Exact Finite-Difference Schemes

by Sergey Lemeshevsky (Editor), Piotr Matus (Editor), Dmitriy Poliakov (Editor)

Overview

Exact Finite-Difference Schemes is a first overview of the topic also describing the state-of-the-art in this field of numerical analysis. Construction of exact difference schemes for various parabolic and elliptic partial differential equations are discussed, including vibrations and transport problems. After this, applications are discussed, such as the discretisation of ODEs and PDEs and numerical methods for stochastic differential equations.

Contents:
Basic notation
Preliminary results
Hyperbolic equations
Parabolic equations
Use of exact difference schemes to construct NSFD discretizations of differential equations
Exact and truncated difference schemes for boundary-value problem
Exact difference schemes for stochastic differential equations
Numerical blow-up time
Bibliography

Product Details

ISBN-13:	9783110489644
Publisher:	De Gruyter
Publication date:	09/26/2016
Pages:	246
Product dimensions:	6.69(w) x 9.45(h) x (d)
Age Range:	18 Years

About the Author

Sergey Lemeshevsky, Piotr Matus and Dmitrii Polyakov Institute of Mathematics, National Academy of Sciences, Belarus.

Table of Contents

Basic notation xiii

Preliminary results Sergey Lemeshevsky Piotr Matus Dmitriy Poliakov 1

1 Exact finite-difference schemes for ordinary differential equations 1

2 Effective computations of integrals 4

Bibliography 6

Hyperbolic equations Sergey Lemeshevsky Piotr Matus Dmitriy Poliakov 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

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

Parabolic equations Sergey Lemeshevsky Piotr Matus Dmitriy Poliakov 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

Use of exact difference schemes to construct NSFD discretizations of differential equations Ronald E. Mickens Talitha M. Washington 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

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

Exact and truncated difference schemes for boundary-value problem Ivan Gavrilyuk Myraslav Kutniv Volodymyr Makarov 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

Exact difference schemes for stochastic differential equations Silvia Jerez Saúl Díaz-Infante 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

Numerical blow-up time Ryszard Kozera Agnieszka Paradzinska Denis Schadinskii 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