Finite Difference Schemes and Partial Differential Equations.pdf

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Finite Difference Schemes and Partial Differential Equations

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Finite Difference Schemes and Partial Differential Equations Second Edition John C. Strikwerda University of Wisconsin-Madison Madison, Wisconsin slam Society for Industrial and Applied Mathematics Philadelphia

Copyright © 2004 by the Society for Industrial and Applied Mathematics. This SIAM edition is a second edition of the work first published by Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1989. 1 0 9 8 76 5 4 3 21 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3f.OO University City Science Center, Philadelphia, PA 19104-2688. Library of Congress Cataloging-in-Publication Data Strikwerda, John C, 1947- Finite difference schemes and partial differential equations / John C. Strikwerda. — 2nd ed. p. cm. Includes bibliographical references and index. ISBN 0-89871-567-9 1. Differential equations, Partial—Numerical solutions. 2. Finite differences. I. Title. QA374.S88 2004 518'.64—dc22 2004048714 5LLELJTL is a registered trademark.

Contents Preface to the Second Edition Preface to the First Edition 1 Hyperbolic Partial Differential Equations 1.1 Overview of Hyperbolic Partial Differential Equations 1.2 Boundary Conditions 1.3 Introduction to Finite Difference Schemes 1.4 Convergence and Consistency 1.5 Stability 1.6 The Courant-Friedrichs-Lewy Condition 2 Analysis of Finite Difference Schemes 2.1 Fourier Analysis 2.2 Von Neumann Analysis 2.3 Comments on Instability and Stability 3 Order of Accuracy of Finite Difference Schemes 3.1 Order of Accuracy 3.2 Stability of the Lax-Wendroff and Crank-Nicolson Schemes 3.3 Difference Notation and the Difference Calculus 3.4 Boundary Conditions for Finite Difference Schemes 3.5 Solving Tridiagonal Systems 4 Stability for Multistep Schemes 4.1 Stability for the Leapfrog Scheme 4.2 Stability for General Multistep Schemes 4.3 The Theory of Schur and von Neumann Polynomials 4.4 The Algorithm for Schur and von Neumann Polynomials ix xi 1 1 9 16 23 28 34 37 37 47 58 61 61 76 78 85 88 95 95 103 108 117 v

vi Contents 5 Dissipation and Dispersion 5.1 Dissipation 5.2 Dispersion 5.3 Group Velocity and the Propagation of Wave Packets 6 Parabolic Partial Differential Equations 6.1 Overview of Parabolic Partial Differential Equations 6.2 Parabolic Systems and Boundary Conditions 6.3 Finite Difference Schemes for Parabolic Equations 6.4 The Convection-Diffusion Equation 6.5 Variable Coefficients 7 Systems of Partial Differential Equations in Higher Dimensions 7.1 Stability of Finite Difference Schemes for Systems of Equations 7.2 Finite Difference Schemes in Two and Three Dimensions 7.3 The Alternating Direction Implicit Method 8 Second-Order Equations 8.1 Second-Order Time-Dependent Equations 8.2 Finite Difference Schemes for Second-Order Equations 8.3 Boundary Conditions for Second-Order Equations 8.4 Second-Order Equations in Two and Three Dimensions 9 Analysis of Well-Posed and Stable Problems 9.1 The Theory of Well-Posed Initial Value Problems 9.2 Well-Posed Systems of Equations 9.3 Estimates for Inhomogeneous Problems 9.4 The Kreiss Matrix Theorem 121 121 125 130 137 137 143 145 157 163 165 165 168 172 187 187 193 199 202 205 205 213 223 225 10 Convergence Estimates for Initial Value Problems 235 235 248 252 259 262 267 270 10.1 Convergence Estimates for Smooth Initial Functions 10.2 Related Topics 10.3 Convergence Estimates for Nonsmooth Initial Functions 10.4 Convergence Estimates for Parabolic Differential Equations 10.5 The Lax-Richtmyer Equivalence Theorem 10.6 Analysis of Multistep Schemes 10.7 Convergence Estimates for Second-Order Differential Equations

Contents 11 Well-Posed and Stable Initial-Boundary Value Problems 11.1 Preliminaries 11.2 Analysis of Boundary Conditions for the Leapfrog Scheme 11.3 The General Analysis of Boundary Conditions 11.4 Initial-Boundary Value Problems for Partial Differential Equations 11.5 The Matrix Method for Analyzing Stability 12 Elliptic Partial Differential Equations and Difference Schemes 12.1 Overview of Elliptic Partial Differential Equations 12.2 Regularity Estimates for Elliptic Equations 12.3 Maximum Principles 12.4 Boundary Conditions for Elliptic Equations 12.5 Finite Difference Schemes for Poisson's Equation 12.6 Polar Coordinates 12.7 Coordinate Changes and Finite Differences 13 Linear Iterative Methods 13.1 Solving Finite Difference Schemes for Laplace's Equation in a Rectangle 13.2 Eigenvalues of the Discrete Laplacian 13.3 Analysis of the Jacobi and Gauss-Seidel Methods 13.4 Convergence Analysis of Point SOR 13.5 Consistently Ordered Matrices 13.6 Linear Iterative Methods for Symmetric, Positive Definite Matrices 13.7 The Neumann Boundary Value Problem 14 The Method of Steepest Descent and the Conjugate Gradient Method 14.1 The Method of Steepest Descent 14.2 The Conjugate Gradient Method 14.3 Implementing the Conjugate Gradient Method 14.4 A Convergence Estimate for the Conjugate Gradient Method 14.5 The Preconditioned Conjugate Gradient Method A Matrix and Vector Analysis A.I Vector and Matrix Norms A.2 Analytic Functions of Matrices vii 275 275 281 288 300 307 311 311 315 317 322 325 333 335 339 339 342 345 351 357 362 365 373 373 377 384 387 390 399 399 406

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