Difference methods for singular perturbation problems / Grigory I. Shishkin, Lidia P. Shishkina.

Format:

Book

Author/Creator:

Shishkin, G. I.

Contributor:

Shishkina, Lidia P.

Series:

Chapman & Hall/CRC monographs and surveys in pure and applied mathematics ; 140.

Chapman & Hall/CRC monographs and surveys in pure and applied mathematics ; 140

Language:

English

Subjects (All):

Singular perturbations (Mathematics).

Difference equations--Numerical solutions.

Algebra, Abstract.

Physical Description:

xv, 393 pages ; 25 cm.

Place of Publication:

Boca Raton : CRC Press, [2009]

Summary:

Difference Methods for Singular Perturbation Problems focuses on the development of robust difference schemes for wide classes of boundary value problems. It justifies the [epsilon]-uniform convergence of these schemes and surveys the latest approaches important for further progress in numerical methods.

The first part of the book explores boundary value problems for elliptic and parabolic reaction-diffusion and convection-diffusion equations in n-dimensional domains with smooth and piecewise-smooth boundaries. Containing material published mainly in the last four years, the second section focuses on problems with boundary layers and additional singularities generated by nonsmooth data, unboundedness of the domain, and the perturbation vector parameter.

Co-authored by the creator of the Shishkin mesh, this book presents a systematic, detailed development of approaches to construct [epsilon] uniformly convergent finite difference schemes for broad classes of singularly perturbed boundary value problems.

Contents:

I Grid approximations of singular perturbation partial differential equations 1

1.1 The development of numerical methods for singularly perturbed problems 3

1.2 Theoretical problems in the construction of difference schemes 6

1.3 The main principles in the construction of special schemes 8

1.4 Modern trends in the development of special difference schemes 10

1.5 The contents of the present book 11

1.6 The present book 12

1.7 The audience for this book 16

2 Boundary value problems for elliptic reaction-diffusion equations in domains with smooth boundaries 17

2.1 Problem formulation. The aim of the research 17

2.2 Estimates of solutions and derivatives 19

2.3 Conditions ensuring [epsilon]-uniform convergence of difference schemes for the problem on a slab 26

2.3.1 Sufficient conditions for [epsilon]-uniform convergence of difference schemes 26

2.3.2 Sufficient conditions for [epsilon]-uniform approximation of the boundary value problem 29

2.3.3 Necessary conditions for distribution of mesh points for [epsilon]-uniform convergence of difference schemes. Construction of condensing meshes 33

2.4 Monotone finite difference approximations of the boundary value problem on a slab. [epsilon]-uniformly convergent difference schemes 38

2.4.1 Problems on uniform meshes 38

2.4.2 Problems on piecewise-uniform meshes 44

2.4.3 Consistent grids on subdomains 51

2.4.4 [epsilon]-uniformly convergent difference schemes 57

2.5 Boundary value problems in domains with curvilinear boundaries 58

2.5.1 A domain-decomposition-based difference scheme for the boundary value problem on a slab 58

2.5.2 A difference scheme for the boundary value problem in a domain with curvilinear boundary 67

3 Boundary value problems for elliptic reaction-diffusion equations in domains with piecewise-smooth boundaries 75

3.1 Problem formulation. The aim of the research 75

3.2 Estimates of solutions and derivatives 76

3.3 Sufficient conditions for [epsilon]-uniform convergence of a difference scheme for the problem on a parallelpiped 85

3.4 A difference scheme for the boundary value problem on a parallelepiped 89

3.5 Consistent grids on subdomains 97

3.6 A difference scheme for the boundary value problem in a domain with piecewise-uniform boundary 102

4 Generalizations for elliptic reaction-diffusion equations 109

4.1 Monotonicity of continual and discrete Schwartz methods 109

4.2 Approximation of the solution in a bounded subdomain for the problem on a strip 112

4.3 Difference schemes of improved accuracy for the problem on a slab 120

4.4 Domain-decomposition method for improved iterative schemes 125

5 Parabolic reaction-diffusion equations 133

5.1 Problem formulation 133

5.2 Estimates of solutions and derivatives 134

5.3 [epsilon]-uniformly convergent difference schemes 145

5.3.1 Grid approximations of the boundary value problem 146

5.3.2 Consistent grids on a slab 147

5.3.3 Consistent grids on a parallelepiped 154

5.4 Consistent grids on subdomains 158

5.4.1 The problem on a slab 158

5.4.2 The problem on a parallelepiped 161

6 Elliptic convection-diffusion equations 165

6.1 Problem formulation 165

6.2 Estimates of solutions and derivatives 166

6.2.1 The problem solution on a slab 166

6.2.2 The problem on a parallelepiped 169

6.3 On construction of [epsilon]-uniformly convergent difference schemes under their monotonicity condition 176

6.3.1 Analysis of necessary conditions for [epsilon]-uniform convergence of difference schemes 177

6.3.2 The problem on a slab 180

6.3.3 The problem on a parallelepiped 183

6.4 Monotone [epsilon]-uniformly convergent difference schemes 185

7 Parabolic convection-diffusion equations 191

7.1 Problem formulation 191

7.2 Estimates of the problem solution on a slab 192

7.3 Estimates of the problem solution on a parallelepiped 199

7.4 Necessary conditions for [epsilon]-uniform convergence of difference schemes 206

7.5 Sufficient conditions for [epsilon]-uniform convergence of monotone difference schemes 210

7.6 Monotone [epsilon]-uniformly convergent difference schemes 213

II Advanced trends in [epsilon]-uniformly convergent difference methods 219

8 Grid approximations of parabolic reaction-diffusion equations with three perturbation parameters 221

8.2 Problem formulation. The aim of the research 222

8.3 A priori estimates 224

8.4 Grid approximations of the initial-boundary value problem 230

9 Application of widths for construction of difference schemes for problems with moving boundary layers 235

9.2 A boundary value problem for a singularly perturbed parabolic reaction-diffusion equation 237

9.2.1 Problem (9.2), (9.1) 237

9.2.3 The aim of the research 240

9.3 A priori estimates 241

9.4 Classical finite difference schemes 243

9.5 Construction of [epsilon]-uniform and almost [epsilon]-uniform approximations to solutions of problem (9.2), (9.1) 246

9.6 Difference scheme on a grid adapted in the moving boundary layer 251

9.7 Remarks and generalizations 254

10 High-order accurate numerical methods for singularly perturbed problems 259

10.2 Boundary value problems for singularly perturbed parabolic convection-diffusion equations with sufficiently smooth data 261

10.2.1 Problem with sufficiently smooth data 261

10.2.2 A finite difference scheme on an arbitrary grid 262

10.2.3 Estimates of solutions on uniform grids 263

10.2.4 Special [epsilon]-uniform convergent finite difference scheme 263

10.2.5 The aim of the research 264

10.3 A priori estimates for problem with sufficiently smooth data 265

10.4 The defect correction method 266

10.5 The Richardson extrapolation scheme 270

10.6 Asymptotic constructs 273

10.7 A scheme with improved convergence for finite values of [epsilon] 275

10.8 Schemes based on asymptotic constructs 277

10.9 Boundary value problem for singularly perturbed parabolic convection-diffusion equation with piecewise-smooth initial data 280

10.9.1 Problem (10.56) with piecewise-smooth initial data 280

10.9.2 The aim of the research 281

10.10 A priori estimates for the boundary value problem (10.56) with piecewise-smooth initial data 282

10.11 Classical finite difference approximations 285

10.12 Improved finite difference scheme 287

11 A finite difference scheme on a priori adapted grids for a singularly perturbed parabolic convection-diffusion equation 289

11.2 Problem formulation. The aim of the research 290

11.3 Grid approximations on locally refined grids that are uniform in subdomains 293

11.4 Difference scheme on a priori adapted grid 297

11.5 Convergence of the difference scheme on a priori adapted grid 303

12 On conditioning of difference schemes and their matrices for singularly perturbed problems 309

12.2 Conditioning of matrices to difference schemes on piecewise-uniform and uniform meshes. Model problem for ODE 311

12.3 Conditioning of difference schemes on uniform and piecewise-uniform grids for the model problem 316

12.4 On conditioning of difference schemes and their matrices for a parabolic problem 323

13 Approximation of systems of singularly perturbed elliptic reaction-diffusion equations with two parameters 327

13.2 Problem formulation. The aim of the research 328

13.3 Compatibility conditions.

Some a priori estimates 330

13.4 Derivation of a priori estimates for the problem (13.2) under the condition (13.5) 333

13.5 A priori estimates for the problem (13.2) under the conditions (13.4), (13.6) 341

13.6 The classical finite difference scheme 343

13.7 The special finite difference scheme 345

13.8 Generalizations 348

14 Survey 349

14.1 Application of special numerical methods to mathematical modeling problems 349

14.2 Numerical methods for problems with piecewise-smooth and nonsmooth boundary functions 351

14.3 On the approximation of solutions and derivatives 352

14.4 On difference schemes on adaptive meshes 354

14.5 On the design of constructive difference schemes for an elliptic convection-diffusion equation in an unbounded domain 357

14.5.1 Problem formulation in an unbounded domain. The task of computing the solution in a bounded domain 357

14.5.2 Domain of essential dependence for solutions of the boundary value problem 359

14.5.3 Generalizations 363

14.6 Compatibility-conditions for a boundary value problem on a rectangle for an elliptic convection-diffusion equation with a perturbation vector parameter 364

14.6.1 Problem formulation 365

14.6.2 Compatibility conditions 366.

Notes:

"A Chapman & Hall book."

Includes bibliographical references (pages 371-388) and index.

ISBN:

9781584884590

1584884592

OCLC:

232327369