007. [9] R. E. Mickens and T. M. Washington. Nsfd discretizations of interacting population models satisfying conservation laws. Comput. Math. Appl., 66:2307–2316, 2013. [10] Z. Zwillinger. Handbook of differential equations. Academic Press, 1989. [11] F. B. Hildebrand. Finite-difference equations and simulations. Prentice-Hall, Englewoods Cliffs, NJ, 1968. [12] D. Potter. Computational physics. Wiley, New York, 1973.

164 | Ronald E. Mickens and Talitha M. Washington

[13] R. E. Mickens, K. Oyedeji, and S. Rucker. Exact finite-difference scheme for second-order, linear odes having constant coefficients. J. Sound Vib., 287:1052–1056, 2005. [14] P. F. Byrd and M. D. Friedman. Handbook of elliptic integrals for engineers and physicists. Springer-Verlag, Berlin, 1954. [15] I. Kovacic and M. J. Brennan (editors). The duffing equation: Nonlinear oscillators and their behaviours. Wiley, New York, 2011. [16] A. Lickler, J. Moser, J. Chaste, M. Zdrojek, I. Wilson-Rae, and A. Bachtold. Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene. Nat. Nano-technol., 6:339–342, 2011. [17] R. E. Mickens and T. M. Washington. A note on exact finite-difference schemes for the differential equations satisfied by the Jacobi cosine and sine functions. J. Differ. Equ. Appl., 19: 2013. [18] G. D. Smith. Numerical solution of partial differential equations: Finite- difference methods, 2nd edition. Clarendon Press, Oxford, 1978. [19] H. J. Stetter. Analysis of discretization methods for ordinary differential equations. Springer-Verlag, Berlin, 1973. [20] R. E. Mickens. Nonstandard finite-difference schemes for differential equations. J. Differ. Equ. Appl., 8:823–847, 2002. [21] R. Buckmire. Application of a mickens finite-difference scheme to the cylindrical bratu-gelfand problem. Numer. Methods Partial Differ. Equ., 20:327–337, 2004. [22] R. E. Mickens. Properties of finite-difference models of nonlinear conservative oscillators. J. Sound Vib., 124:194–198, 1988. [23] R. E. Mickens. Construction of finite-difference schemes for coupled nonlinear oscillators derived from a discrete energy function. J. Diff. Equ. Appl., 2:185–193, 1996. [24] R. E. Mickens and I. Ramadhani. Finite-difference scheme for the numerical solution of the schrödinger equation. Phys. Rev. A, 45:2074–2075, 1992. [25] R. Chen, Z. Xu, and L. Sun. Finite-difference scheme to solve schrödinger equations. Phys. Rev. E, 47:3799–3802, 1993. [26] M. B. Allen III, I. Herrera, and G. F. Pinder. Numerical modeling in science and engineering. Wiley-Interscience, New York, 1988. [27] R. E. Mickens. Analysis of a new finite-difference scheme for the linear advection-diffusion equation. J. Sound Vib., 146:342–344, 1991. [28] D. T. Kimitrov and H. V. Kojouharov. Dynamically consistent numerical methods for productive-destructive systems. J. Differ. Equ. Appl., 17:1721–1736, 2011. [29] D. T. Wood and D. T. Dimitrov. A nsfd method for n-dimensional prductive-destructive system. J. Differ. Equ. Appl., 21:240–254, 2015. [30] R. E. O’Malley, Jr. Thinking about ordinary differential equations. Cambridge University Press, New York, 1997. [31] J. D. Murray. Mathematical biology. Springer-Verlag, Berlin, 1989. [32] P. Wilmott, S. Howison, and J. Dewynne. The mathematics of financial derivatives. Cambridge University Press, New York, 2002. [33] R. E. Mickens, J. Munyakazi, and T. M. Washington. A note on the exact discretization for a Cauchy–Euler equation: application to the Black–Scholes equation. J. Differ. Equ. Appl., 21:547–552, 2015.

Ivan Gavrilyuk, Myroslav Kutniv, and Volodymyr Makarov

Exact and truncated difference schemes for boundary-value problem Everything should be made as simple as possible, but not simpler. A. Einstein

1 Introduction Consider the non-linear BVP u = f (x, u),

x ∈ (0, 1),

u(0) = μ1 ,

u(1) = μ2 .

(1)

In order to numerically solve problem (1) on the interval [0, 1], we introduce a uniform grid ω h = {x j ∈ [0, 1], j = 0, 1, . . . , N, h = 1/N } and replace u (x j ) by the second-order difference derivative u x¯ x,j =

u j+1 − 2u j + u j−1 h2

given on a three-point stencil (x j−1 , x j , x j+1 ). Then we obtain the classical three-point difference scheme of the second order of accuracy (provided the exact solution is smooth enough) y x¯ x,j = f (x j , y j ),

j = 1, 2, . . . , N − 1,

y0 = μ1 ,

y N = μ2 ,

(2)

where y j ≈ u(x j ). The following question arises: How one can construct difference schemes of higher accuracy order? An evident answer is: one can enlarge the stencil. For example, a five-point difference scheme given on a uniform grid of the form −y j+2 + 16y j+1 − 30y j + 16y j−1 − y j−2 = f (x j , y j ), 12h2 j = 1, 2, . . . , N − 1, y0 = μ1 , y N = μ2

(3)

possesses the fourth order of accuracy on solutions smooth enough. The practical disadvantage of such approach is the fact that the complexity of the scheme and the computational costs increase. The second approach is based on the following arguments. We note that the expansion of the second-order three-point difference derivative in a Taylor series in

166 | Ivan Gavrilyuk, Myroslav Kutniv, and Volodymyr Makarov

a vicinity of the point x j has the form u x¯ x,j , = uj +

h2 (4) h4 (6) u + u + O(h6 ). 12 j 360 j

(4)

Then, by differentiating twice the differential equation (1), we obtain the equality u(4) = f  (x, u). This yields the difference equation y x¯ x,j = f (x j , y j ) +

h2  f (x j , y j ), 12

j = 1, 2, . . . , N − 1.

Replacing the second derivative f  (x j , y j ) by the second-order difference derivative f  (x j , y j ) ≈

1 [f (x j+1 , y j+1 ) − 2f (x j , y j ) + f (x j−1 , y j−1 )], h2

we obtain the well-known Numerov’s scheme y x¯ x,j =

 1  f (x j+1 , y j+1 ) + 10f (x j , y j ) + f (x j−1 , y j−1 ) , 12 j = 1, 2, . . . , N − 1, y0 = μ1 , y N = μ2 ,

(5)

which has the fourth order of accuracy. Since the difference scheme for the non-linear BVP is a system of N non-linear algebraic equations, its solution is sought with the use of iteration methods such as, the Newton method, which requires one to solve a system of linear algebraic equations on each iteration. This system has a pentadiagonal matrix for the difference scheme (3) and a tridiagonal matrix for scheme (5). To solve the system of linear equations with a tridiagonal matrix by elimination method required 8N + 1 operations (see, e.g., [1, p. 64]), whereas the solution of the system with pentadiagonal matrix required 19N − 10 operations (see, e.g., [1, p. 92]). Hence, the three-point difference scheme (5) requires a significantly smaller number of operations than scheme (3). If a difference scheme approximating the ODEs of the order k, it is uses only k + 1 values of the grid function, i.e., it is k + 1-point, we call it compact (see, e.g., [2]). A solution of compact difference schemes can be found by using the minimum number of operations. In addition, it was shown in ref. [2] that a difference scheme is stable if it is compact. In order to attain a high accuracy, the grid step h must be sufficiently small, which leads to a large size of the matrix. The application of schemes of high-order accuracy allows one to decrease the matrix size. It is impossible to extend the second approach for the construction of three-point difference schemes with an accuracy order higher than the fourth one. Indeed, if we replace the derivative u (6) in series (4) by f (4) , there exists no three-point difference approximation f (4) in terms of values of the right-hand side of the differential equation at points of the grid. Since the solution of the boundary-value problem can vary more rapidly or slowly on different parts of the interval [0, 1], it is important to be able to construct

Exact and truncated FDS for BVP

| 167

 ˆ¯ h = x j ∈ (0, 1), j = 0, 1, . . . , N, h j = x j − difference schemes on a non-uniform grid ω  x j−1 > 0, h1 + h2 + . . . + h N . The classical three-point difference scheme for problem (1) on a non-uniform grid takes the form y x¯ xˆ ,j = f (x j , y j ),

j = 1, 2, . . . , N − 1,

y0 = μ1 ,

y N = μ2 ,

where / 6 1 y j+1 − y j y j − y j−1 − , y x¯ xˆ ,j = j h j+1 hj

j =

h j + h j+1 . 2

However, it is known that this scheme has only the first order of accuracy. Thus, the construction of difference schemes of high-order accuracy on non-uniform grids is a highly complicated problem. In this connection, we encounter the question on methods of construction of compact difference schemes of high-order accuracy on any non-uniform grid. One can easily see that due to expansion (4), the solution of the three- point difference scheme (2) (3DS) coincides with the exact solution (ES) of the difference equation on the grid provided that the exact solution belongs to the class of polynomials of degree less or equal to 3. We say in this case that the difference scheme is exact on this class of solutions (EDSoC) of the differential problem. There exist also exact-difference schemes on classes of solutions different from polynomials. For example, it is well known that the eigenvalue problem u + λu = 0, u(0) = 0, u(1) = 0

(6)

possesses the exact solution λ = λ n = n2 π 2 , u = u n = sin nπx, n = 1, 2, . . .. The corresponding difference eigenvalue problem on an equidistant grid y x¯ x + λ h y(x) = 0, x ∈ ω h = {x i = ih, i = 1, 2, . . . , N − 1} y(0) = 0, y(1) = 0 possesses the solution λ h,n = the difference scheme

4 h2

sin2

nπh 2 ,

(7)

y n (x) = sin nπx, x ∈ ω h . Thus, the solution of

nπh 4 sin2 y(x) = 0, x ∈ ω h = {x i = ih, i = 1, 2, . . . , N − 1} 2 2 h y(0) = 0, y(1) = 0

y x¯ x +

(8)

coincides on the grid with the exact solution u(x) = sin nπx of the BVP [3, p. 102] u + n2 π 2 u(x) = 0, u(0) = 0, u(1) = 0.

(9)

168 | Ivan Gavrilyuk, Myroslav Kutniv, and Volodymyr Makarov

Since the eigenvalues of the difference scheme with a fixed number do not coincide with the exact ones but approximate these only, this difference scheme is called the 3DS with explicit spectrum. It is possible for some differential eigenvalue problems on some grids to construct discrete approximations with exact eigenvalues and exact eigenfunctions, which are called the best grid schemes with exact spectrum. Such schemes were proposed for the first time in ref. [4] and then applied in refs. [5, 6] and can be considered as forerunner of modern spectral methods. Contrary to the usual difference schemes, they are not local. These simple examples suggest the question whether for a given boundary problem for an ODE, there exists a difference scheme (EDS) on an arbitrary grid with the solution that coincides with the exact one independent of the function classes. Below we will show that the answer is positive and we will construct such schemes. These EDs are implicit in the sense that their coefficients are some functionals from the input data of the differential problem and can be calculated explicitly for some particular cases only. But it is possible to give an algorithm (through solutions of certain Cauchy problems on small intervals), which computes the corresponding input data of the difference scheme, which we will call the truncated difference scheme (TDS), through the input data of the differential problem within an arbitrary given accuracy. We prove that the solution of the truncated difference scheme approximates the exact solution of the differential problem with the accuracy defined by the user. EDS for linear second-order ODEs were proposed for the first time in ref. [7] where the algorithm for the coefficients of TDS was based on some series expansions. EDS and TDS for non-linear ODEs were proposed for the first time in ref. [8]. There was proposed an another and more efficient algorithm for the coefficients of TDS based on the solvers of some initial value problems (IVP) on small intervals of the length proportional to the mesh grid step. Further we will present EDS and associated TDS for arbitrary ODEs.

2 Three-point difference schemes of high-order accuracy for linear boundary-value problems For the first time, an EDS for linear boundary-value problem(BVP) was proposed by A. N. Tikhonov and A. A. Samarskii in the 1960s. They together with their disciples, developed the theory of exact three-point difference scheme (E3DS) and truncated three-point difference schemes (T3DS) of any (defined by user) order of accuracy (see [7, 9–15]).

Exact and truncated FDS for BVP | 169

Consider the BVP L(k,q) u ≡

 du d k(x) dx dx

− q(x)u = −f (x),

u(0) = μ1 ,

x ∈ (0, 1),

(10)

u(1) = μ2

under the conditions 0 < c1 ≤ k(x),

q(x) ≤ c2 ,

k(x), q(x), f (x) ∈ Q(0) [0, 1],

(11)

where Q(0) [0, 1] is the class of piecewise continuous functions with a finite number of discontinuity points of the first kind. For problem (10), (11), an E3DS on a uniform grid was constructed in ref. [15]. Moreover, the algorithmic realization of an E3DS in terms of T3DS of the rank n was developed and substantiated. These results were generalized in ref. [7] for a non-uniform grid and boundary conditions of the third kind and for the input data smooth enough. The truncated schemes constructed in refs. [7, 13, 15] allow one to attain any order of accuracy for arbitrary piecewise continuous functions k(x), q(x), and f (x). In ref. [16] E3DS and truncated 3DS were proposed and justified for the third BVP for the second-order ODE under minimal smoothness requirements and for the generalized solution. E3DS and T3DS for one-dimensional variational inequalities were proposed and substantiated in refs. [17, 18]. However, the practical application of such schemes in the case of variable coefficients requires calculation of multiple integrals over each subinterval of the grid and some series, which is a shortcoming. ˆ h so that the discontinuity points of the We introduce a non-uniform grid ω ˆ h . We denote the set of all functions k(x), q(x), f (x) coincide with nodes of the grid ω discontinuity points by ρ and assume that the number of grid nodes N is such that ˆ h . At the discontinuity points of coefficients, the solution of problem (10), (11) ρ := ω satisfies the continuity condition du du u(x i − 0) = u(x i + 0), k(x) = k(x) ∀x i ∈ ρ. dx x=x i −0 dx x=x i +0 Next, we define stencil functions v jα (x), α = 1, 2, as the solutions of the IVP (see ref. [7]) L(k,q) v jα (x) = 0, v jα (x j+(−1)α ) = 0,

x j−2+α < x < x j−1+α , dv jα k(x) = (−1)α+1 , dx

(12)

x=x j+(−1)α

α = 1, 2,

j = 1, 2, . . . , N − 1,

which must satisfy the conditions of continuity of the solution v jα (x) and the flow k(x)dv jα (x)/dx.

170 | Ivan Gavrilyuk, Myroslav Kutniv, and Volodymyr Makarov

Then the E3DS (see ref. [7]) has the following form (au x¯ )xˆ − du = −φ(x),

ˆ h, x∈ω

u(0) = μ1 ,

u(1) = μ2 ,

(13)

where u i − u i−1 u − ui 1 , u xˆ ,i = i+1 , i = (h j + h j+1 ), hi i 2 / 6−1 1 j v (x ) , d(x j ) = Tˆ x j (q), φ(x j ) = Tˆ x j (f ), a(x j ) = hj 1 j u x¯ ,i =

−1  j x



ˆ xj

T (w) =

j v1j (x j )

v1j (ξ )w(ξ )dξ

 +

−1 j+1 v2j (ξ )w(ξ )dξ . x

(14)

j v2j (x j )

x j−1

xj

In refs. [12, 14] it was shown that its coefficients a(x) and d(x) and the right-hand side ˆ h can be written in terms of the solutions φ(x) at an arbitrary node x j of the grid ω of four auxiliary IVPs. As follows from (14), the coefficients a(x) and d(x) can be represented as /

1 j v (x ) a(x j ) = hj 1 j

6−1 ,

d(x j ) = −1 j

2  [v jα (x j )]−1 [(−1)α+1 m jα (x j ) − 1],

(15)

α=1

where m jα (x) = k(x)

dv jα (x) , dx

α = 1, 2.

In order to obtain a representation for φ(x), we introduce two new auxiliary functions w jα (x), α = 1, 2 as the solutions of the following two IVPs: L(k,q) w jα (x) = −f (x), dw jα j w α (x j+(−1)α ) = dx

x j−2+α < x < x j−1+α , (16) = 0,

α = 1, 2.

x=x j+(−1)α

Now, in terms of the functions w jα (x), α = 1, 2, we can write the right-hand side φ(x) of the E3DS: 6 / 2  w jα (x j ) j −1 α j , (17) (−1) l α (x j ) − m α (x j ) j φ(x j ) = j v α (x j ) α=1 where l jα (x) = k(x)

dw jα (x) , dx

α = 1, 2.

Exact and truncated FDS for BVP

| 171

Formulas (15) and (17) show that, for the determination of the coefficients a(x j ), d(x j ) ˆ h , it is necessary to solve four IVPs and the right-hand side φ(x j ) of the E3DS ∀x j ∈ ω (12), (16) with smooth coefficients: for α = 1 on the interval [x j−1 , x j ] (forward) and for α = 2 on the interval [x j , x j+1 ] (backward). These IVPs can be approximately solved by executing only one step with an arbitrary one-step method (the Taylor series or a Runge–Kutta method) of the order of accuracy n¯ = 2(n + 1)/2, where · denotes the entire part of the argument in these brackets. The obtained approximations, as distinct from the exact values, will be distinguished by the index n: v(n)j α (x j ), ¯ ¯ (x j ), w(αn)j (x j ), l(n)j m(αn)j α (x j ). Instead of the E3DS (13), (15), (17), we now consider a 3DS of the form (n) (n) = −φ(n) (x), (a(n) y(n) x¯ )xˆ − d y

y(n) (0) = μ1 ,

ˆ h, x∈ω

(18)

y(n) (1) = μ2 ,

whose coefficients are determined by the formulas /

a

(n)

1 (n)j (x j ) = v (x j ) hj 1

d(n) (x j ) = −1 j

6−1 ,

2  ¯ (n)j −1 α (−1)α+1 [v(n)j α (x j )] [m α (x j ) + (−1) ], α=1

⎡ ⎤ 2 ¯ (n)j  w (x ) j α ¯ (n)j ( n)j ⎦ (−1)α ⎣l α (x j ) − m α (x j ) (n)j φ(n) (x j ) = −1 j v α (x j ) α=1

(19)

and can be really computed. The next statement characterizes the order of accuracy of this scheme (see [14]). Theorem 2.1. Let the conditions k(x) ∈ Q n+1 [0, 1], q(x), f (x) ∈ Q n [0, 1] be satisfied and let the homogeneous BVP (10), (11) have the trivial solution. Then the error of scheme (18), (19) for sufficiently small h satisfies the estimate ⎧ ⎫ # # #* ⎨# # # ⎬ # # (n) n du # # # # # dy # (n) = max #y(n) − u# , #k ≤ M h , −k # #y − u# dx dx # ⎩ ⎭ ˆh ˆh # 1,∞, ω 0,∞, ω ˆh 0,∞, ω

where y 0,∞, ωˆ h = max |y(ξ )|, ˆh ξ ∈ω

⎤−1 2 2    dy(n) (x j ) ¯ ¯ (n)j ⎦ k(x j ) m(αn)j (x j )y(n) (x j+(−1)α ) = ⎣ (−1)α v(n)j α (x j )m 3−α (x j ) dx ⎡

α=1

α=1



(

¯ ¯ ¯ ¯ (n)j + (−1)α+1 m(1n)j (x j )m(2n)j (x j )w(αn)j (x j ) + m(αn)j (x j )v(n)j 3−α (x j )l 3−α (x j )

,

172 | Ivan Gavrilyuk, Myroslav Kutniv, and Volodymyr Makarov Q p [0, 1] is the class of functions with piecewise continuous derivatives up to order p with a finite number of discontinuity points of first kind, and the constant M does not depend on h . In ref. [9], an E3DS and its algorithmic realization in terms of a 3DS of any order of accuracy were given for systems of linear second-order ODEs  du d K(x) − Q(x)u(x) = −f(x), x ∈ (0, 1) (20) L(K,Q) u ≡ dx dx with boundary Dirichlet conditions u(0) = u(1) = 0.

(21)

The matrices K(x) = [k ij (x)]di,j=1 and Q(x) = [q ij (x)]di,j=1 and the vector f(x) = {f i (x)}di=1 satisfy the conditions C1 v2 ≤ (K(x)v, v), k ij (x) ∈ Q n+1 [0, 1],

C 1 > 0 ∀v ∈ R d

q ij (x) ∈ Q n [0, 1],

∀x ∈ [0, 1],

(22)

f i (x) ∈ Q n [0, 1],

(23)

where (u, v) is inner product in Rd . At the discontinuity points x i of the coefficients, the solution of problem (20), (21) satisfies the conditions du du u(x i − 0) = u(x i + 0), K(x) = K(x) . dx x=x i −0 dx x=x i +0 Let us introduce the matrices and vectors V αj (x), V¯ αj (x), Wjα (x), α = 1, 2 as the solutions of the following problems: L(K,Q) V αj (x) = 0, V αj (x j+(−1)α ) = 0,

x ∈ (x j−2+α , x j−1+α ), dV αj K(x) = (−1)α+1 I, dx x=x j+(−1)α

/

6 d d V¯ αj (x) K(x) − V¯ αj (x)Q(x) = 0, dx dx ¯ αj d V j V¯ α (x j+(−1)α ) = 0, K(x) dx

x ∈ (x j−2+α , x j−1+α ), (24) = (−1)α+1 I,

x=x j+(−1)α

L(K,Q) Wjα (x) = −f(x), Wjα (x j+(−1)α ) = 0,

x ∈ (x j−2+α , x j−1+α ), dWjα K(x) = 0, α = 1, 2, dx x=x j+(−1)α

where I is the identity matrix.

(25)

Exact and truncated FDS for BVP

|

173

Then the E3DS (see ref. [9]) has the following form j j −1 j (A (x j )ux,j − B (x j )ux¯ ,j ) − D j u(x j ) = −Φ j ,

ˆ h, xj ∈ ω

(26)

u0 = uN = 0, where coefficients A j (x j ), B j (x j ), D j , and Φ j can be represented by the solutions of the IVPs (24), (25) in the form  −1 ¯j A j (x j ) = h−1 , j+1 V 2 (x j )

 −1 ¯j B j (x j ) = h−1 , j V 1 (x j )

⎡ ⎤ −1 ¯ j d V 1¯j 2 V (x ) ⎣−I − K(x j + 0)⎦ Dj = j 2 j dx x=x j ⎡ ⎤ −1 d V¯ j 1¯j 1 V (x ) ⎣ K(x j − 0) − I ⎦, + j 1 j dx Φj =

2 

x=x j

' (−1)

Ljα (x j ) −

α



V¯ αj (x j )

−1

( j j ¯ M α (x j )Wα (x j ) ,

α=1

M αj (x) = K(x)

dV αj (x) , dx

¯j ¯ αj (x) = d V α (x) K(x), M dx

Ljα (x) = K(x)

dWjα . dx

Starting from the E3DS, we can now introduce in analogy to the scalar case the truncated 3DS (n) (n) (n)j (n)j (x j )y(n) (x j )y(n) −1 j (A x,j − B x¯ ,j ) − D j yj = −Φ j ,

ˆ h, xj ∈ ω

y0(n) = y(n) N = 0, where  −1 ¯ (n)j A(n)j (x j ) = h−1 , j+1 V 2 (x j ) D(n) j =

2 −1   1  ¯ (n)j ¯ (n)j V α (x j ) −I − (−1)α M α (x j ) , j α=1

Φ(n) j

=

 −1 ¯ (n)j B(n)j (x j ) = h−1 , j V 1 (x j )

2 

(−1)

' α

L(n)j α (x j ) −



−1

V¯ α(n)j (x j )

( ¯ ¯ (n)j (n)j ¯ M α (x j )Wα (x j ) ,

α=1 ¯ ¯ ¯ (αn)j (x), W(αn)j (x), L(n)j and V¯ α(n)j (x), M α (x), α = 1, 2, are approximations of the solutions of ¯ the IVPs (24), (25) obtained by an arbitrary one-step method of the order of accuracy n. An a priori estimate for the truncated T3DS of rank n gives the following theorem (see ref. [9]).

174 | Ivan Gavrilyuk, Myroslav Kutniv, and Volodymyr Makarov

Theorem 2.2. Under the assumptions (22) and (23) the following error estimate holds: ⎧ ⎫ # # ⎨# # ⎬ #* # # (n) # n dy du # # # # # # (n) = max #u − y(n) # , #K ≤ M h , −K # #u − y # dx # ⎩ ⎭ ˆh ˆ h # dx 1,∞, ω 0,∞, ω ˆh 0,∞, ω

where y0,∞, ωˆ h = max |y(ξ )|,

K(x j )

dy(n) j dx

⎧ 2 ⎨

=

⎩

ˆh ξ ∈ω

(−1)

α+1

α=1

⎫−1 ⎬

¯ M (αn)j (x j )V α(n)j (x j )

⎭

' ( 2  −1  ¯ ¯ (n)j (n)j (n)j (n)j α × (−1) Lα (x j ) − M α (x j ) V α (x j ) Wα (x j ) α=1 + L(n)j 2 (x j ),

and h is small enough.

3 Exact and truncated difference schemes for non-linear boundary-value problems For the first time, an approach to the construction of an E3DS on a uniform grid for the non-linear BVP 1 2 du d k(x) = −f (x, u), x ∈ (0, 1), u(0) = μ1 , u(1) = μ2 , dx dx was announced in ref. [8], where the algorithmic realization of these schemes in terms of a T3 3T3DS was proposed. Then these results were developed and completely substantiated in refs. [19, 20] and, for monotone ODEs, in [21]. In refs. [22, 23], the results obtained for scalar non-linear ODEs were generalized to the case of systems of second-order non-linear ODEs. A new algorithmic realization of the exact E3DS on a non-uniform grid in terms of a 3DS of the accuracy order n¯ was proposed in [24]. In ref. [25], the theory of the E3DS for non-linear ODEs is developed.

Exact and truncated FDS for BVP |

175

4 Two-point difference schemes for systems of first-order ODEs Exact two-point difference schemes (E2DS) and two-point difference schemes (2DS) of any order of accuracy for systems of ordinary first-order differential equations with non-separable boundary conditions: u (x) + A(x)u = f(x, u), A(x), B0 , B1 , ∈ R

d×d

x ∈ (0, 1),

,

B0 u(0) + B1 u(1) = d,

rank[B0 , B1 ] = d,

(27) d

f(x, u), d, u(x) ∈ R ,

were first constructed and substantiated in refs. [26, 27]. √ In what follows, we denote by u = uT u the Euclidian norm of u ∈ Rd and we will use the subordinate matrix norm generated by this vector norm. For # # vector-functions u(x) ∈ C[0, 1], we define the norm u0,∞,[0,1] = maxx∈[0,1] #u(x)#. Let us make the following assumptions: (PI) The linear homogeneous problem corresponding to (27) possesses only the trivial solution. d  (PII) For the elements of the matrix A(x) = a ij (x) i,j=1 , it holds a ij (x) ∈ C[0, 1], i, j = 1, 2, . . . , d. d  (PIII) The vector-function f(x, u) = f j (x, u) j=1 satisfies the conditions # #     f j (x, u) ∈ C([0, 1] × Ω [0, 1], r(·) ), #f(x, u)# ≤ K for all u ∈Ω [0, 1], r(·) , # #   #f(x, u) − f(x, v)# ≤ Lu − v for all x ∈ [0, 1] and u, v ∈ Ω [0, 1], r(·) ,   r(x) = K exp(c1 x) x + H  exp(c1 ) , where d     Ω [0, 1], r(·) = v(x) = v j (x) j=1 , v j (x) ∈ C[0, 1], j = 1, 2, . . . , d, ( # # # # (0) −1 #v(x) − u (x)# ≤ r(x), x ∈ [0, 1] , H = Q B1 , u(0) (x) = U(x, 0)Q−1 d,

Q = B0 + B1 U(1, 0),

# # c1 = max #A(x)#, x∈[0,1]

U(x, ξ ) ∈ Rd×d is the fundamental matrix of the linear part of the ODE (27). Sufficient conditions under which the problem (27) possesses a unique solution in the sphere Ω([0, 1], r(·)) are given in ref. [26]. The E2DS for problem (27) has the following form uj = Y(x j , uj−1 ),

j = 1, 2, . . . N,

B0 u0 + B1 uN = d,

176 | Ivan Gavrilyuk, Myroslav Kutniv, and Volodymyr Makarov

where Y(x j , uj−1 ) is the solution of the IVP

 dYj (x, uj−1 ) + A(x)Yj (x, uj−1 ) = f x, Yj (x, uj−1 ) , dx Yj (x j−1 , uj−1 ) = uj−1 , j = 1, 2, . . . N.

x ∈ (x j−1 , x j ], (28)

If the initial value problem (28) is solved numerically with the help of any one-step method of the accuracy order n, then, instead of the E2DS, we obtain the truncated 2DS of rank n (n)j (x j , y(n) y(n) j =Y j−1 ),

j = 1, 2, . . . N,

B0 y0(n) + B1 y(n) N

= d,

(29) (30)

where Y(n)j (x j , yj−1 ) is the numerical solution of the IVP (28), which can be determined by some one-step method of the order n Y(n)j (x j , yj−1 ) = yj−1 + h j Φ(x j−1 , yj−1 , h j ).

(31)

For example in the case of an explicit Runge–Kutta method, we have Φ(x j−1 , yj−1 , h j ) = b1 k1 + b2 k2 + · · · + b s ks ,   k1 = f x j−1 , yj−1 − A(x j−1 )yj−1 ,     k2 = f x j−1 + c2 h j , yj−1 + h j a21 k1 − A(x j−1 + c2 h j ) yj−1 + h j a21 k1 , .. .

  ks = f x j−1 + c s h j , yj−1 + h j a s1 k1 + a s2 k2 + · · · + a s,s−1 ks−1

  − A(x j−1 + c s h j ) yj−1 + h j a s1 k1 + a s2 k2 + · · · + a s,s−1 ks−1 , with the corresponding real parameters c i , a ij , i = 2, 3, . . . , s, j = 1, 2, . . . , s − 1, and b i , i = 1, 2, . . . , s. Let the method (31) be of the accuracy order n. If we assume that the increment function Φ(x, u, h) is sufficiently smooth, the entries a ps (x) of the matrix A(x) belong to Cn [0, 1] and there exists a real number ∆ > 0 such that f p (x, u) ∈ Ck,n−k ([0, 1] × Ω([0, 1], r(·) + ∆)), with k = 0, 1, . . . , n − 1 and p = 1, 2, . . . , d, then + O(h n+2 ). Yj (x j , yj−1 ) = Y(n)j (x j , yj−1 ) + ψ j (x j , yj−1 )h n+1 j j It was proven in ref. [27] that the difference scheme (29), (30) possesses the order of accuracy n and the a priori estimate # #

 # (n) # (n) h n = max − u y ≤ M #y − u# j j  0,∞, ω h

0≤j≤N

holds, where the constant M does not depend on h = max1≤j≤N h j .

Exact and truncated FDS for BVP

| 177

Newton’s method applied to the difference equations (29), (30) has the form (n,k−1) (n)j (n,k) − ∂Y (x j , yj−1 ) y(n,k) j−1 ∂u

y j

= Y(n)j (x j , y(n,k−1) ) − y(n,k−1) , j−1 j B0 y(n,k) + B1 y(n,k) = 0, 0 N y(n,k) = y(n,k−1) + y(n,k) , j j j

j = 1, 2, . . . , N,

(32)

k = 1, 2, . . . , j = 0, 1, . . . , N,

k = 1, 2, . . . ,

where k is the iteration number, ∂Y(n)j (x j , y(n) j−1 ) ∂u

= I + hj

∂Φ(x j−1 , y(n) j−1 , h j ) ⎡

= I + hj ⎣

and

∂f(x j−1 , y(n) j−1 ) ∂u Setting

∂u ∂f(x j−1 , y(n) j−1 ) ∂u

⎤

 − A(x j−1 )⎦ + O h2j ,

is the Jacobian of the vector-function f(x, u) at the point (x j−1 , y(n) j−1 ).

Sj =

∂Y(n)j (x j , y(n,k−1) ) j−1 ∂u

system (32) can be written in the following equivalent form = −B1 φ, [B0 + B1 S]  y(n,k) 0 = y(n,k−1) + y(n,k) , y(n,k) 0 0 0

(33)

with S = S N S N−1 · · · S1 , (n)j

φ j = S j φ j−1 + Y

φ = φN ,

(x j , y(n,k−1) ) − y(n,k−1) , j−1 j

φ0 = 0,

j = 1, 2, . . . , N.

The coefficient matrix of the linear system (33) has the dimension d × d. After the solution of (33) is determined (this requires O(N) arithmetical operations since the dimension d is small in comparison with N), one calculates the solution of the system (32) by  y(n,k) = S j S j−1 · · · S1  y(n,k) + φj , 0 j

y(n,k) = y(n,k−1) + y(n,k) , j j j

j = 1, 2, . . . , N.

178 | Ivan Gavrilyuk, Myroslav Kutniv, and Volodymyr Makarov

Tab. 1. Numerical results for the 2DS with n = 6. EPS −4

10 10−6 10−8

N

NFUN

64 64 128

5712 5712 12432

   (n)  y − u

0,∞,ω h

0.453 · 10−8 0.453 · 10−8 0.267 · 10−11

Example 4.1. Let us consider the periodic BVP u = −0.05u − 0.02u2 sin x + 0.00005 sin x cos2 x − 0.05 cos x − 0.0025 sin x, u(0) = u(2π),



x ∈ (0, 2π),



u (0) = u (2π),

which has the exact solution u(x) = 0, 05 cos x. In Table 1 we give the error of the 2DS (29), (30) on the uniform grid ¯ h = {x j = jh, j = 0, 1, . . . , N, h = 2π/N } ω for n = n¯ = 6 and the number of right-hand sides evaluations of the differential equations (NFUN) for a given tolerance EPS using a posteriori strategy h − h/2 and the Runge–Kutta method of the accuracy order 6 (see, e.g., [28, p. 202]). The results of numerical experiments [27] show that the proposed 2DS are more efficient than the shooting method in the solution of boundary-value problems with large Lipschitz constants, boundary-value problems for stiff systems of ODEs, and problems with small parameter. A new algorithm of numerical solution of the BVP (27) with the use of 2DS with given accuracy and automatic choice of points of a grid was developed in ref. [29].

5 Three-point difference schemes for non-linear second-order ODEs ˆ h constructed in An E3DS and a 3DS of the accuracy order n¯ on non-uniform grid ω ref. [30] for BVP 1 2  du du d k(x) = −f x, u, , x ∈ (0, 1), u(0) = μ1 , u(1) = μ2 , (34) dx dx dx under the following conditions 0 < c1 ≤ k(x) ≤ c2

∀x ∈ [0, 1] ,

k(x) ∈ Q1 [0, 1],

(35)

Exact and truncated FDS for BVP

  f uξ (x) ≡ f x, u, ξ ∈ Q0 [0, 1] ∀u, ξ ∈ R1 ,     f x u, ξ ≡ f x, u, ξ ∈ C(R2 ) ∀x ∈ [0, 1],   1 f x, u, ξ − f0 (x) ≤ c(|u|)[g(x) + |ξ |] ∀x ∈ [0, 1], u, ξ ∈ R ,      f x, u, ξ − f x, v, η (u − v)

2  ≤ c3 |u − v|2 + ξ − η 0 ≤ c3
0 such that ∀ h j j=1

 N f (x, u, ξ ) ∈ ∪ Cn [x j−1 , x j ] × R2 , j=1

: h = max h j ≤ h0 , the 3DS (44) possesses a unique 1≤j≤N

solution with the error estimate ⎡ #* # #2 # # # ¯ # # (n) ¯ = ⎣#y(n) − u# #y − u#

# #2 # # ¯ du # # dy(n) + #k −k # dx dx # ˆh # 0,2, ω

ˆh 1,2, ω

⎤1/2 ⎦

n¯ ≤ M h ,

ˆ¯ h 0,2, ω

where y 20,2, ωˆ h =

 ˆh ξ ∈ω

(ξ )u2 (ξ ),

y 20,2, ωˆ¯ =



h

ˆ¯ h ξ ∈ω

(ξ )u2 (ξ ),

Exact and truncated FDS for BVP |

 ¯ ¯ (n)0 ¯ (n)

 x − y(0n) , y Y 0 2 dy ¯ k(x) + , = Z2(n)0 x0 , y(n) ¯ dx V1(n)1 (x1 ) x=x0

 ¯ ¯ ¯

 y(j n) x j , y(n) − Y1(n)j ¯ dy(n) (n)j ¯ + , j = 1, 2, . . . , N, = Z1 x j , y(n) k(x) ¯ dx V (n)j (x j )

181

¯ (n)

(45)

1

x=x j

and the constant M is independent of h . In other words, this theorem asserts that the truncated T3DS for the BVP under consideration possesses the same accuracy order as the IVP-solver used for the computation of its coefficients and based on E3DS. Moreover, the approximation of ¯ too. flow (45) at the grid nodes has this accuracy order (i.e, n) From a practical point of view to find a solution, 3DS (44) will eventually need to use modified Newton method   ¯ dy(n,m−1) ¯ (n,m−1) ∂f x j , y j , dx 

x=x j ¯ ¯ ¯ ∇y(x¯n,m) + ∇y(j n,m) a(n) ∂u xˆ ,j   ¯ dy(n,m−1) ¯ (n,m−1) ∂f x j , y j , dx x=x j ¯ + ∇y(x¯n,m) ,j ∂ξ



 ¯ ¯ ¯ (n,m−1) ¯ (n) (n,m−1) (n) xj , y − a y x¯ = −φ , j = 1, 2, . . . , N − 1, xˆ ,j

¯ ∇y(0n,m)

= 0,

¯ ¯ ¯ = y(j n,m−1) + ∇y(j n,m) , y(j n,m)

¯ ∇y(Nn,m)

= 0,

j = 0, 1, . . . , N,

m = 1, 2, . . .

For a non-linear ODEs 1 2  du du d k(x) = −f x, u, , dx dx dx

x ∈ (0, 1),

with a boundary conditions of the third kind k(0)

du(0) − β1 u(0) = −μ1 , dx

−k(1)

du(1) − β2 u(1) = −μ2 , dx

the 3DS of order of accuracy n¯ (see refs. [31, 32]) has the following form ¯ (n) ¯ ¯ y x¯ ¯ )xˆ ,j = −φ(n) (x j , y(n) ), (a(n)

j = 1, 2, . . . , N − 1,

 1 (n) ¯ ¯ ¯ ¯ n) a1¯ y(x,0 = −φ(n) − β1 y0(n) (x0 , y(n) ), 0  1 (n) ¯ ¯ ¯ ¯ a N¯ y(x¯n) = −φ(n) − + β2 y(Nn) (x N , y(n) ), ,N N

182 | Ivan Gavrilyuk, Myroslav Kutniv, and Volodymyr Makarov

where / 6 ¯ (x0 , u) − u1 Y2(n)0 1 (n)0 + μ1 , Z (x0 , u) + φ (x0 , u) = ¯ 0 2 V2(n)0 (x0 ) / 6 ¯ (x N , u) − u N−1 Y1(n)N 1 ¯ (n) (n)N + μ2 . φ (x N , u) = −Z1 (x N , u) + ¯ N V (n)N (x N ) ¯ (n)

1

The three-point approximation of flow has the next form dy k(x) dx ¯ (n)

¯ dy(n) k(x) dx





¯ x j , y(n) + (−1)α+1 = Z (n)j α

 ¯ ¯ ¯ x j , y(n) y(j n) − Y α(n)j ¯ V α(n)j (x j )

x=x j

,

α = 1, 2,

x=x0

j = 1, 2, . . . , N − 1, ¯ dy(n) ¯ ¯ (n) = β1 y0 − μ1 , k(x) = −β2 y(Nn) + μ2 . dx x=x N

Similar results are obtained for the non-linear BVP for the system of the second-order differential equations of the kind 1 2  du du d K(x) = −f x, u, , x ∈ (0, 1), u(0) = μ1 , u(1) = μ2 , (46) dx dx dx   where K(x) ∈ Rd×d , f x, u, ξ , μ1 , μ2 ∈ Rd are the given and u(x) ∈ Rd is the unknown vectors. The E3DS for this problem (see ref. [33]) has the next form (Aux¯ )xˆ = −φ(x, u),

ˆh x∈ω

u(0) = μ1 ,

u(1) = μ2 ,

(47)

where  −1 A(x j ) = h j V1j (x j ) , φ(x j , u) = −1 j

2 

 (−1)α Zjα (x j , u)

α=1

+ (−1)

α



−1 

V αj (x j )

Yjα (x j , u) − uj+(−1)α

( .

(48)

The functions Yjα (x, u), Zjα (x, u), α = 1, 2 are the solutions of the following initial value problems dYjα (x, u) = K −1 (x)Zjα (x, u), dx 

dZjα (x, u) = −f x, Yjα (x, u), K −1 (x)Zjα (x, u) , dx

(49) x j−2+α < x < x j−1+α ,

Exact and truncated FDS for BVP

Yjα (x j+(−1)α , u) = uj+(−1)α ,

Zjα (x j+(−1)α , u) =

j = 2 − α, 3 − α, . . . , N + 1 − α,

du K(x) dx x=x

|

183

, j+(−1)α

α = 1, 2,

the matrix functions V αj (x), α = 1, 2 are the solutions of the IVPs dV αj (x) = (−1)α+1 K −1 (x), dx V αj (x j+(−1)α ) = 0,

x j−2+α < x < x j−1+α ,

j = 2 − α, 3 − α, . . . , N + 1 − α,

α = 1, 2.

Solving these IVPs for the coefficients of the E3DS by an IVP-solver of an accuracy ¯ we arrive at the following truncated T3DS of the same accuracy order n: ¯ order n, ¯ (n) ¯ ¯ ˆ h, yx¯ ¯ )xˆ = −φ(n) (x, y(n) ), x ∈ ω (A(n)  −1 ¯ ¯ (x j ) = h j V1(n)j (x j ) , A(n) ¯ φ(n) (x j , u) = −1 j

2 

¯ y(n) (1) = μ2 ,

(50)

 (−1)α Z(n)j α (x j , u)

α=1

+ (−1)

¯ y(n) (0) = μ1 ,

α



¯ V α(n)j (x j )

−1 

¯ Y(αn)j (x j , u) − uj+(−1)α

( .

(51)

Example 5.1. Let us consider the problem d2 u1 = λ2 u1 + u2 + x + (1 − λ2 ) exp(−x), dx2 d2 u2 = −u1 + exp(u2 ) + exp(−λx), dx2 u1 (0) = 2, u2 (0) = 0,

u1 (1) = exp(−λ) + exp(−1), u2 (1) = −1

(52) (53)

with the exact solution u1 (x) = exp(−λx) + exp(−x),

u2 (x) = −x.

Let us note that the Lipschitz’s constant of the right-hand side of (52) in a neighbour√ hood of the exact solution is L = λ4 + 3 > 1. In order to solve problem (52), (53) numerically on the equidistant grid ω h = {x j = jh, j = 0, 1, . . . , N, h = 1/N } we use the difference scheme (50), (51) of the accuracy order n = n¯ = 6. We solve the IVPs (49) by the Runge–Kutta method of the accuracy order 6 [28, p. 202]. For the numerical solution of the difference scheme, we use the Newton method. Tables 2 and 3 contain the numerical results obtained by the 3DS (50), (51) with a given tolerance EPS. Our adaptive algorithm for the BVP (52), (53) with λ = 500 and λ = 1000 uses the ‘h − h/2-strategy’ for the step-size control.

184 | Ivan Gavrilyuk, Myroslav Kutniv, and Volodymyr Makarov

Tab. 2. Numerical results for problem (52), (53) with λ = 500. EPS −4

10 10−6 10−8

N 1000 2000 4000

 *  (6)  y − u

1,2,ω h

0, 4406 · 10−5 0, 1315 · 10−6 0, 1380 · 10−9

Tab. 3. Numerical results for problem (52), (53) with λ = 1000. EPS

N

10−4 10−6 10−8

2000 4000 8000

 *  (6)  y − u

1,2,ω h

0, 4411 · 10−5 0, 1316 · 10−7 0, 2239 · 10−9

6 Three-point difference schemes for non-linear BVPs on the half-axis Let us consider the following non-linear BVP d2 u − m2 u = −f (x, u), x ∈ (0, ∞), dx2 u(0) = μ1 , lim u(x) = 0, x→∞

(54)

 0 is a real constant. For this problem an E3DS was constructed on a finite where m = non-uniform grid in refs. [34, 35]. In addition, a practical realization of the E3DS through a T3DS of the accuracy order n¯ was proposed, and a new adaptive algorithm of numerical solution of problem (54) with a given tolerance was developed. The results of numerical experiments confirm the efficiency of the proposed approach. On the interval [0, ∞), we introduce the non-uniform closed grid

ˆ¯ N = {x j ∈ [0, ∞), j = 0, 1, . . . , N : ω x0 = 0, h j = x j − x j−1 > 0, h1 + · · · + h N = x N }. Let |h| = h max and h min denote the maximum and minimum step size, respectively. We ˆ¯ N satisfy assume that the step- sizes h j and the grid ω c1 ≤

h max ≤ c2 , h min

(55)

Exact and truncated FDS for BVP

|

185

where c1 and c2 are real constants. In order to arrive the maximal convergence order of our difference scheme, we set 1 1 ≤ xN ≤ . h max h min

(56)

Due to (55), (56) we further obtain (see ref. [34]) c h max ≤ √2 , N

h min ≥

√

1

√ ,

c2 N

N

c2

√

≤ h min N ≤ x N ≤ h max N ≤ c2 N.

Thus, we have h max → 0, x N → ∞ as N → ∞. Then, for problem (54), there exists a E3DS (see ref. [34]), which is of the form   (au x¯ )xˆ ,j − d x j u j = −φ(x j , u), j = 1, 2, . . . , N − 1, (57) u0 = μ1 , −a(x N )u x¯ ,N = β2 u N − μ2 (x N , u), where h j + h j+1 , 2     exp mh N − 1 mh j    , j = 1, 2, . . . , N, β2 = m  , a xj = sinh mh j sinh mh N " !       m cosh mh j − 1 cosh mh j+1 − 1    +  , j = 1, 2, . . . , N − 1, d xj = j sinh mh j sinh mh j+1 ⎡ 

    m cosh mh j Y1j x j , u − u j−1  1⎢ j       φ x j , u = ⎣Z2 x j , u − Z1j x j , u + j sinh mh j u x¯ ,j =

u j − u j−1 , hj

u xˆ ,j =

u j+1 − u j , j

j =

⎤

    m cosh mh j+1 Y2j x j , u − u j+1 ⎥   + ⎦, sinh mh j+1

(58)

(59) j = 1, 2, . . . , N − 1,

μ2 (x N , u) = Z2N (x N , u) + mY2N (x N , u) − Z1N (x N , u) 

  m cosh mh N Y1N (x N , u) − u N−1   + , sinh mh N

(60)

Y αj (x, u), Z αj (x, u), j = 2 − α, 3 − α, . . . N + 1 − α, α = 1, 2 are solutions of the IVPs: dY αj (x, u) = Z αj (x, u), dx

 dZ αj (x, u) − m2 Y αj (x, u) = −f x, Y αj (x, u) , x j+α−2 < x < x j+α−1 ,

dx 



 du Y αj x j+(−1)α , u = u x j+(−1)α , Z αj x j+(−1)α , u = , dx x=x α j+(−1)

(61)

186 | Ivan Gavrilyuk, Myroslav Kutniv, and Volodymyr Makarov and Y2j (x, u), Z2j (x, u) is the solution of the BVP dY2N (x, u) = Z2N (x, u), dx

 dZ2N (x, u) − m2 Y2N (x, u) = −f x, Y2N (x, u) , dx Y2N (x N , u) = u(x N ),

x > xN ,

(62)

lim Y2N (x, u) = 0.

x→∞

For the solution of problems (61), we use a one-step method (e.g., Taylor series method or Runge–Kutta method) of the accuracy order n¯ with the corresponding increment function Φ = (Φ1 , Φ2 )T : ¯ (x j , u) = u j+(−1)α Y α(n)j

 + (−1)α+1 h j−1+α Φ1 x j+(−1)α , u j+(−1)α , uj+(−1)α , (−1)α+1 h j−1+α ,

 Z (n)j α (x j , u) = u j+(−1)α

 + (−1)α+1 h j−1+α Φ2 x j+(−1)α , u j+(−1)α , uj+(−1)α , (−1)α+1 h j−1+α . (63)

j Thus, the value Z (n)j α (x j , u) approximates Z α (x j , u) at least with the accuracy order n ¯ (n)j j ¯ and Y α (x j , u) approximates Y α (x j , u) with the order of accuracy n. Now, let us find an approximation for the solution of problem (62). We make the following two assumptions: f (x, u) is analytical in a neighbourhood of the point (∞, 0), and lim f u (x, 0) = 0. We represent the exact solution of problem (62) in the form x→∞

Y2N (x, u) =

∞  A i=1

i xi

+ r(x),

Z2N (x, u) = −

∞  iA i + r (x), x i+1

(64)

i=1

where r(x) ∈ C∞ [x N , ∞) and lim x n r(x) = 0 for all n ∈ N0 . Inserting Y2N (x, u) into (62), x→∞

we obtain ∞ ∞     i(i + 1)A i Ai 2 F x, {A} = − m x i+2 xi i=1 i=1 ⎛ ⎞ ∞  Ai + f ⎝x, + r(x)⎠ + r (x) − m2 r(x) = 0, xi i=1

  where {A} = A1 , A2 , . . . We change the variables by t = 1/x and denote F˜ t, {A} = F(1/t, {A}). Taking into account our assumptions on the function f (x, u), for the coefficients A i , i = 1, 2, . . ., we get the following recurrent system of equations   d k F˜ t, {A} (65) = 0, k = 1, 2, . . . dt k t=0

Exact and truncated FDS for BVP |

187

Note, if the differential equation in (54) is autonomous, then system (65) possesses only the trivial solution. A look at formula (64) suggests to use the following ansatz for the approximate solution of problem (62): A1 A2 A n−1 ¯ , + 2 + · · · + n−1 x x x¯ A 2A (n¯ − 2)A n−2 ¯ ¯ Z2(n−1)N (x, u) = − 21 − 32 − · · · − . ¯ x x x n−1

¯ (x, u) = Y2(n−1)N

(66)

can successively be determined from (65), The unknown coefficients A1 , A2 , . . . , A n−1 ¯ whereby two cases are possible. In the first case, all the coefficients A1 , A2 , . . . , A n−1 ¯ ¯ ¯ are equal to zero. Then Y2(n−1)N (x, u) ≡ 0, Z2(n−1)N (x, u) ≡ 0. In the second case, at least one of the coefficients A1 , A2 , . . . , A n−1 is not equal to zero. ¯ The method just described can be realized with a computer algebra program. Often, however, the coefficients A i are known from the given problem. Moreover, it is also possible to compute the coefficients numerically. For instance, in the paper [36], a numerical algorithm is proposed by which the coefficients can be computed automatically through the input data of the differential problem. Now, instead of the E3DS (57), and (60) we can use the 3DS of the rank n¯ 



  ¯ ¯ ¯ ¯ x j , y(n) , j = 1, 2, . . . N − 1, − d x j y(j n) = −φ(n) ay(x¯n) xˆ ,j (67)

 ¯ ¯ ¯ ¯ (n) (n) ¯ (n) x , = μ1 , −a(x N )y(x¯n) = β y − μ , y y(0n) 2 N 2 ,N N where 1  1 (n)j     ¯  φ(n) Z2 x j , u − Z1(n)j x j , u xj , u = j 

  ¯   x j , u − u j−1 m cosh mh j Y1(n)j   + sinh mh j

  ¯  m cosh(mh j+1 )Y2(n)j x j , u − u j+1 2   . + sinh mh j+1

(68)

If at least one of the coefficients A1 , A2 , . . . , A n−1 is not equal to zero, we have ¯ ¯ μ(2n) (x N , u) =

− (n¯ − 2)A n−2 mA1 mA2 − A1 mA n−1 ¯ ¯ + + ··· + ¯ xN x2N x n−1 N 

¯ m cosh(mh N )Y1(n)N (x N , u) − u N−1 − Z1(n)N (x N , u) + . sinh(mh N )

(69)

188 | Ivan Gavrilyuk, Myroslav Kutniv, and Volodymyr Makarov

Otherwise it holds ¯ (n)N μ(2n) (x N , u) = −Z1 (x N , u) +



¯ m cosh(mh N )Y1(n)N (x N , u) − u N−1 sinh(mh N )

.

(70)

It was shown in ref. [34] that there exists a constant N0 > 0 such that for all N ≥ N0 , the 3DS (67)–(70), (63) possesses the accuracy order n¯ and the following estimate holds ⎧ ⎫ # # #* ⎨# # ⎬ # # # (n) ¯ du # # # ¯ # # (n) # dy ¯ = max #y(n) − u# ,# − #y − u# # + + dx # ⎩ ⎭ ˆ N # dx ˆN 0,∞, ω 1,∞, ω ˆ¯ N 0,∞, ω

n¯ ¯ ≤ M h ≤ M N −n/2 . where

y 0,∞, ωˆ + = max y j , N

1≤j≤N



y 0,∞, ωˆ¯ N = max y j , 0≤j≤N



   ¯ ¯ ¯ m cosh mh1 Y2(n)0 x0 , y(n) − y(0n) dy (x0 ) ¯ (n)0 (n)   x0 , y + , = Z2 dx sinh mh1 

   ¯ ¯ ¯   x j , y(n) − Y1(n)j m cosh mh j y(j n) ¯ n) (

 dy x j ¯   + , = Z1(n)j x j , y(n) dx sinh mh j

¯ (n)



j = 1, 2, . . . , N and the constant M does not depend on h and 1/x N . For the non-linear BVP on a semiaxis d2 u du + m2 = −f (x, u), x ∈ (0, ∞), dx dx2 u(0) = μ1 , lim u(x) = 0

m=  0,

x→∞

(71)

ˆ h in refs. [37, 38]. The E3DS an E3DS was constructed on a finite non-uniform grid ω (see ref. [37]) for problem (71) has the next form  1 b u − a j u x¯ ,j = −φ(x j , u), j j x,j u0 = μ1 ,

j = 1, 2, . . . , N − 1,

(72)

−a N u x¯ ,N = m2 u N − μ2 (x N , u),

where aj =

m2 h j e

m2 h

j

−1

,

bj =

m2 h j+1 1 − e−m

2h

j+1

,

j = 1, 2, . . . , N,

(73)

Exact and truncated FDS for BVP | 189

⎡ φ(x j , u) =

1⎢ j j ⎣Z (x , u) − Z1 (x j , u) + j 2 j

 m2 Y1j (x j , u) − u j−1

⎤ m2 Y2j (x j , u) − u j+1 ⎥ + ⎦, 2 1 − e−m h j+1 μ2 (x N , u) =Z2N (x N , u) − Z1N (x N , u) +

em

2h

j

−1 (74)

j = 1, 2, . . . , N − 1,

 m2 Y1N (x N , u) − u N−1 e m2 h N − 1

(75)

.

The functions Y αj (x j , u), Z αj (x j , u), α = 1, 2 are the solutions of the IVPs

 dZ αj (x, u) + m2 Z αj (x, u) = −f x, Y αj (x, u) , dx x j−2+α < x < x j−1+α , du j j Y α (x j+(−1)α , u) = u j+(−1)α , Z α (x j+(−1)α , u) = , dx

dY αj (x, u) = Z αj (x, u), dx

x=x j+(−1)α

j = 2 − α, 3 − α, . . . , N + 1 − α,

α = 1, 2,

and Y2N (x, u), Z2N (x, u) is the solution of the IVP 

dY2N (x, u) dZ2N (x, u) + m2 Y2N = Z2N (x, u), = −f x, Y2N (x, u) , dx dx Y2N (x N , u) = u(x N ), lim Y2N (x, u) = 0.

x > xN

(76)

x→∞

The approximate solution of problem (76) is sought in the form (66). The coefficients A1 , A2 , . . . , A n−1 can successively be determined from (65), where ¯ ⎛ ⎞ ∞ ∞ ∞      i(i + 1)A i t i+2 − m2 iA i t i+1 + f ⎝1/t, A i t i + r(1/t)⎠ F˜ t, {A} = i=1

i=1 

i=1

2 

+ r (1/t) + m r (1/t) = 0. The T3DS of the accuracy order n¯ has the following form  1 (n) ¯ ¯ ¯ ¯ b j y x,j = −φ(n) − a j y(x¯n) (x j , y(n) ), j = 1, 2, . . . , N − 1, ,j j ¯ ¯ ¯ ¯ ¯ 2 (n) (n) (n) = μ1 , −a N y(x¯n) y(0n) ,N = m y N − μ 2 (x N , y ), where ¯ (x j , u) = φ(n)

1  (n)j Z (x j , u) − Z1(n)j (x j , u) j 2



⎤ ¯ ¯ (x j , u) − u j−1 (x j , u) − u j+1 m2 Y1(n)j m2 Y2(n)j ⎥ + + ⎦. 2 2 e m hj − 1 1 − e−m h j+1

190 | Ivan Gavrilyuk, Myroslav Kutniv, and Volodymyr Makarov

If at least one of the coefficients A1 , A2 , . . . , A n−1 is not equal to zero, we have ¯ ¯ μ(2n) (x N , u) =

− (n¯ − 2)A n−2 m2 A1 m2 A2 − A1 m2 A n−1 ¯ ¯ + + ··· + ¯ 2 xN xN x n−1 N

 ¯ (n)N 2 m Y1 (x N , u) − u N−1 − Z1(n)N (x N , u) + . e m2 h N − 1

Otherwise it holds that ¯ (n)N μ(2n) (x N , u) = −Z1 (x N , u) +

 ¯ (x N , u) − u N−1 m2 Y1(n)N e m2 h N − 1

.

The values

 ¯ ¯ ¯ (x0 , y(n) ) − y0(n) m2 Y2(n)0

¯ (n)

(x0 ) ¯ )+ , = Z2(n)0 (x0 , y(n) dx 1 − e−m2 h1

 ¯ ¯ ¯ ¯ − Y1(n)j (x j , y(n) ) m2 y(j n) dy(n) (x j ) ¯ )+ , j = 1, 2, . . . , N. = Z1(n)j (x j , y(n) 2 dx e m hj − 1 dy

ˆ¯ N with order of accuracy n. ¯ approximate du/dx at the point of grid ω Example 6.1. The problem 4 7 d2 u − 4u = − + − u2 , dx2 (1 + x)2 (1 + x)4 u(0) = 1,

x ∈ (0, ∞),

lim u(x) = 0,

(77)

x→∞

possesses the exact solution u(x) = 1/(1 + x)2 . We have solved this problem by the difference scheme (67)–(69) with n = n¯ = 4. Solving systems (65) exactly, we get A1 = 0, A2 = 1, A3 = −2. # #* # # Table 4 represents the error ER = #y(4) − u# and the number of ODE + ˆN 1,∞, ω

calls (NFUN) for given tolerance EPS using the algorithm [35] and the embedded Runge–Kutta–Nystrom methods RKN6(4)(see ref. [39], Table 5).

Tab. 4. Numerical results for Example 6.1. EPS −3

10 10−5 10−7

N

NFUN

ER

27 117 573

2604 14052 82524

0.380 · 10−5 0.940 · 10−7 0.985 · 10−9

Exact and truncated FDS for BVP

| 191

7 Three-point difference schemes for singular non-linear BVPs The stationary diffusion or heat conduction equation div (k(x, y, z) grad u) = −f (x, y, z, u) takes in the cylindrical coordinate system the form 1 2 du 1 d xk(x) = −f (x, u), x ∈ [0, R], x dx dx in the case of the axial symmetry. When x = 0, we impose the boundedness condition |u(0)| < ∞ being equivalent to the requirements lim xk(x) x→0

du = 0. dx

When x = R, the usual boundary condition holds u(R) = μ2 . There exists an E3DS (see ref. [40]), which is of the next form u x,0 = −φ(x0 , u), a2 u x,1 /(1 x1 ) = −φ(x1 , u), 1 (au x¯ )xˆ ,j = −φ(x j , u), j = 2, 3, . . . , N − 1, u N = μ2 , xj where j = 2, 3, . . . , N − 1, /

6−1 1 j V (x ) , a j = a(x j ) = hj 1 j  1

u0 − Y11 (x1 , u) , φ(x0 , u) = h1 6 / Y21 (x1 , u) − u2 1 1 1 , Z (x , u) − Z1 (x1 , u) + φ(x1 , u) = 1 2 1 x1 V21 (x1 ) ⎤ ⎡ j 2 Y (x , u) − u α 1 j α j+(−1) ⎦, φ(x j , u) = (−1)α ⎣Z αj (x j , u) + (−1)α j x j V αj (x j ) α=1

j = 2, 3, . . . , N − 1.

192 | Ivan Gavrilyuk, Myroslav Kutniv, and Volodymyr Makarov The functions Y11 (x1 , u), Z11 (x1 , u), Y αj (x j , u), Z αj (x j , u), α = 1, 2 are solutions of the IVPs dY11 (x, u) Z11 (x, u) , = dx k(x) dZ11 (x, u) Z 1 (x, u) = −f (x, Y11 (x, u)) − 1 , dx x Y11 (0, u) = u0 ,

0 < x < x1 ,

lim Z11 (x, u) = 0,

(78)

x→0

dY αj (x, u) Z αj (x, u) , = dx k(x) Z j (x, u) dZ αj (x, u) = −f (x, Y αj (x, u)) − α , dx x Y αj (x j+(−1)α , u) = u j+(−1)α ,

x j−2+α < x < x j−1+α , du j Z α (x j+(−1)α , u) = k(x) , dx x=x α

(79)

j+(−1)

and V αj (x), α = 1, 2 are solutions of the IVPs dV αj (x) (−1)α+1 , = dx xk(x) V αj (x j+(−1)α ) = 0,

x j−2+α < x < x j−1+α ,

j = 2 − α, 3 − α, . . . , N + 1 − α,

α = 1, 2.

(80)

For the solution of problems (79), (80), we use a one-step method (e.g., the Taylor ¯ Let us note that the IVPs (78), series method or a Runge–Kutta method) of the order n. is a singular and the Taylor series method for this problem has the following form f (0, u0 ) 4k(x0 ) ⎡ ⎛ ⎞ p−2 n¯  h1p ⎢ p − 1 p − j − 1 d p−j−2 f (x, u) ⎠ − ⎣ ⎝ p! p−j dx p−j−2 j

¯ (x1 , u) = u0 − h21 Y1(n)1

p=3

j=0

d 1 dx j k(x) x=0

n  h1p h1 d p−1 f (x, u) (n)1 Z1 (x1 , u) = − f (0, u0 ) − 2 (p − 1)!(p + 1) dx p−1 p=2

⎤

j

⎥ ⎦, x=0

. x=0

Now, instead of the E3DS, we can use the 3DS of the order accuracy n¯ (see ref. [41]) ¯ ¯ ¯ ¯ (n) ¯ ¯ ¯ n) = −φ(n) (x0 , y(n) ), a2(n) y x,1 /(1 x1 ) = −φ(n) (x1 , y(n) ), y(x,0  1 (n) ¯ ¯ ¯ ¯ a ¯ y(x¯n) = −φ(n) (x j , y(n) ), j = 2, 3, . . . , N − 1, y(Nn) = μ2 , xj xˆ ,j

Exact and truncated FDS for BVP

|

193

where /

6−1  1 (n)j 1

¯ ¯ ¯ u0 − Y1(n)1 V1 (x j ) , φ(n) (x0 , u) = (x1 , u) , a (x j ) = hj h1 6 / ¯ (x1 , u) − u2 Y2(n)1 1 ¯ (n) (n)1 (n)1 , Z (x1 , u) − Z1 (x1 , u) + φ (x1 , u) = ¯ 1 2 x1 V2(n)1 (x1 ) ⎡ ⎤ ¯ (n)j 2 1 ¯ (n) α ⎣ (n)j α Y α (x j , u) − u j+(−1)α ⎦ . (−1) Z α (x j , u) + (−1) φ (x j , u) = ¯ j x V (n)j (x ) ¯ (n)

j

α=1

j

α

The three-point approximation of flow xk(x)du/dx of order accuracy n¯ has the following form 6 / ¯ dy(n) ¯ = x1 Z1(n)1 (x1 , y(n) ), xk(x) dx x=x1

/

¯ (n)

xk(x)

dy dx

6





¯ + = x j Z1(n)j x j , y(n)

 ¯ ¯ ¯ x j , y(n) − Y1(n)j y(j n)

x=x j

¯ V1(n)j (x j )

,

j = 2, 3, . . . , N.

If a solution to the diffusion (or heat conduction) equation in the spherical coordinate system is centrally symmetric, the function u(x) satisfies the equation 1 2 1 d 2 du x = −f (x, u), x ∈ [0, R], k(x) dx x2 dx At the points x = 0 and x = R, we may impose the conditions lim x2 k(x) x→0

du = 0, dx

u(R) = μ2 .

9 h was In ref. [42], for this problem the following E3DS on the irregular grid ω constructed: u x,0 = −φ(x0 , u),

a2 u x,1 /(1 x21 ) = −φ(x1 , u),

1 (au x¯ )xˆ ,j = −φ(x j , u), j = 2, 3, . . . , N − 1, u N = μ2 , x2j / 6−1 1 j V (x ) , j = 2, 3, . . . , N − 1, a j = a(x j ) = hj 1 j

(81)

(82)

194 | Ivan Gavrilyuk, Myroslav Kutniv, and Volodymyr Makarov Y20 (0, u) − u1 , h 6 / 1 Y21 (x1 , u) − u2 1 1 1 , Z (x , u) − Z1 (x1 , u) + φ(x1 , u) = 1 2 1 x21 V21 (x1 ) ⎤ ⎡ j 2 1 α⎣ j α Y α (x j , u) − u j+(−1)α ⎦ φ(x j , u) = , (−1) Z α (x j , u) + (−1) j x2 V αj (x j ) φ(x0 , u) =

α=1

j

j = 2, 3, . . . , N − 1.

(83)

where Y20 (0, u), Z20 (0, u), Y11 (x1 , u), Z11 (x1 , u), Y αj (x j , u), Z αj (x j , u), α = 1, 2 are solutions of the IVPs dY20 (x, u) Z20 (x, u) , = dx k(x) 2Z 0 (x, u) dZ20 (x, u) = −f (x, Y20 (x, u)) − 2 , 0 < x < x1 , dx x du , Y20 (x1 , u) = u1 , Z20 (x1 , u) = k(x) dx x=x1 dY11 (x, u) Z11 (x, u) , = dx k(x) dZ11 (x, u) 2Z 1 (x, u) = −f (x, Y11 (x, u)) − 1 , dx x Y11 (0, u) = u0 , lim Z11 (x, u) = 0,

(84)

0 < x < x1 , (85)

x→0

dY αj (x, u) Z αj (x, u) , = dx k(x) 2Z j (x, u) dZ αj (x, u) = −f (x, Y αj (x, u)) − α , dx x Y αj (x j+(−1)α , u) = u j+(−1)α ,

x j−2+α < x < x j−1+α , du j Z α (x j+(−1)α , u) = k(x) , dx x=x α j+(−1)

j = 3 − α, 4 − α, . . . , N + 1 − α,

α = 1, 2,

(86)

and V αj (x) are solutions of the IVPs dV αj (x) (−1)α+1 , = 2 dx x k(x) V αj (x j+(−1)α ) = 0,

x j−2+α < x < x j−1+α ,

j = 2 − α, 3 − α, . . . , N + 1 − α,

α = 1, 2.

The practical realization of the E3DS for all x j , j = 1, 2, . . . N − 1 can be achieved by the integration of the four auxiliary IVPs: two non-linear ODEs and two linear ODEs on the intervals [x j−1 , x j ] (forward) and [x j , x j+1 ] (backward). These IVPs can be solved by executing only one step with an arbitrary one-step method (e.g., the

Exact and truncated FDS for BVP

|

195

Taylor series method or a Runge–Kutta method). Then instead of the E3TS, we obtain a practically realizable T3DS. The accuracy of the T3DS is determined by the accuracy of the corresponding IVP-solvers using for the calculation of its coefficients. Example 7.1. Let us consider the problem 1 2 1 d 2 du x = −u5 , x ∈ (0; 1), dx x2 dx

lim x x→0

2 du

dx

√

= 0,

u(1) =

3 , 2

(87)

with the exact solution u(x) = (1 + x2 /3)−1/2 . We solve numerically the auxiliary IVPs (84)–(86), e.g. using the Taylor series method of the fourth accuracy order. We denote the numerical solutions of these IVPs by Y α(4)j (x j , u), Z (4)j α (x j , u) and apply the Newton iteration method to find the numerical solution y(4) ≈ u j of difference scheme (81)–(83) on the regular grid j   ¯ h = x j = jh, j = 0, N, h = 1/N . The Taylor series method for this problem has the ω following form h2 h3 df (0, u0 ) h4 d2 f (0, u0 ) , f (0, u0 ) − − 6 12 dx 40 dx2 h h2 df (0, u0 ) h3 d2 f (0, u0 ) h4 d3 f (0, u0 ) − . Z1(4)1 (x1 , u) = − f (0, u0 ) − − 3 4 dx 10 36 dx2 dx3

Y1(4)1 (x1 , u) = u0 −

The results of solving problem (4) are given in Table 5. Here # #* # # E1 = #y(4) − u#

¯h 1,∞, ω

/ x2

(4)

dy dx

,

#* # # # (4) #y − u# ¯h 1,∞, ω p = log2 # , #* # # (4) #y − u# ¯ h/2 1,∞, ω

6

= x2j Z1(4)j (x j , y(4) ),

j = 1, 2, . . . , N.

x=x j

Thus, this difference scheme has the fourth order of accuracy.

Tab. 5. Numerical results for problem (4.) N 10 20 40 80 160

E1

p −5

0.3955·10 0.1477·10−6 0.9432·10−8 0.5915·10−9 0.3699·10−10

4.7 4.0 4.0 4.0

196 | Ivan Gavrilyuk, Myroslav Kutniv, and Volodymyr Makarov

Tab. 6. Comparison of numerical methods. Numerical methods

E2 0.18 · 10−3 0.60 · 10−4 0.57. 10−5

difference scheme [43] collocation method [43] difference scheme (81)–(83) of forth accuracy order

The numerical results for problem (4) by different numerical methods are given in the ¯h= Table 6, where E2 = y − u0,∞,ω−h = max |y j − u j |, y j is a numerical solution and ω 0≤j≤N−1   x j = jh, j = 0, 8, h = 1/8 . It should be noted that the numerical solution obtained by the scheme (81)–(83) is more accurate. Moreover, this difference scheme provides not only the solution u(x), but also the flow x2 k(x)du/dx at the grid nodes with the same accuracy order.

8 Exact- difference schemes for PDEs Contrary to ODEs, there does not exist a closed description (‘general solution’) of the whole solution set of a PDE. But, e.g., the Lie group theory allows to describe various subsets of the whole set of solutions. Let us consider the following non-linear BVP ∂U ∂2 U − Φ (U), = ∂t ∂ξ 2 U(ξ , 0) = U0 (ξ ), lim U(ξ , t) = U1 ,

ξ →−∞

ξ ∈ (−∞, ∞),

(88)

∂U(ξ , 0) ¯ = U0 (ξ ), ∂t

(89)

lim U(ξ , t) = U2 ,

(90)

ξ →∞

where U1 , U2 are local minima of the function Φ(U), where Φ(U1 ) = Φ(U2 ) = 0. This BVP describes various physical processes, in particular the proton transfer process in systems with hydrogen bonds [44]. In the case of the one- component, model the potential function Φ(U) can be of the form : Φ(u) = (1 − u2 )2 4, Φ(u) = (1 − |u|)2 , Φ(u) = 1 − cos u, 1

cos(u/2) − λ Φ(u) = 2 1 − ν(cos(u/2) − λ)

22 ,

0 ≤ λ < 1,

−(1 + λ)−1 < ν < (1 − λ)−1 .

Exact and truncated FDS for BVP

| 197

One can show that ξ − vt, U are constants, i.e., invariants under transformations of the Lie subgroup with the infinitesimal generator U = V2 =

∂ ∂ +v , ∂t ∂x

(91)

Using these two invariants we introduce two new invariant variables instead of three variables ξ , t, U x = ξ − vt, u = u(x) = U(x, t)

(92)

and obtain the ODE (1 − v2 )u = Φ (u),

x ∈ (−∞, ∞)

(93)

lim u(x) = U2 .

(94)

with the boundary conditions lim u(x) = U1 ,

x→−∞

After changing of variables x˜ = x

:7

x→∞

2(1 − v2 ) problem (93)–(94) is transformed to

u = 2Φ (u), lim u(x) = U1 ,

x→−∞

x ∈ (−∞, ∞), lim u(x) = U2 .

x→∞

(95) (96)

: Let Φ(u) = (1 − u2 )2 4, then u = −2u(1 − u2 ), x ∈ (−∞, ∞), lim u(x) = −1, lim u(x) = 1.

x→−∞

x→∞

The solution u(x) is odd, i.e. the problem can be considered on the half-axis: u = −2u(1 − u2 ), x ∈ (0, ∞), u(0) = 0, lim u(x) = 1. x→∞

After variables transform u˜ (x) = u(x) − 1 we obtain the problem d2 u − 4u = 2u3 + 6u2 , dx2 u(0) = −1,

x ∈ (0, ∞),

lim u(x) = 0,

x→∞

(97) (98)

with the exact solution u(x) = th(x) − 1. The BVP (97)–(98) is of the form d2 u − m2 u = −f (x, u), dx2

x ∈ (0, ∞),

(99)

198 | Ivan Gavrilyuk, Myroslav Kutniv, and Volodymyr Makarov Traveling wave grid 1 0.8 0.6 0.4

t

0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1

−0.5

0

0.5

1

ξ

Fig. 1. Traveling wave grid.

u(0) = μ1 ,

lim u(x) = 0,

x→∞

(100)

where m = 2, f (x, u) = −2u3 − 6u2 , μ1 = −1. This is exactly the form that was treated in Subsection 6, and we can apply the T3DS. Now we can introduce an arbitrary grid ω x,N = {x1 , x2 , . . . , x N },

(101)

and solve the ODE numerically by a truncated difference schemes (T3DS) of an arbitrary given accuracy order. The approximate solution of the PDE will be then defined on the grid ω N = {(t i , ξ i ), i = 1, 2, . . . , N },

(102)

where u i = u(x i ) = U(t i , ξ i ), x i are arbitrary given and t i , ξ i are such that ξ i − vt i = x i . This means that we obtain the contour lines of the approximate solution at points (t i , ξ i ) lying on the characteristics (see Fig. 1). To obtain the solution within a given tolerance ε using an adaptive algorithm with the T3DS of the fourth (n = n¯ = 4) and of sixth (n = n¯ = 6) accuracy order of the form ¯ ¯ ¯ ¯ )xˆ ,j − d(x j )y(j n) = −φ(n) (x j , y(n) ), (ay (x¯n) ¯ = μ1 , y(0n)

j = 1, 2, . . . , N − 1,

¯ ¯ ¯ ¯ (n) (n) (n) −a(x N )y(x¯n) ,N = β 2 y N − μ 2 (x N , y ),

(103) (104)

Exact and truncated FDS for BVP

| 199

where y x¯ ,j =

y j − y j−1 , hj

y xˆ ,j =

y j+1 − y j , j

j =

h j + h j+1 , 2

mh j exp(mh N ) − 1 , j = 1, 2, . . . , N, β2 = m , sh(mh j ) sh(mh N ) " ! m ch(mh j ) − 1 ch(mh j+1 ) − 1 d(x j ) = + , j = 1, 2, . . . , N − 1, j sh(mh j ) sh(mh j+1 )  ¯ Z2(n)j (x j , u) − Z1(n)j (x j , u) (x j , u) = −1 φ(n) j a(x j ) =

⎤ ¯ ¯ (x j , u) − u j−1 ) m(ch(mh j+1 )Y2(n)j (x j , u) − u j+1 m(ch(mh j )Y1(n)j ⎦, + + sh(mh j ) sh(mh j+1 ) ¯ (x N , u) = μ(2n)

(105) (106)

(107)

mA1 mA2 − A1 mA n−1 − (n¯ − 2)A n−2 ¯ ¯ + +...+ − ¯ xN x2N x n−1 N −Z1(n)N (x N , u) +

¯ (x N , u) − u N−1 ) m(ch(mh N )Y1(n)N , sh(mh N )

(108)

provided that at least one of the coefficients A i , i = 1, 2, . . . , n¯ − 1 is not equal to zero, and ¯ (x N , u) = −Z1(n)N (x N , u) + μ2(n)

¯ (x N , u) − u N−1 ) m(ch(mh N )Y1(n)N , sh(mh N )

(109)

¯ (n)j provided that all A i = 0, i = 1, 2, . . . , n¯ − 1. The values Z (n)j α , Y α , α = 1, 2 are the numerical solutions of the Cauchy problems

dY αj (x, u) = Z αj (x, u), dx

 dZ αj (x, u) − m2 Y αj (x, u) = −f x, Y αj (x, u) , dx

(110)

x j−2+α < x < x j−1+α , Y αj (x j+(−1)α , u) =

u j+(−1)α ,

Z αj (x j+(−1)α , u) =

du dx x=x

j = 2 − α, 3 − α, . . . , N + 1 − α,

,

(111)

j+(−1)α

α = 1, 2,

¯ obtained by an arbitrary one-step method of the accuracy n. ¯ is given by Since our ODE (97) is autonomous, we have A i = 0, i = 1, 2, . . . and μ(2n) (109). The Cauchy problems (110) and (111) were solved by the imbedded Runge–Kutta– Nyström methods of the accuracy orders 4 and 6.

200 | Ivan Gavrilyuk, Myroslav Kutniv, and Volodymyr Makarov

Tab. 7. Numerical results for BVP (97). ε 10−3 10−5 10−7 10−9

N

NFUN

4 19 75 417

1320 5280 46164 225948

er 0, 552 · 10−3 0, 704 · 10−6 0, 352 · 10−8 0, 398 · 10−10

¯ The solution of the truncated 3DS y(j n) , j = 1, . . . , N − 1 was computed by the iteration method ¯ ¯ ¯ ¯ )xˆ ,j − d(x j )y(j n,k) = −φ(n) (x j , y(n,k−1) ), (ay(x¯n,k) ¯ = μ1 , y(0n,k)

j = 1, 2, . . . , N − 1,

¯ ¯ ¯ ¯ (n,k) −a(x N )y(x¯n,k) − μ(2n) (x N , y(n,k−1) ), ,N = β 2 y N

¯ (x j ) = μ1 exp(−mx j ), y(n,0)

k = 1, 2, . . . ,

¯ dy(n,0) (x j ) = −mμ1 exp(−mx j ), dx

(112) (113)

j = 0, 1, . . . , N.

The grid approximation to the derivative of the solution was computed using the formulas ¯ ¯ ¯ ¯ m · ch(mh1 )(Y2(n)0 (x0 , y(n,k) ) − y(0n,k) ) dy(n,k) (x0 ) ¯ ¯ (x0 , y(n,k) )+ , = Z2(n)0 dx sh(mh1 ) ¯ dy(n,k) (x j ) dx

¯ ¯ = Z1(n)j (x j , y(n,k) )+

¯

¯ ¯ m·ch(mh j )(y(j n,k) −Y1(n)j (x j ,y(n,k) )) , sh(mh j )

j = 1, 2, . . . , N,

k = 1, 2, . . .

The system of linear algebraic equations (112), (113) with a tridiagonal matrix was solved by the sweep method. The numerical results are given ! " by Table 7, where er = # # # #* # # # # ¯ (n) # (n) # # # dy ¯ # = max #y(n) − u# ,# − du is the computed error, #y ¯ − u# dx # + + # dx ˆN 1,∞, ω

ˆN 0,∞, ω

ˆ¯ N 0,∞, ω

NFUN is the number of calls of the right-side of ODE and ε is the given tolerance.

Bibliography [1] [2] [3]

A. A. Samarskii and E. S. Nikolaev. Numerical Methods for Grid Equations, volume 1. Birkhäuser Verlag, Basel, 1989. H. B. Keller and A. B. White Jr. Difference methods for boundary value problems in ordinary differential equations. SIAM J. Numer. Anal., 12:791–802, 1975. A. A. Samarskii. The theory of difference schemes. Marcel Dekker Inc., New York and Basel, 2001.

Exact and truncated FDS for BVP | 201

[4] [5] [6] [7] [8]

[9]

[10] [11]

[12]

[13] [14]

[15] [16]

[17] [18] [19]

[20]

[21]

[22]

V. L. Makarov and Gavrilyuk I. P. The construction of best difference schemes with an exact spectrum. DAN USSR, Ser. A, (12):1078–1081, 1975. I. P. Gavrilyuk and V. L. Makarov. A method for solving boundary value problems for nonlinear parabolic equations. DAN USSR, Ser. A, (7):10–14, 1980. I. P. Gavrilyuk. An economical method for solving boundary value problems for a class of quasilinear equations containing the laplace operator. Vychisl. Prikl. Mat., (42):109–117, 1980. A. N. Tihonov and A. A. Samarskii. Homogeneous difference schemes of a high order of accuracy on non-uniform nets (in Russian). Zh. Vychisl. Mat. i Mat. Fiz., 1:425–440, 1961. V. L. Makarov and A. A. Samarskii. Exact three-point difference schemes for nonlinear ordinary differential equations of second order, and their implementation. Sov. Math. Dokl., 41: 495–500, 1990. V. L. Makarov and V. V. Guminskii. A three-point difference schemes of a high order of accuracy for a system of second-order ordinary differential equations (the nonselfadjoint case). Differ. Equ., 30:457–465, 1994. V. L. Makarov and S. G. Gocheva. Finite-difference schemes of arbitrary accuracy for second-order differential equations on a half-line. Differ. Equ., 17:367–377, 1981. V. L. Makarov, I. L. Makarov, and V. G. Prikazchikov. Exact difference schemes and schemes of any order of accuracy for systems of second-order differential equations (in Russian). Differ. Uravn., 15:1194–1205, 1979. A. A. Samarskii and V. L. Makarov. On the realization of exact three-point difference schemes for second-order ordinary differential equations with piecewise smooth coefficients. Sov. Math. Dokl., 41:463–467, 1990. A. A. Samarskii, R. D. Lazarov, and V. L. Makarov. Difference schemes for differential equations with generalized solutions (in Russian). Nauka, Moscow, 1987. A. A. Samarskii and V. L. Makarov. Realization of exact three-point difference schemes for second-order ordinary differential equations with piecewise-smooth coefficients. Differ. Equ., 26:922–930, 1991. A. N. Tihonov and A. A. Samarskii. Homogeneous difference schemes (in Russian). Zh. Vychisl. Mat. i Mat. Fiz., 1:5–63, 1961. I. P. Gavrilyuk. Exact difference schemes and difference schemes of arbitrary given degree of accuracy for generalized one-dimensional third boundary value problems. Z. Anal. Anwend., 12(3):549–566, 1993. I. P. Gavrilyuk. A class of one-dimensional variational inequalities and difference schemes of arbitrary given degree of accuracy. Z. Anal. Anwend., 12(4):751–758, 1993. I. P. Gavrilyuk. Algorithm for solving a class of one-dimensional variational inequalities. J. Sov. Math., 66(3):24–30, 1988. M. V. Kutniv, V. L. Makarov, and A. A. Samarskii. Accurate three-point difference schemes for second-order nonlinear ordinary differential equations and their implementation. Comput. Math. Math. Phys., 39:45–60, 1999. M. V. Kutniv. Three-point difference schemes of high accuracy order for second order nonlinear ordinary differential equations with the boundary conditions of the third type (in Ukrainian). Visnyk Lviv Univ. Ser. Appl. Math. Comput. Sci., 4:61–66, 2002. M. V. Kutniv. Accurate three-point difference schemes for second-order monotone ordinary differential equations and their implementation. Comput. Math. Math. Phys., 40:368–382, 2000. M. V. Kutniv. Three-point difference schemes of high accuracy order for systems of nonlinear ordinary differential equations of the second order. Comput. Math. Math. Phys., 41:860–873, 2001.

202 | Ivan Gavrilyuk, Myroslav Kutniv, and Volodymyr Makarov

[23] M. V. Kutniv. High-order accurate three-point difference schemes for systems of second-order ordinary differential equations with a monotone operator. Comput. Math. Math. Phys., 42(5): 724–738, 2002. [24] M. V. Kutniv. Modified three-point difference schemes of high-accuracy order for second order nonlinear ordinary differential equations. Comput. Methods Appl. Math., 3:287–312, 2003. [25] I. P. Gavrilyuk, M. Herman, V. L. Makarov, and M. V. Kutniv. Exact and truncated difference schemes for boundary value ODEs. Springer Basel AG, 2011. [26] V. L. Makarov, I. P. Gavrilyuk, M. V. Kutniv, and M. Hermann. A two-point difference scheme of arbitrary given accuracy order for BVPs for systems of first order non-linear ODEs. Comput. Methods Appl. Math., 4:464–493, 2004. [27] I. P. Gavrilyuk, M. Hermann, M. V. Kutniv, and V. L. Makarov. Difference schemes for non-linear BVPs using Runge–Kutta IVP-solvers. Adv. Differ. Equ., 2006:1–29, 2006. [28] E. Hairer, S. P. Nørsett, and G. Wanner. Solving ordinary differential equations I, nonstiff problems. Springer Verlag, Berlin, Heidelberg, New York, 1993. [29] I. P. Gavrilyuk, M. Hermann, M. V. Kutniv, and V. L. Makarov. Two-point difference schemes of an arbitrary given order of accuracy for nonlinear bvps. Appl. Math. Lett., 23:585–590, 2010. [30] L. B. Gnativ, M. V. Kutniv, and A. I. Chukhrai. Generalized three-point difference schemes of high order of accuracy for nonlinear ordinary differential equations of the second order. J. Math. Sci., 167:62–75, 2010. [31] L. B. Gnativ, M. Król, and M. V. Kutniv. Exact three-point difference schemes for second order nonlinear differential equations with boundary conditions of the third kind. J. Comput. Appl. Math. (Series “Numerical Mathematics”), 3 (109):34–52, 2012. [32] M. Król and M. Kutniv. Three-point difference schemes of high-order accuracy for second-order nonlinear ordinary differential equations with boundary conditions of third kind. J. Comput. Appl. Math. (Series “Numerical Mathematics”), 2 (116):43–62, 2014. [33] L. B. Gnativ, M. V. Kutniv, and V. L. Makarov. Generalized three-point difference schemes of high-order accuracy for systems of second order nonlinear ordinary differential equations. Differ. Equ., 45:998–1019, 2009. [34] I. P. Gavrilyuk, M. Hermann, M. V. Kutniv, and V. L. Makarov. Difference schemes for nonlinear bvps on the semiaxis. Comput. Methods Appl. Math., 7:25–47, 2007. [35] I. P. Gavrilyuk, M. Hermann, M. V. Kutniv, and V. L. Makarov. Adaptive algorithms based on exact difference schemes for nonlinear bvps on the half-axis. Appl. Numer. Math., 59: 1529–1536, 2009. [36] N. S. Nedialkov and J. D. Pryce. Solving differential-algebraic equations by Taylor series (i): Computing Taylor coefficients. BIT Numer. Math., 45:561–591, 2005. [37] M. V. Kutniv and O. I. Pazdrii. Exact three-point difference schemes for nonlinear boundary value-problem on the semiaxis. J. Math. Sci., 181:383–400, 2012. [38] M. V. Kutniv and O. I. Pazdrii. Three-point difference schemes of high accuracy order for second-order nonlinear ordinary differential equations on the semiaxis (in Ukrainian). Visnyk Lviv Univ. Ser. Appl. Math. Comput. Sci., 17:10–21, 2011. [39] J. R. Dormand, E. A. El-Mikkawy, and P. J. Prince. Families of Runge-–Kutta—Nystrom formulae. IMA J. Numer. Anal., 7:235—250, 1987. [40] A. V. Kunynets and M. V. Kutniv. Exact three-point difference scheme for nonlinear stationary differential equations in cylindrical coordinates. J. Comput. Appl. Math. (Series “Numerical Mathematics”), 2 (105):51–68, 2011. [41] A. V. Kunynets and M. V. Kutniv. Three-point difference scheme of high order accuracy for nonlinear ordinary differential equations of the second order in cylindrical coordinates (in Ukrainian). J. Natl Univ. “Lvivska Politechnika” (Phys. Math. Sci.), 768:85–99, 2013.

Exact and truncated FDS for BVP

|

203

[42] M. Król, A. V. Kunynets, and M. V. Kutniv. Exact three-point difference scheme for singular nonlinear boundary value problems. J. Comput. Appl. Math., 298:175–189, 2016. [43] R. D. Russell and F. Shampine. Numerical methods for singular boundary value problems. SIAM J. Numer. Anal., 12 (1):13–36, 1975. [44] M. V. Kutniv and O. I. Pazdrii. Numerical solution of nonlinear bvps on the half-axis describing the proton transfer process in systems with hydrogen bonds. Phys.-Math. Modelling Inf. Technol., 13:95–102, 2011.

Silvia Jerez and Saúl Díaz-Infante

Exact difference schemes for stochastic differential equations In this chapter, we propose a new way to construct explicit numerical methods for stochastic differential equations (SDEs) via the Steklov mean. The foundations for this new family of methods are based on the exact finite-difference scheme for the deterministic version of the SDEs. First, we construct a scheme for the scalar case that is named Steklov method and present the principal results about its convergence and stability for multiplicative and additive SDEs. Next, we extend the previous method towards a multi-dimensional set up with coefficients of the SDEs under locally Lipschitz and monotone growth conditions. This method is constructed on the basis that the drift function can be rewritten in a linearized form, hence its name, the linear Steklov (LS) method. Also we provide numerical evidence of the accuracy and efficiency of the Steklov family of schemes versus several methods for SDEs.

1 Introduction Nowadays, stochastic differential modelling is used to describe the random environmental variations occurring in any nature phenomenon. However, an analytical solution is only obtained for few SDEs; therefore, the use of accurate stochastic numerical approximations represent an option to analyse and confirm (by simulation) the dynamics of a stochastic model. Actually, the Euler–Maruyama (EM) scheme is one of the most popular methods to solve SDEs, due to its low computational cost and its acceptable accuracy. But recent research has shown that this scheme diverges for SDEs with non-globally Lipschitz conditions and super-linear growth of the coefficients [1, 2]. Moreover, in applications of Monte Carlo simulations and Brownian Dynamics, the EM requires pint-size time steps to assure its stability. From, previous arguments, new methods have been developed to solve SDEs under more general conditions [3–6]. In this direction, we propose a new family of methods that are based on the exact discretization of the associated deterministic equation to the SDE. It is important to mention that the key tool used to construct the new methods is the Steklov mean. In this section, we present the Steklov and LS methods as the basic schemes of this new family and display their construction. It is important to point out that by using the Steklov mean, we obtain implicit finite-difference schemes; hence, we impose some extra conditions to the coefficients of SDEs in order to develop explicit methods. The reason to propose explicit methods is determined by the high computational cost that

EDS for Stochastic Differential Equations | 205

represents the use of implicit methods in applications like Monte Carlo simulations and Brownian Dynamics. Thus, here we lay the foundations of explicit methods for the Steklov family of schemes. This chapter is organized as follows: In 6.3, Section, 2, we give the standard setup and necessary theoretical assumptions. In Section, 3 we construct the explicit Steklov scheme and state some theorems about its strong convergence and non-linear stability. In Section 6, we present the linear Steklov method and prove its almost sure stability. Finally, in Section 7 we show numerical simulations of the Steklov and LS methods and other known methods for scalar and vector SDEs even with non-linear diffusion coefficients.

2 General settings Throughout this chapter, we consider the following stochastic differential equation dy(t) = f (y(t))dt + g(y(t))dW(t),

0 ≤ t ≤ T,

y(0) = y0 ,

(1)

where y(t) is the solution process that is governed by the drift function f = (f (1) , . . . , f (d) )T : Rd → Rd and the diffusion function g = (g (j,i) )j∈{1,...,d},i∈{1,...,m} : Rd → Rd×m . We use a standard setup, so let (Ω, F , (Ft )t∈[0,T] , P) a filtered and complete probability space with the filtration (Ft )t∈[0,T] generated by a m-dimensional Brownian process W(t). Denoting the norm of a vector y ∈ Rd by |y| and the Frobenious norm of a matrix G ∈ Rd×m by |G|. The usual scalar product of two vectors x, x ∈ Rd is denoted by < x, x >. Moreover, we use x to denote a deterministic variable and y to denote a stochastic process. In order to assure existence and uniqueness of the solution of SDE (1), we assume one and only one of these hypotheses [7, 8]. Hypothesis 2.1. The coefficients of SDE (1) satisfy the conditions: (i) The functions f , g are of class C1 (Rd ). (ii) Global Lipschitz condition. There is a positive constant L such that |f (x) − f (x  )|2 ≤ L |x − x  |2

and

|g(x) − g(x )|2 ≤ L |x − x  |2 ,

∀x, x  ∈ Rd .

(iii) Linear growth condition. There exists a positive constant M such that |F(x, t)| ≤ M(1 + |x | + |t |),

|G(x, t)| ≤ M(1 + |x | + |t |),

Hypothesis 2.2. The coefficients of SDE (1) satisfy the conditions: (i) The functions f , g are of class C1 (Rd ).

∀ x ∈ Rd .

206 | Silvia Jerez and Saúl Díaz-Infante

(ii) Local, global Lipschitz condition. For each integer n, there is a positive constant L f = L f (n) such that |f (x) − f (x  )|2 ≤ L f |x − x  |2

∀x, x  ∈ Rd ,

max{x, x } ≤ n,

and there is a positive constant L g such that |g(x) − g(x )|2 ≤ L g |x − x  |2 ,

∀x, x  ∈ Rd .

(iii) Monotone condition. There exist two positive constants α and β such that 1 < x, f (x) > + |g(x)|2 ≤ α + β|x|2 , 2

∀ x ∈ Rd .

Hypothesis 2.1 is more restrictive than Hypothesis 2.2, and this last one have been proved to be more useful on applications.

3 Steklov method Here, we develop the Steklov method for a scalar SDE, i.e., d = 1 in (1), and we start by discretization of the time domain [0, T] with a constant time-step h = T/N such that t k = kh for k = 0, . . . , N. Denoting by Y k the approximation to the solution state y(t ) and W k the discrete standard Brownian motion satisfying ∆W k := (W t k+1 − W t k ) = √ k h V k with V k ∼ N (0, 1). Using the integral formulation of SDE (1), we can rewrite the solution process at time t k+1 as follows: t k+1 t k+1 f (s, y(s))ds + g(s, y(s))dW(s), y(t k+1 ) = y(t k ) + tk

t k ≤ t ≤ t k+1 .

(2)

tk

Applying different numerical integration to the Lebesgue and Itô integrals of (2), we obtain different finite-difference schemes to numerically solve (1). For example, using a backward approximation on the drift and diffusion coefficients, we have the well -known Euler–Maruyama scheme: Y k+1 = Y k + f (t k , Y k ) h + g(t k , Y k )∆W k ,

k = 1, 2, . . . , N − 1,

taking Y0 = y0 . So we can extend to stochastic differential equations several deterministic finite-difference methods such as the Runge–Kutta and Taylor methods among others [9]. In particular, here we focus on adapting to SDEs the exact FDS developed for ODEs proposed in ref. [10]. Assuming that the drift function can be rewritten of the form f (t, x) = f1 (t)f2 (x),

(3)

EDS for Stochastic Differential Equations | 207

and approximating the deterministic integral of scalar SDE (2) by the Steklov mean t k+1 f1 (s)f2 (x)ds ≈ ϕ1 (t k )ϕ2 (x k , x k+1 ) h,

(4)

tk

where

ϕ1 (t k ) =

1 t k+1 − t k

t k+1 f1 (s)ds

⎛ and

⎜ ϕ2 (x k , x k+1 ) = ⎝

tk

1 x k+1 − x k

x k+1 xk

⎞−1 du ⎟ ⎠ . f2 (u)

Thus, the associated ODE to SDE (1) has the following exact FD scheme: x k+1 − x k = ϕ1 (t k )ϕ2 (x k , x k+1 )h.

(5)

In order to solve for x k+1 , we rewrite (5) as follows: x k+1 − x k = ϕ1 (t k )

(x k+1 − x k ) h, H(x k+1 ) − H(x k )

where x H(x) := 0

du f2 (u)

and assuming the existence of the function H −1 , we give the following explicit scheme for the SDE (1): Y k+1 = Ψ h (t k , Y k ) + g(t k , Y k )∆W k ,

k = 1, . . . , N − 1,

(6)

with initial condition Y0 = y0 and where Ψ h (t k , Y k ) := H −1 [H(Y k ) + hϕ1 (t k )]. Scheme (6) is called the Steklov method. It is worth noting that the Steklov method tries to preserve qualitative properties of the deterministic solution using an exact FD discretization for the drift term. The diffusion term is approximated as in the EM method. Next, we present some examples of the Steklov algorithm. Example 3.1. Let us first consider the linear SDE dy(t) = λy(t) dt + ξy(t) dW(t),

(7)

208 | Silvia Jerez and Saúl Díaz-Infante where λ, ξ ∈ C and y0 =  0 with probability one. Using the Steklov mean (4), we approximate the drift integral as follows:  x k+1 λudu ≈ xk



1

x ln k+1 xk λ(x k+1 − x k )

−1 h,

k = 1, . . . , N − 1.

(8)

Our next step is to obtain the exact FDS for the ODE associated with SDE (7), i.e., (x − x k ) x k+1 − x k

x  , = λ k+1 h ln k+1 xk which is equivalent to the explicit scheme x k+1 = exp(λh)x k .

(9)

Thus, the Steklov scheme for (7) is given by Y k+1 = exp(λh)Y k + ξY k ∆W k

k = 1, . . . , N − 1.

Example 3.2. Now we consider a SDE with non-linear drift and diffusion coefficients proposed by Appleby and Kelly in ref. [11]: dy(t) = −y3 (t)dt + 

1

1.1 dW(t), log(t + 1)

(10)

Using the Steklov mean, we can get the approximation x k+1 x2 x2 u3 du ≈ 2 k+1 k , x k+1 + x k

xk

and the exact FD scheme for the dx = −x3 becomes x2 x2 x k+1 − x k = −2 k+1 k . h x k+1 + x k By algebraic manipulation, we get an equivalent explicit exact FDS x , x k+1 = 8 k 1 + 2x2k h and the Steklov method for (10) is Yk 1 + Y k+1 = 8 1.1 ∆W k . 2 log(t 1 + 2hY k k + 1)

(11)

EDS for Stochastic Differential Equations | 209

We will later return to these examples, since the linear Steklov scheme is based on the approximation (8), and in the section of numerical simulations, we analyse both Steklov methods for the examples proposed on this section. Next we give some important results about the global discretization error and stability properties.

4 Convergence and Stability In this subsection, we state some theorems to assure convergence and give stability conditions to preserve the dynamical structure of the Steklov solution for long time periods. Convergence gives us information about the behavior of a scheme on a fixed time interval, whereas the stability analysis allow us to understand the behavior of the approximation for a fixed step-size when the time interval approaches to infinity. For simplicity these properties are studied for a scalar autonomous SDE dy(t) = f (y(t)) dt + g(y(t)) dW(t).

(12)

For details of the proofs of the following theorems, we refer the reader to [12]. Theorem 4.1. Assume Hypothesis 2.1 holds, then the Steklov approximation (6) of the scalar SDE (12) with Y0 = y0 is strongly convergent to the solution y(t), that is, lim E y(T) − Y N = 0, h→0

for the end time T. There are different ways of analysing the non-linear stability for stochastic numerical methods; a popular one is the use of Lyapunov functions [13] and another one based on the theory of random dynamical systems [14], which is more general than the previous one. In the following, we provide sufficient conditions for the non-linear stability of the Steklov method applied to (12) for both multiplicative and additive cases. Let us start stating the notion of the non-linear asymptotic stability in a quadratic mean-square sense [15]. 9 k two different numerical recurrences for SDE (12) with Definition 4.1. Let Y k and Y 90 . We shall say that multiplicative noise and corresponding initial process Y0 and Y a discrete time, Y k is numerically zero-stable in quadratic mean-square sense if given ϵ > 0, there are positive constants h0 and δ = δ(ϵ, h0 ) such that for all h ∈ (0, h0 ) and 2 90 < δ then positive integers k ≤ T/h whenever E Y0 − Y 2 9 k < ϵ. ρ k := E Y k − Y

210 | Silvia Jerez and Saúl Díaz-Infante

If the method is stable and ρ k → 0 when k → ∞, then the method is asymptotically zero-stable in the quadratic mean-square sense. Theorem 4.2. Assume Hypothesis 2.1 holds, then the Steklov approximation (6) has a global Lipschitz constant L for its functions Ψ h and g and, it is zero-stable in quadratic mean square sense for the multiplicative SDE (12). In addition, if L < 1, then the Steklov method is asymptotically zero-stable stable in quadratic mean-square sense. Following the work of Caraballo and Kloeden [16], we showed stability in a path-wise sense for the Steklov method for an additive SDE in ref. [12], i.e, the process (6) attracts all the Steklov pathwise approximations forwards in time. Theorem 4.3. Assume Hypothesis 2.1 holds, if the Steklov function Ψ h satisfies (i) (Contractive Lipschitz condition). There exists a constant L 1 ∈ (0, 1) such that |Ψ h (x) − Ψ h (x  )| ≤ L 1 |x − x  |

∀x, x  ∈ R,

(ii) (Contractive one sided Lipschitz condition). There exists a constant L 2 such that Ψ h (x) − Ψ h (x  ), x − x   ≤ −L 2 |x − x  |2

∀x, y ∈ R,

(iii) (Linear growth bound). There exists a constant L 3 such that |Ψ h (x)| ≤ L 3 (1 + h + |x |)

∀x ∈ R,

and the condition L3 0 such that for all 0 < h < h* , the Steklov method (6) has a unique stochastic stationary solution that is path-wise asymptotically stable for the SDE (12) with additive noise. In the next subsection, we construct the linear Steklov scheme for which the requirement (11) on the drift function is relaxed.

5 Linear Steklov method For simplicity, we consider the autonomous case of the vector Itô stochastic differential equation (1) and in order to approximate the solution using the Steklov mean without solving a multiple integral, we assume that each component f (j) of the drift term, f = (f 1 , . . . , f d ) can be linearized as f (j) (x) = a j (x)x(j) + b j (x(−j) ),

j = 1, . . . , d,

(13)

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| 211

where a j and b j are two scalar functions from Rd to R and x(−j) = (x(1) , . . . , x(j−1) , x(j+1) , . . . x(d) ). Once the linearized form (13) is obtained, we use the Steklov mean to get an exact FD system defined by (j)

Y k* = Y k(j) + h φ f (j) (Y k(j) , (Y k* )(j) ),

(14)

where ⎞−1

⎛ Y k*(j)

 ⎟ ⎜ 1 du ⎟ ⎜ φ f (j) (Y k(j) , (Y k* )(j) ) = ⎜ ⎟ , a j,k u + b j,k ⎠ ⎝ Y *(j) − Y (j) k

k

Y k(j)

for a j,k = a j (Y k ) = a j (Y k(1) , . . . , Y k(d) ) and b j,k = b j (Y k(−j) ). Substituting the definition of

 (j) *(j) φ(j) Y into (14) and after some algebraic manipulations, we can derive the , Y k f k following explicit formula: ⎤ ⎡⎛ ⎞ (j)

(j) (j) e ha j (Y k ) − 1 ⎠ ⎥ ⎢ Y k* = e ha j (Y k ) Y k(j) + ⎣⎝h 1E cj + h1E j ⎦b j (Y k(−j) ), a j (Y k(j) )

with the set E j := {y ∈ Rd : a j (y) = 0} and denoting by E cj its complement and 1{E} the (j)

indicator function of the set E. Now substituting the value of Y k* in equation (j)

(j) Y k+1 = Y k* + g (j) (Y k ) ∆W k ,

where g (j) is the j-th row of the diffusion term, we obtain an explicit finite- difference for SDE (1) that with a vector formulation is written as follows: (2) Y k+1 = A(1) k,h Y k + A k,h b(Y k ) + g(Y k ) ∆B k ,

k = 1, . . . , N − 1.

(15)

where ⎛ ⎜ := ⎜ A(1) k,h ⎝

e ha1 (Y k )

0 .. .

0

e ha d (Y k )

⎛ ⎜ ⎜ ⎜ ⎜ (2) A k,h := ⎜ ⎜ ⎜ ⎜ ⎝

⎞ ⎟ ⎟, ⎠

e ha1 (Y k ) − 1 1{E1c } a1 (Y k ) .. . 0

(16)

⎞



0 

 e ha d (Y k ) − 1 1{E cd } a d (Y k )

⎟ ⎛ ⎟ 1{E1 } ⎟ ⎟ ⎜ .. ⎟ + h⎜ . ⎟ ⎝ ⎟ ⎟ 0 ⎠

0 1{E d }

⎞ ⎟ ⎟, ⎠

212 | Silvia Jerez and Saúl Díaz-Infante with b(Y k ) = (b1 (Y k(−1) ), . . . , b d (Y k(−d) ))T . Scheme (15)–(16) is named the linear Steklov method. The LS method is a robust and accurate algorithm, and its strong convergence is proved to SDEs with locally Lipschitz coefficients in ref. [17] using the Higham-Mao-Stuart technique [18]. Next, we state this result. Theorem 5.1. Assume Hypothesis 2.2 holds and considering sufficient regularity of the function a(x), then the linear Steklov method has a unique solution defined by (15)–(16) satisfying that the functions F h (·), φ f h (·) and g h (·) defined by F h (x) = x(j) + h φ f (j) (x),

φ f h (x) = φ f (F h (x)),

g h (x) = g(F h (x)),

are local Lipschitz functions and for all u ∈ Rd and each h fixed, there is a positive constant L φ such that |φ f h (x)| ≤ L φ |f (x)|.

Moreover, for each h fixed, there are positive constants α* and β* such that < φ f h (x), x > ∨ |g h (x)|2 ≤ α* + β* |x|2 ,

∀ x ∈ Rd .

Thus, the linear Steklov method is strongly convergent with a standard order of one-half, i.e., 6 / E sup |Y(t) − y(t)|2 = O(h). 0≤t≤T

where Y(t) is a continuous-time extension of the LS method. In the following, we give some examples to display the construction of this new Steklov method. Example 5.1. Here, we apply the LS method to scalar SDE (10). First, we get the linearized form (13) of the drift term, that for function f (x) = −x3 is given by a(x) = −x2 ,

b = 0,

(17)

and substituting (17) in (16), we obtain the LS approximation for (10) written as follows: 2

Y k+1 = Y k e−hY k + 

1 1.1 ∆W k . log(t k + 1)

(18)

Example 5.2. Now we consider the following generalized stochastic van der Pol oscillator [19]: (19) dy1 (t) = y2 (t)dt, 1 2 

dy2 (t) = −ω2 y1 (t) + σ 1 − μ1 (y1 (t))2 − μ2 (y2 (t))2 y2 (t) dt + σy2 (t)dW(t),

EDS for Stochastic Differential Equations

| 213

for which the linearized drift (13) is defined by a1 (x1 , x2 ) = 0, a2 (x1 , x2 ) =

b1 (x1 , x2 ) = x2 ,

γ(1 − μ1 x21 − μ2 x22 ),

b2 (x1 , x2 ) = −ω2 x1 .

Given E2 := {(x1 , x2 ) ∈ R2 : μ1 x21 + μ2 x22 = 1}, then the LS method is written as follows: Y k+1 = A(1) (h, Y k )Y k + A(2) (h, Y k )b(Y k ) + (0, σY k(2) ∆W k )T ,

(20)

where 

 1 0 , A (h, Y k ) = 0 e ha2 (Y k ) ⎛ ⎞  0 0 1 ⎜ ⎟ (2) ha (Y ) 2 k A (h, Y k ) = ⎝ ⎠+h e −1 0 0 1{E2 } a2 (Y k ) (1)

(21) 0 1{E2 }

 .

6 Almost sure stability In this section, we study the globally almost surely asymptotic stability (as-stability) of the linear Steklov method (15)–(16) for an autonomous scalar SDE assuming that the drift coefficient satisfies f (x) = a(x)x, for some suitable non-linear function a : R → R. Here, we will follow the same technique reported by Mao and Szprunch in ref. [8]. Let us first state a condition that assures the as-stability for a solution process of SDE (1). Theorem 6.1 ([8]). Assume Hypothesis 2.2 holds and suppose that there exists a function z ∈ C (Rd , R+ ) such that 1 < x, f (x) > + |g(x)|2 ≤ −z(x), 2

∀ x ∈ Rd ,

then (i) for any y0 ∈ Rn the solution of the SDE (1), y(t), satisfies lim sup |y(t)|2 ≤ ∞

a.s.

and

t→∞

lim z(y(t)) = 0

t→∞

a.s.

(ii) additionally, if z(x) = 0 only when it is evaluated at x = 0, then lim y(t) = 0

t→∞

a.s.

∀ y 0 ∈ Rd .

Denoting by {Z →} the set of all ω ∈ Ω such that the scalar process Z satisfies that limk→∞ Z k exists and is finite, we give the following lemma that is necessary for our purposes.

214 | Silvia Jerez and Saúl Díaz-Infante Lemma 6.1 ([20]). Let Z = {Z k } be a non-negative semimartingale with E(|Z |) < ∞ and Doob decomposition Z = Z0 + A(1) − A(2) + M, where A(1) := {A(1) } and A(2) := {A(2) } are a.s. non-decreasing predictable k k∈N k k∈N (1) (2) processes with A0 = A0 = 0 and M := {M k }k∈N is a local {Fk }-martingale with M0 = 0. Then     A(1) → := A(2) → ∩ {Z →} a.s. We can now state the as-stability for the linear Steklov method. Theorem 6.2. Assume Hypothesis 2.2 holds and suppose that there is a function z ∈ (Rn , R+ ) and a step-size h* > 0 such that for all x ∈ R and for all h in (0, h* ), 1 < x, f (x) > + |g(x)|2 ≤ −z(x), 2 e2ha(x) − 1 |x |2 + |g h (x)|2 ≤ −z(x). h

(22)

Then the LS method defined by (15)–(16) satisfies lim sup |Y k |2 < ∞

and

k→∞

lim z(Y k ) = 0.

k→∞

In addition, if z(x) = 0 only when x = 0, then lim Y k = 0. k→∞

Proof. In order to use Lemma 6.1, we proceed to construct a conveniently semimartingale as follows: |Y k+1 |2 = |Y k |2 + h2 |φ kfh |2 + |g hk ∆W k |2 + 2h < Y k , φ kfh >

+ 2 < Y k , g hk ∆W k > +2h < φ kfh , g hk ∆W k >,

(23)

where φ kfh = φ f h (Y k , Y k+1 ) and g hk = g h (Y k ). Let us consider ∆M k+1 = |g hk ∆W k+1 |2 − |g hk |2 h + 2 < Y k , g hk ∆W k+1 > +2h < φ f h (Y k ), g hk ∆W k+1 >, which is a local martingale. Fixing N ∈ N, we can rewrite (23) as |Y N+1 |2 = |Y 0 |2 −

N  k=0

Bk h +

N  k=0

∆M k+1 ,

(24)

EDS for Stochastic Differential Equations

| 215

  where B k = − 2 < Y k , φ kfh > +|g hk |2 + h|φ kfh |2 . To prove that (24) is the required decomposition to apply Lemma 6.1, we define φ f h (x) = x

e ha(x) − 1 . h

By algebraic manipulations, we get / 6 2ha(Y k ) − 1) 2 (e 2 + |g h (Y k )| , B k = − |Y k | h

k = 0, . . . , N.

Given that inequality (22) holds, we can deduce that B k ≥ z(Y k ) ≥ 0,

k = 0, . . . N.

5 := Nk=0 B k h is a non-decreasing process. Taking A(1) = 0, Z = |Y k |2 and Thus, A(2) 5NN M N = k=0 ∆M k+1 . By Lemma 6.1, we have {A(1) →} = Ω, and lim sup |Y k |2 < ∞ a.s., k→∞

and

∞ 

z(Y k ) ≤

k=0

∞ 

B k h < ∞.

k=0

Therefore, limk→∞ z(Y k ) = 0, and the theorem follows.

7 Numerical simulations Here, we show the behavior of the explicit Steklov and LS methods for some of SDEs proposed as examples previously. Moreover, we compare both Steklov methods with other popular schemes like the Euler–Maruyama scheme, the Tamed Euler–Maruyama (TEM) scheme [21] and the Partial Linear Implicit Euler Maruyama (PLIE) scheme [19]. Example 7.1. Appleby and Kelly [11] proved that the EM method does not satisfy the almost sure stability for SDEs of the type: dy(t) = −βy(t)|y(t)|p dt + σ(t)|y(t)|ρ dW(t), and the EM approximation explodes to infinity on a finite time when p + 1 > 2ρ. In particular for SDE (10), they deduce conditions for the step-size h and initial condition y(t0 ) = y0 in order to claim with high probability when the EM scheme is as-stable or diverges. More specifically, given h < 0.0384

8   √ 8 √  (i) if Y0 ∈ − 2h + 7 h, 2h − 7 h , then P lim Y kEM = 0 > 0.95, k→∞

216 | Silvia Jerez and Saúl Díaz-Infante

20

5 4 3

15

2 y

1 10

0 −1

5

0

0

5

10

15

20

t

EM h=0.2 TEM h=0.2

LS h = 0.2 Steklov h=0.2

Exact h=2e-05

Fig. 1. Numerical approximation of SDE (10).

8

 √  ; 8 2 √ (ii) if Y0 ∈ − ∞, − 2h − 7 h + 7 h, ∞ , then h   P lim sup Y kEM = ∞ or lim inf Y kEM = −∞ > 0.95. n→∞

n→∞

Now we perform a simulation of SDE (10) with step-size h = 0.2 using the EM, TEM, Steklov and LS schemes with unstable EM initial conditions. The TEM approximation for (10) is given by Y k+1 = Y k −

h Y k3 1 + 1.1 ∆W k . 1 + h |Y k3 | log(t k + 1)

and we recall the Steklov and LS iterations proposed in (11) and (18) respectively. Notice in Fig. 1 how the EM scheme produces spurious solutions, meanwhile the other approximations reproduce the asymptotic behavior. We also show a close-up of the numerical solutions for an initial time period to exhibit the differences between the numerical solutions, and we can observe a better initial precision of the LS approximation in comparison to the other methods. Example 7.2. Finally, we present the numerical results of the LS method for the generalized Van der Pol Oscillator (19). The Van der Pol equation is a classical model

| 217

y2

y1

EDS for Stochastic Differential Equations

4 3 2 1 0 −1 −2

0

2

4

6

8

10

6 4 2 0 −2 −4 −6 −8 0

2

4

t

6

Exact h = 0.0001 EM h = 0.1

8

10

PLIE h= 0.1 LS h= 0.1

Fig. 2. Numerical approximation of SDE (19).

6 Exact h = 0.0001 PLIE h= 0.1 LS h= 0.1

4 2

y2

0 −2 −4 −6 −8 −1.0

−0.5

0.0

0.5

1.0

1.5 y1

Fig. 3. Phase plane of SDE (19).

2.0

2.5

3.0

3.5

218 | Silvia Jerez and Saúl Díaz-Infante

to describe excitable/oscillatory processes, and it is a benchmark test to prove the efficiency of numerical methods. It is known that for the deterministic case,it is common practice to use implicit algorithms because of its stability properties. Now we solve numerically the stochastic version (19) using the LS method (20)–(21) and the PLIE method given by (1) = Y k(1) + hY k(2) , Y k+1 / (2) Y k+1

=

Y k(2) +

−ω

2

Y k(1) + γY k(2) − γ





μ1 Y k(1)

2



+ μ2 Y k(2)

2

6 (2) Y k+1

h + ∆W k .

We take as parameters values; ω = 5, γ = 1, μ1 = 0.05, μ2 = 0.25, σ = 0.5 and initial condition y(0) = (3, 2), and compare the results with a reference solution obtained by the EM scheme with a tiny step-size. In Fig. 2, we can observe that the more accurate method is the LS scheme since the EM and PLIE approximations present a different amplitude that the reference solution as well as some lags on their cycles. A phase portrait of SDE (19) is shown in Fig. 3.

Bibliography [1]

[2]

[3]

[4]

[5]

[6]

[7] [8]

M. Hutzenthaler, A. Jentzen, and P. E. Kloeden. Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally lipschitz continuous coefficients. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467:1563–1576, 2011. Numerical Analysis. M. Hutzenthaler, A. Jentzen, and P. E. Kloeden. Divergence of the multilevel monte carlo Euler method for non-linear stochastic differential equations. Ann. Appl. Probab., 23:1913–1966, 2013. Numerical Analysis. M. Hutzenthaler and A. Jentzen. Numerical approximations of stochastic differential equations with non-globally lipschitz continuous coefficients. Am. Math. Soc., 236:No.1112, 2015. Numerical Analysis. W. Liu and X. Mao. Strong convergence of the stopped Euler–Maruyama method for non-linear stochastic differential equations. Appl. Math. Comput., 223:389–400, 2013. Mathematical Modeling and Computational Methods. X. Wang and S. Gan. The tamed milstein method for commutative stochastic differential equations with non-globally lipschitz continuous coefficients. J. Differ. Equ. Appl., 19(3): 466–490, 2013. Mathematical Modeling and Computational Methods. S. Sabanis. Euler approximations with varying coefficients: the case of superlinearly growing diffusion coefficients. to appear in Ann. Appl. Probab. Mathematical Modeling and Computational Methods. X. Mao. Stochastic differential equations and application. Elsevier, 2007. Mathematical Modeling and Computational Methods. X. Mao and L. Szpruch. Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally lipschitz continuous coefficients. J. Comput. Appl. Math., 238:140–28, 2013. Mathematical Modeling and Computational Methods.

EDS for Stochastic Differential Equations | 219

[9] [10]

[11]

[12]

[13] [14] [15]

[16]

[17]

[18]

[19]

[20] [21]

P. E. Kloeden and E. Platen. Numerical solution of stochastic differential equations. Springer science & business media, 2011. Stochastic Modelling and Applied Probability. P. Matus, U. Irkhin, and M. Lapinska-Chrzczonowicz. Exact difference schemes for time-dependent problems. Comput. Methods Appl. Math., 5:422–448, 2005. Mathematical Modeling and Computational Methods. J. Appleby and C. Kelly. On the local dynamics of polynomial difference equations with fading stochastic perturbations, dynamics of continuous. Dyn. Continuous Discrete Impulsive Syst. Ser. A, 17:401–430, 2010. Mathematical Modeling and Computational Methods. S. Diaz-Infante and S. Jerez. Convergence and asymptotic stability of the explicit steklov method for stochastic differential equations. J. Comput. Appl. Math., 291:36–47, 2016. ISSN 0377-0427. http://dx.doi.org/10.1016/j.cam.2015.01.016. URL http://www.sciencedirect. com/science/article/pii/S037704271500028X. Mathematical Modeling and Computational Methods. R. Khasminskii and G. N. Milstein. Stochastic stability of differential equations. Springer science & business media, volume 66. 2011. Stochastic Modelling and Applied Probability. A. Ludwig. Random dynamical systems. Springer monographs in mathematics, 1998. Mathematical Modeling and Computational Methods. C. T. H. Baker and E. Buckwar. Numerical analysis of explicit one-step methods for stochastic delay differential equations. LMS J. Comput. Math., 3:315–335, 2010. Mathematical Modeling and Computational Methods. T. Caraballo and P. Kloeden. The pathwise numerical approximation of stationary solutions of semilinear stochastic evolution equations. Appl. Math. Optim., 54:401–415, 2006. Mathematical Modeling and Computational Methods. S. Diaz-Infante and S. Jerez. The linear steklov method for sdes with non-globally lipschitz coefficients: Strong convergence and simulation. submitted to J. Comput. Appl. Math.. Mathematical Modeling and Computational Methods. D. J. Higham, X. Mao, and A. M. Stuart. Strong convergence of euler-type methods for non-linear stochastic differential equations. SIAM J. Numer. Anal., 40:1041–1063, 2002. Numerical Analysis. H. Schurz. General theorems for numerical approximation of stochastic processes on the hilbert space. Electron. Trans. Numer. Anal., 16:50–69, 2003. Mathematical Modeling and Computational Methods. R. Liptser and A. N. Shiryaye. Theory of martingales. Springer science & business media, volume 49. 2012. Probability. M. Hutzenthaler, A. Jentzen, and P. E. Kloeden. Strong convergence of an explicit numerical method for sdes with nonglobally lipschitz continuous coefficients. Ann. Appl. Probab., 22:1611–1641, 2013. Numerical Analysis.

Ryszard Kozera, Agnieszka Paradzinska, and Denis Schadinskii

Numerical blow-up time In this section, the EDSs are used for determinations of the blow-up effects in numerical methods approximating IBVP for quasilinear parabolic equations. Discrete analogues of the comparison theorems are proved and used [1, 2].

1 Introduction The comparison theorems are often used to study the properties of solution of partial differential equations [3–5]. Differential inequalities were the first time considered by S. A. Chaplygin in the first half of the 20th century [4]. We know just some discrete analogs of these inequalities. For example, there is discrete analog of the Bihari lemma for explicit FDS [5]. Some generalizations of this Lemma for the implicit FDS one can find in ref. [6]. In works [7–10], interval methods for obtaining two-sided estimates of solution of the IVPs for ordinary differential equation and partial differential equation were considered. There is a growing interest in methods to get two-sided estimates [11, 12]. These methods allow to determine the interval, which contains exact solution and are consistent with the order of accuracy of the numerical method. In this section, we prove the difference analogues of the comparison theorems for solutions of the Cauchy problem for a non-linear ordinary differential equation. These theorems are based on the properties of implicit and explicit FDS [13]. In problems of the parabolic type whose solution develops a singularity [6], such theorems are very important in studying of solution behaviour, stability in the context of the application for the reconstruction of the maria of the Moon [14] and blow-up time [15]. Here, the blow-up is phenomenon when solution tends to infinity in finite time. We find a blow-up condition for solution of the FDS approximating these problems. We present numerical results for non-linear parabolic equation with Neumann boundary condition, whose solution blows up in finite time. Note, the blow-up time of implicit FDS converges to blow-up time of differential problem when the mesh size tends to zero. Last subsection is devoted to the method for obtaining two-sided estimates of solution of parabolic linear equations. The disadvantage of this method is the requirement of a constant sign of the input data derivatives. However, using FDS with variable weights [16], we may be able to generalize the proposed method to the case of functions of alternating signs.

Numerical blow-up time

| 221

2 The Cauchy problem for ODE 3 Statement of the problem and FDS We assume the existence of a classical solution u(t) ∈ C1 (0, T] ∩ C[0, T] of the problem du = f (t, u), dt

0 < t ≤ T,

u(0) = u0 .

Let α(t), β(t) be lower and upper solutions [17] satisfying the following inequalities dα ≤ f (t, α(t)), dt dβ ≥ f (t, β(t)), dt

0 < t ≤ T,

α(0) = u0 ,

(1)

0 < t ≤ T,

β(0) = u0 .

(2)

Then α(t) ≤ u(t) ≤ β(t), for all t belonging to the whole interval of existence [4, 17]. Let us introduce the range of the exact solutions D u = {u(t) : u1 ≤ u(t) ≤ u2 , 0 ≤ t ≤ T } and its neighbourhood, D ε (u) = {u˜ : |u˜ − u| < ε}, which can be sufficiently small. Now let us consider the case when the function f (t, v) satisfies the condition given by the inequalities f2 (t)g2 (v) ≤ f (t, v) ≤ f1 (t)g1 (v), where f k (t), g k (v), k = 1, 2, are continuous and monotonically growing functions for all t ∈ [0, T], v ∈ D ε (u). According to Corollary 4.2 in Chapter III [17], in this case function u(t) is a lower and upper solution of the following differential problem dv1 = f1 (t)g1 (v1 ), dt v k (0) = u0 ,

dv2 = f2 (t)g2 (v2 ), dt

(3)

k = 1, 2,

i.e. v1 (t) ≤ u(t) ≤ v2 (t). ¯ τ = ω τ ∪ {0}, ω τ = {t n+1 = t n + τ n , τ n > 0, n = Let us introduce a non-uniform grid ω 0, 1, . . . , N T −1, t0 = 0, t N T = T } on the interval [0, T]. We approximate problems (1)–(3) using the following FDSs n v n+1 k − vk = φ k (t n+1 )q k (v n+1 v0k = u0 , k = 1, 2, k ), τn n α n+1 τ − ατ ≤ f1 (t n )g1 (α nτ ), α0τ = u0 , τn

(4) (5)

222 | Ryszard Kozera, Agnieszka Paradzinska, and Denis Schadinskii

and n β n+1 τ − βτ ≥ f2 (t n+1 )g2 (β n+1 τ ), τn

β0τ = u0 ,

(6)

where 1 φ k (t n+1 ) = τn ⎡ ⎢ ⎢ q k (v n+1 k )=⎣

t n+1 f k (t)dt, tn

⎤−1

n+1

v k

1 v n+1 − v nk k

v nk

dv ⎥ ⎥ g k (v) ⎦

are chosen from the condition of EDS (see Section 1), and α nτ , β nτ are lower and upper approximate solutions of the Cauchy problem, satisfying differential inequalities (1)–(2). It is worth mentioning that the corresponding discrete analogues of the lower estimation α ≤ v1 and the upper estimation β ≥ v2 can be obtained with use of the explicit and implicit approximation, respectively. Note that the non-uniform grids ¯ τ in each of problems (4)–(6) can be different and the nodes t n do not coincide. ω Lemma 3.1. For any y k , y n ∈ D ε (u)(y n > y k ) and t k , t n ∈ ω τ (t n > t k ) the following estimates ⎡ ⎤−1 n t n y ⎢ 1 1 du ⎥ ⎥ f1 (t)dt⎢ f1 (t k )g1 (y k ) ≤ ⎣ n k tn − tk g1 (u) ⎦ y −y tk

yk

and

f2 (t n )g2 (y n ) ≥

1 tn − tk

t n

⎡ ⎢ f2 (t)dt⎢ ⎣

1 yn − yk

tk

y

⎤−1

n

du ⎥ ⎥ g2 (u) ⎦

yk

hold. The proof follows directly from the mean value theorem. Lemma 3.2. For the solution of problem (4), the following equality takes place F k (v k (t n )) = Φ k (t n ),

tn ∈ ωτ ,

k = 1, 2,

where n

F k (v nk ) =

v k u0

dv , g k (v)

t n Φ k (t n ) =

f k (t)dt. 0

Numerical blow-up time

| 223

Proof. Formula (4) with n = m can be rewritten as m+1

vk 

vm k

dv = g k (v)

t m+1  f k (t)dt. tm

Summation of the last equalities by indices m = 0, 1, . . . , n − 1, yields the required relations. The lemma is proved. Theorem 3.1. Let the classic solution of the Cauchy problem (3) exists with k = 1 and inequalities (5) hold with α nτ ∈ D ε (u), n = 0, 1, . . . , N T . Then m αm τ ≤ v1 ,

m = 0, 1, . . . , N T .

(7)

Proof. For m = 0 we have α0τ = v01 . Let inequality (7) be true for all m = 1, 2, . . . , n. We show that it holds for m = n + 1 as well. In fact, using (5), Lemma 3.1 and the assumption of mathematical induction we get the estimation ≤ α nτ + τ n f1 (t n )g1 (α nτ ) ≤ v1n + τ n f1 (t n )g1 (v1n ) α n+1 τ ≤ v1n + τ n φ1 (t n+1 )q1 (v1n+1 ) = v1n+1 . Theorem is proved. Remark 3.1. In refs. [5, 18] one can find the proof of Theorem 3.1 for f1 (t) = 1 in more complicated way. Theorem 3.2. Let the classic solution of the Cauchy problem (3) exist with k = 2 and the grid function β nτ ∈ D ε (u), n = 0, 1, . . . , N T satisfies inequality (6). Then m βm τ ≥ v2 ,

m = 0, 1, . . . , N T .

(8)

Proof. For m = 0 we have β 0τ = v02 . Let condition (8) be satisfied for all m = 1, 2, . . . , n. We prove that it is true for m = n + 1. Then, using Lemmas 3.1 and 3.2 and (6), we obtain the following inequalities n+1

βτ 

β nτ

dv ≥ g2 (v)

β n+1 τ



u0

dv ≥ g2 (v)

t n+1 f2 (t)dt, tn

t n+1 f2 (t)dt = Φ2 (t n+1 ), 0

224 | Ryszard Kozera, Agnieszka Paradzinska, and Denis Schadinskii n+1

n+1

v2 u0

dv + g2 (v)

βτ 

dv ≥ g2 (v)

v2n+1

t n+1 f2 (t)dt, 0

β n+1 τ



dv ≥ 0, g2 (v)

≥ v2n+1 . β n+1 τ

v2n+1

Theorem is therefore proved. Corollary 3.1. Let for the FDS y αt = f1 (t n )g1 (y nα ),

y βt = f2 (t n+1 )g2 (y n+1 β ),

y0α = y0β = u0 , solutions y α , y β ∈ D ε (u) exist and unique and inequalities (5), (6) be satisfied. Then for ¯ τ the following relations all t n ∈ ω α nτ ≤ y nα ≤ v1n ,

β nτ ≥ y nβ ≥ v2n ,

m = 0, 1, . . . , N T ,

(9)

hold. Example 3.1. Let f (t, u) = t r u p , r > 0, p > 1, u0 > 0. Then in (9) we have v1n = v2 = u(t n ) =

u0 p−1 r+1 1 − p−1 r+1 u 0 t n

1/(p−1) .

Consider important corollaries of the discrete analogues of Theorems 3.1 and 3.2. First from inequalities (7), (9) it follows the boundedness of the difference solution on the arbitrary segment [0, t1 ]  t1 < T blow−up ,

T blow−up =

r+1 (p − 1)u0p−1

1  r+1

.

The occurrence of blow-up phenomena in the case of (7) is only possible. From the second estimate u0

y nβ ≥ 

p − 1 p−1 r+1 1− u t r+1 0 n

it follows that global solution of the FDS exists [19].

1 p−1

,

Numerical blow-up time

| 225

4 Solvability of the implicit FDS for ODE In this subsection, we prove the existence and the uniqueness of the solution of implicit FDS n β n+1 τ − βτ = f1 (t n+1 )f2 (β n+1 τ ), τn

β0τ = u0 ,

(10)

u(0) = u0 ,

(11)

approximating the solution of the Cauchy problem du = f1 (t)f2 (u), dt

0 ≤ t ≤ T,

f2 (u) ∈ C1 (D ε (u)). Here, we use known results for proving existence and uniqueness conditions. We present this proof only for obtaining bound of τ n under which solution of implicit FDS exists and is unique. According to (10) let us consider the non-linear equation of the form x − β nτ = f1 (t n+1 )f2 (x). τn

(12)

in the set R arises. For the clarity Here, the question about the existence of root x = β n+1 τ of presentation, let us rewrite the equation (12) as x = Φ(x),

(13)

where Φ(x) = β nτ + τ n f1 (t n+1 )f2 (x). Let us introduce the segment U r (a) = {x : |x − a| ≤ r} {D}ε (u). Note that Φ(x) ∈ C1 (D ε (u)). Below we use the result from [20]. Lemma 4.1. If the following condition is satisfied 

|Φ (x)| ≤ q < 1,

|Φ(a) − a | ≤ (1 − q)r,

(14)

then equation (13) has a unique solution x * in the interval U r (a) and the simple iteration method x k+1 = Φ(x k ),

k = 0, 1, . . .

226 | Ryszard Kozera, Agnieszka Paradzinska, and Denis Schadinskii converges to x * for an arbitrary initial point x0 ∈ U r (a). Moreover, the following estimation |x k − x* | ≤ q k |x0 − x* |,

k = 0, 1, 2 . . . .

holds. We will show that the conditions of the lemma are fulfilled when 1

Θ(τ n ) = f1 (t n + τ n )τ n
0, n = 0, 1, . . . , N T − 1, t0 = 0, t N T = T } ω h = {x i = ih, h = l/N, n = 0, 1, . . . , N }. Now consider the FDS y), y t = (ay x¯ )(σ) x + f (ˆ

(x, t) ∈ ω,

ω = ωh × ωτ ,

(23)

where y(x, 0) = u0 (x),

a i = 0.5(k(y i−1 ) + k(y i )),

h (a1 y x,0 )(σ) − (y t,0 + f (ˆy0 )) 2 h = (a N y x¯ ,N )(σ) + (y t,N + f (ˆy N )) = 0, 2

(24)

which approximates problem (18)–(19) with an order O(h2 + τ). Multiplying the difference equation (23) by h and summing the result over the internal nodes ω h , we obtain N−1 

hy t = (a N y x¯ ,N )(σ) − (a1 y x,0 )(σ) +

i=1

N−1 

hf (ˆy i ).

i=1

Using the homogeneous boundary conditions (24) we rewrite the last expression as ⎛ ⎞ N−1  h 1⎝h hy t + y t,N ⎠ y + l 2 t,0 2 i=1 ⎞ ⎛ N−1  h 1⎝h hf (ˆy i ) + f (ˆy N )⎠. = f (ˆy ) + l 2 0 2 i=1

230 | Ryszard Kozera, Agnieszka Paradzinska, and Denis Schadinskii

Since the sum of the coefficients of the function f is equal to 1 we can apply the discrete analogue Jensen’s inequality (21). The following estimate ⎞ ⎞ ⎛ ⎛ N−1 N−1   h h h 1⎝h y i + y N ⎠ ≥ f ⎝ yˆ 0 + h y +h yˆ i + yˆ N ⎠, l 2 0 2 2 2 i=1

is valid. For v h =

1 l



h 2 y0

+

N−1 5 i=1

i=1

t

 hy i + 2h y N , the last estimate can be rewritten in the

form of v ht ≥ f (ˆv h ). Using the Theorem 1 we arrive at the estimate v nh ≥ w(t n ),

(25)

where w(t) is the solution of the following Cauchy problem dw = f (w), dt

w(0) = v0h .

Consider the following initial-boundary value problem  ∂u Ls ∂u ∂ Ls u + u2 , − < x < , = ∂t ∂x ∂x 2 2

(26)

0 < t < T0 ,

u(x, 0) = u0 (x), 

   Ls Ls ∂u ∂u − , t = k(u) k(u) , t = 0, ∂x 2 ∂x 2

which has solution [15] u(x, t) = (T0 − t)−1 u0 (x), T0 = T blow−up = 1, ⎧ ⎪ ⎨ 4 cos2 πx , |x| < L s /2; Ls u0 (x) = 3 , ⎪ ⎩0, |x | ≥ L s /2. √

where L s = 2 2π. When T is finite and the solution u develops a singularity in a finite time, namely lim ||u(t)||∞ = ∞, t→T

Numerical blow-up time

| 231

Tab. 1. Numerical results. Approximation of sources Explicit Implicit

h

N

τN

yN

tN

0.27 0.27

46211 45516

1.01 · 10−152 1.00 · 10−152

1.00 · 10150 1.01 · 10150

1.017597 0.976533

then we say that solution u blows up in a finite time and time T is called the blow-up time of the solution u. Similarly, if ||y n ||∞,h = max |y n | ≥ ω(t n ), ¯h x∈ω

lim ω(t) = ∞,

t→T h

then we say that y ni blows up for finite time T h ≤ T. The theoretical study of blow-up of differential and numerical solutions for quasilinear parabolic equations has been the subject of investigation of many authors [19, 21, 22]. According to (25), (26), solution of the FDS (23)–(24) satisfies max |y(x, t)| ≥ v nh ≥ ω(t n ) =

¯τ (x,t)∈ω

v0h , 1 − v0h t

v0h =

2 . 3

(27)

Inequality (27) implies that solution blows up in finite time. In our numerical experiment, we use the conservative FDS (23)–(24) with σ = 0.5. The time steps are set as τ n = 0.03/||y n ||∞,h . Computing is stopped when y N > 10150 . Numerical result for FDS with the explicit approximation of source with weight σ = 0, 5 is also presented. It is widely known that blow-up time of implicit FDS is less than the blow-up time of differential problem T0 = 1 and vice versa for explicit FDS. Numerical results are presented in Table 1. Here, we can see that solution of implicit FDS very rapidly tends to infinity, and blow-up time is less than T1 in accordance with (27). For the implicit FDS, the blow-up time is less than T1 , and this corresponds to the above estimate (25). Since v0h = v0 , blow-up time of differential problem T0 is less than T1 that follows from (22).

Bibliography [1]

[2]

[3]

P. P. Matus, R. Kozera, A. Paradzinska, and D. A. Schadinskii. Discrete analogs of the comparison theorem and two-side estimates of solution of parabolic equations. AIP Conf. Proc., 1648(660017):1–4, 2015. P. P. Matus, R. Kozera, A. Paradzinska, and D. A. Schadinskii. Discrete analogs of the comparison theorem and two-side estimates of solution of parabolic equations. Appl. Math. Inf. Sci., 2015. A. G. Sveshnikov, A. B. Al’shin, M. O. Korpusov, and Yu. D. Pletner. Linear and non-linear equations of Sobolev type. Fizmatlit, Moskva, 2007.

232 | Ryszard Kozera, Agnieszka Paradzinska, and Denis Schadinskii

[4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

[14] [15] [16] [17] [18] [19] [20] [21] [22]

A. B. Vasil’eva and N. N. Nefedov. Comparison theorems. Chaplygin method of differential inequalities. Phy. Fac. MSU, Moskva, 2007. V. B. Demidovich. Stability criterion for difference equations. Differ. Equ., 5:1247–1255, 1969. S. I. Pokhozhaev. Multidimensional scalar conservation laws. Sbornik: Math., 194:151–164, 2003. B. S. Dobronetc. Interval mathematics. Krasnoyarsk State University, 2004. Yu. I. Shokin. Interval analysis. Nauka, Novosibirsk, 1981. R. E. Moore. Methods and applications of interval analysis. SIAM Studies in Applied Mathematics, Philadelphia, 1979. F. Bierbaum and K.-P. Schwiertz. A bibliography on interval-mathematics. J. Comput. Methods Appl. Math., 4:59–86, 1978. R. E. Moore and J. B. Kioustelidis. A simple test for accuracy of approximate solutions to non-linear (or linear) systems. SIAM J. Numer. Anal., 17:521–529, 1980. N. Bakhvalov. Numerical methods (analysis, algebra, ordinary differential equations). Nauka, Moscow, 1975. P. P. Matus. On the role of conservation laws in the problem on the occurrence of unstable solutions for quasilinear parabolic equations and their approximations. Differ. Equ., 49(7): 883–894, 2013. R. Kozera and R. Klette. Evaluation of numerical solution schemes for differential equations. In Performance characterization in computer vision, pages 153–166. Springer, 2000. A. Samarskii, V. Galaktionov, S. Kurdyumov, and A. Mikhailov. Blow-up in quasilinear parabolic equations. De Gruyter, Berlin, 1995. A. A. Samarskii, P. P. Matus, and P. N. Vabishchevich. Difference schemes with operator factors. Springer, 2002. P. Hartman. Ordinary differential equations. Wiley, New York, 1964. M. Pinto. Integral inequalities of Bihari-type and applications. Funkcialaj Ekvacioj, 33:387–403, 1990. M. Fila, H. Ninomiya, and J. L. Vázquez. Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems. Discrete Continuous Dyn. Syst., 14(1):63, 2006. A. A. Samarskii and A. V. Goolin. Numerical Methods. Nauka, Moscow, 1989. (in Russian). A. Friedman. Remarks on non-linear parabolic equations. Proc. Symp. in Appl. Math. AMS, 13:3–23, 1965. I. M. Nikol’skii. Unbounded solutions of a non-linear parabolic equation. Comput. Math. Model., 20(4):339–347, 2009.

List of contributors Ronald E. Mickens Department of Physics, Clark Atlanta University, Atlanta, GA 30314, USA, [email protected] Talitha M. Washington Department of Mathematics, Howard University, Washington, DC 20059, USA, [email protected]

Ivan Gavrilyuk Berufsakademie Thueringen, Staatliche Studienakademie, Am Wartenberg 2, D-99817 Eisenach, Germany, [email protected]

Myroslav Kutniv Lviv Polytechnic National University, 12 St. Bandery Str., 79013 Lviv, Ukraine, [email protected]

Volodymyr Makarov Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivs’ka Str., 01601 Kyiv-4, Ukraine, [email protected] Silvia Jerez Department of Applied Mathematics, CIMAT, Guanajuato, Gto., 36240, Mexico, [email protected]

Saúl Díaz-Infante Department of Applied Mathematics, CIMAT, Guanajuato, Gto., 36240, Mexico, [email protected] Ryszard Kozera Institute of Mathematics and Computer Science, The John Paul II Catholic University of Lublin, Al. Raclawickie 14, 20-950 Lublin, Poland; Faculty of Applied Informatics and Mathematics, Warsaw University of Life Sciences - SGGW, Nowoursynowska 159, 02-776 Warsaw, Poland, [email protected]

Agnieszka Paradzinska Institute of Mathematics and Computer Science, The John Paul II Catholic University of Lublin, Al. Raclawickie 14, 20-950 Lublin, Poland, [email protected]

Denis Schadinskii Institute of Mathematics, NAS of Belarus, 11 Surganov St., 220072 Minsk, Belarus, [email protected]