Difference Methods for Singular Perturbation Problems: 1st Edition (Hardback) book cover

Difference Methods for Singular Perturbation Problems

1st Edition

By Grigory I. Shishkin, Lidia P. Shishkina

Chapman and Hall/CRC

408 pages

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pub: 2008-09-22
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Description

  Difference Methods for Singular Perturbation Problems focuses on the development of robust difference schemes for wide classes of boundary value problems. It justifies the ε-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. The authors develop a technique for constructing and justifying ε uniformly convergent difference schemes for boundary value problems with fewer restrictions on the problem data.

Containing information 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. This part also studies both the solution and its derivatives with errors that are independent of the perturbation parameters.

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

Reviews

"This book focuses on the development of robust difference schemes for wide classes of boundary value problems. … The book can be of use for scientists, researchers, students, and professionals in the field of developing numerical methods for singularly perturbed problems and also for anybody interested in mathematical modelling or in the fields where the problems with boundary and interior layers arise naturally."

EMS Newsletter, December 2009

Table of Contents

Preface

Part I: Grid Approximations of Singular Perturbation Partial Differential Equations

Introduction

Boundary Value Problems for Elliptic Reaction-Diffusion Equations in Domains with Smooth Boundaries

Boundary Value Problems for Elliptic Reaction-Diffusion Equations in Domains with Piecewise-Smooth Boundaries

Generalizations for Elliptic Reaction-Diffusion Equations

Parabolic Reaction-Diffusion Equations

Elliptic Convection-Diffusion Equations

Parabolic Convection-Diffusion Equations

Part II: Advanced Trends in ε Uniformly Convergent Difference Methods

Grid Approximations of Parabolic Reaction-Diffusion Equations with Three Perturbation Parameters

Application of Widths for Construction of Difference Schemes for Problems with Moving Boundary Layers

High-Order Accurate Numerical Methods for Singularly Perturbed Problems

A Finite Difference Scheme on a priori Adapted Grids for a Singularly Perturbed Parabolic Convection-Diffusion Equation

On Conditioning of Difference Schemes and Their Matrices for Singularly Perturbed Problems

Approximation of Systems of Singularly Perturbed Elliptic Reaction-Diffusion Equations with Two Parameters

Survey

References

About the Series

Monographs and Surveys in Pure and Applied Mathematics

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Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT007000
MATHEMATICS / Differential Equations
SCI040000
SCIENCE / Mathematical Physics