Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems (Classics in Applied Mathematics)

356 Pages·2007·22.43 MB·English

by  Randall Leveque

OT98_LevequeFM2.qxp 6/4/2007 10:20 AM Page 1
Finite Difference Methods
for Ordinary and Partial
Differential Equations
OT98_LevequeFM2.qxp 6/4/2007 10:20 AM Page 2
OT98_LevequeFM2.qxp 6/4/2007 10:20 AM Page 3
Finite Difference Methods
for Ordinary and Partial
Differential Equations
Steady-State and Time-Dependent Problems
Randall J. LeVeque
University of Washington
Seattle, Washington
Society for Industrial and Applied Mathematics • Philadelphia
OT98_LevequeFM2.qxp 6/4/2007 10:20 AM Page 4
Copyright © 2007 by the Society for Industrial and Applied Mathematics.
10 9 8 7 6 5 4 3 2 1
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, 3600
University City Science Center, Philadelphia, PA 19104-2688.
Trademarked names may be used in this book without the inclusion of a trademark symbol.
These names are used in an editorial context only; no infringement of trademark is intended.
MATLAB is a registered trademark of The MathWorks, Inc. For MATLAB product information,
please contact The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098 USA, 508-
647-7000, Fax: [phone], [email protected], www.mathworks.com.
Library of Congress Cataloging-in-Publication Data
LeVeque, Randall J., 1955-
Finite difference methods for ordinary and partial differential equations : steady-state and
time-dependent problems / Randall J. LeVeque.
p.cm.
Includes bibliographical references and index.
ISBN 978-0-898716-29-0 (alk. paper)
1. Finite differences. 2. Differential equations. I. Title.
QA431.L548 2007
515’.35—dc22 2007061732
Partial royalties from the sale of this book are placed in a fund to help students
attend SIAM meetings and other SIAM-related activities. This fund is administered
by SIAM, and qualified individuals are encouraged to write directly to SIAM for
guidelines.
is a registered trademark.
OT98_LevequeFM2.qxp 6/4/2007 10:20 AM Page 5
To my family,
Loyce, Ben, Bill, and Ann
“rjlfdm”
2007/6/1
i ipagevii
i i
Contents
Preface xiii
I BoundaryValueProblemsandIterativeMethods 1
1 FiniteDifferenceApproximations 3
1.1 Truncationerrors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Derivingfinitedifferenceapproximations . . . . . . . . . . . . . . . . 7
1.3 Secondorderderivatives . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Higherorderderivatives . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Ageneralapproachtoderivingthecoefficients . . . . . . . . . . . . . 10
2 SteadyStatesandBoundaryValueProblems 13
2.1 Theheatequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Boundaryconditions. . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Thesteady-stateproblem . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Asimplefinitedifferencemethod . . . . . . . . . . . . . . . . . . . . 15
2.5 Localtruncationerror . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6 Globalerror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.7 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.8 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.9 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.10 Stabilityinthe2-norm . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.11 Green’sfunctionsandmax-normstability. . . . . . . . . . . . . . . . 22
2.12 Neumannboundaryconditions . . . . . . . . . . . . . . . . . . . . . 29
2.13 Existenceanduniqueness . . . . . . . . . . . . . . . . . . . . . . . . 32
2.14 Orderingtheunknownsandequations. . . . . . . . . . . . . . . . . . 34
2.15 Agenerallinearsecondorderequation . . . . . . . . . . . . . . . . . 35
2.16 Nonlinearequations . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.16.1 Discretizationofthenonlinearboundaryvalueproblem . 38
2.16.2 Nonuniqueness. . . . . . . . . . . . . . . . . . . . . . . 40
2.16.3 Accuracyonnonlinearequations . . . . . . . . . . . . . 41
2.17 Singularperturbationsandboundarylayers . . . . . . . . . . . . . . . 43
2.17.1 Interiorlayers . . . . . . . . . . . . . . . . . . . . . . . 46
vii
i i
i i
“rjlfdm”
2007/6/1
i ipageviii
i i
viii Contents
2.18 Nonuniformgrids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.18.1 Adaptivemeshselection . . . . . . . . . . . . . . . . . . 51
2.19 Continuationmethods . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.20 Higherordermethods . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.20.1 Fourthorderdifferencing . . . . . . . . . . . . . . . . . 52
2.20.2 Extrapolationmethods . . . . . . . . . . . . . . . . . . . 53
2.20.3 Deferredcorrections . . . . . . . . . . . . . . . . . . . . 54
2.21 Spectralmethods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3 EllipticEquations 59
3.1 Steady-stateheatconduction . . . . . . . . . . . . . . . . . . . . . . 59
3.2 The5-pointstencilfortheLaplacian . . . . . . . . . . . . . . . . . . 60
3.3 Orderingtheunknownsandequations. . . . . . . . . . . . . . . . . . 61
3.4 Accuracyandstability . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.5 The9-pointLaplacian . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.6 Otherellipticequations . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.7 Solvingthelinearsystem . . . . . . . . . . . . . . . . . . . . . . . . 66
3.7.1 SparsestorageinMATLAB . . . . . . . . . . . . . . . . 68
4 IterativeMethodsforSparseLinearSystems 69
4.1 JacobiandGauss–Seidel. . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 Analysisofmatrixsplittingmethods . . . . . . . . . . . . . . . . . . 71
4.2.1 Rateofconvergence . . . . . . . . . . . . . . . . . . . . 74
4.2.2 Successiveoverrelaxation . . . . . . . . . . . . . . . . . 76
4.3 Descentmethodsandconjugategradients . . . . . . . . . . . . . . . . 78
4.3.1 Themethodofsteepestdescent . . . . . . . . . . . . . . 79
4.3.2 TheA-conjugatesearchdirection . . . . . . . . . . . . . 83
4.3.3 Theconjugate-gradientalgorithm . . . . . . . . . . . . . 86
4.3.4 Convergenceofconjugategradient . . . . . . . . . . . . 88
4.3.5 Preconditioners . . . . . . . . . . . . . . . . . . . . . . 93
4.3.6 IncompleteCholeskyandILUpreconditioners . . . . . . 96
4.4 TheArnoldiprocessandGMRESalgorithm . . . . . . . . . . . . . . 96
4.4.1 Krylovmethodsbasedonthreetermrecurrences . . . . . 99
4.4.2 OtherapplicationsofArnoldi . . . . . . . . . . . . . . . 100
4.5 Newton–Krylovmethodsfornonlinearproblems . . . . . . . . . . . . 101
4.6 Multigridmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.6.1 SlowconvergenceofJacobi . . . . . . . . . . . . . . . . 103
4.6.2 Themultigridapproach . . . . . . . . . . . . . . . . . . 106
II InitialValueProblems 111
5 TheInitialValueProblemforOrdinaryDifferentialEquations 113
5.1 Linearordinarydifferentialequations . . . . . . . . . . . . . . . . . . 114
5.1.1 Duhamel’sprinciple . . . . . . . . . . . . . . . . . . . . 115
5.2 Lipschitzcontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
i i
i i
“rjlfdm”
2007/6/1
i ipageix
i i
Contents ix
5.2.1 Existenceanduniquenessofsolutions. . . . . . . . . . . 116
5.2.2 Systemsofequations . . . . . . . . . . . . . . . . . . . 117
5.2.3 SignificanceoftheLipschitzconstant . . . . . . . . . . . 118
5.2.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.3 Somebasicnumericalmethods . . . . . . . . . . . . . . . . . . . . . 120
5.4 Truncationerrors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.5 One-steperrors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.6 Taylorseriesmethods . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.7 Runge–Kuttamethods . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.7.1 Embeddedmethodsanderrorestimation . . . . . . . . . 128
5.8 One-stepversusmultistepmethods . . . . . . . . . . . . . . . . . . . 130
5.9 Linearmultistepmethods . . . . . . . . . . . . . . . . . . . . . . . . 131
5.9.1 Localtruncationerror . . . . . . . . . . . . . . . . . . . 132
5.9.2 Characteristicpolynomials. . . . . . . . . . . . . . . . . 133
5.9.3 Startingvalues . . . . . . . . . . . . . . . . . . . . . . . 134
5.9.4 Predictor-correctormethods . . . . . . . . . . . . . . . . 135
6 Zero-StabilityandConvergenceforInitialValueProblems 137
6.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.2 Thetestproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.3 One-stepmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.3.1 Euler’smethodonlinearproblems . . . . . . . . . . . . 138
6.3.2 Relationtostabilityforboundaryvalueproblems. . . . . 140
6.3.3 Euler’smethodonnonlinearproblems . . . . . . . . . . 141
6.3.4 Generalone-stepmethods . . . . . . . . . . . . . . . . . 142
6.4 Zero-stabilityoflinearmultistepmethods. . . . . . . . . . . . . . . . 143
6.4.1 Solvinglineardifferenceequations . . . . . . . . . . . . 144
7 AbsoluteStabilityforOrdinaryDifferentialEquations 149
7.1 Unstablecomputationswithazero-stablemethod . . . . . . . . . . . 149
7.2 Absolutestability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.3 Stabilityregionsforlinearmultistepmethods . . . . . . . . . . . . . . 153
7.4 Systemsofordinarydifferentialequations . . . . . . . . . . . . . . . 156
7.4.1 Chemicalkinetics . . . . . . . . . . . . . . . . . . . . . 157
7.4.2 Linearsystems . . . . . . . . . . . . . . . . . . . . . . . 158
7.4.3 Nonlinearsystems . . . . . . . . . . . . . . . . . . . . . 160
7.5 Practicalchoiceofstepsize . . . . . . . . . . . . . . . . . . . . . . . 161
7.6 Plottingstabilityregions . . . . . . . . . . . . . . . . . . . . . . . . . 162
7.6.1 Theboundarylocusmethodforlinearmultistepmethods. 162
7.6.2 Plottingstabilityregionsofone-stepmethods . . . . . . . 163
7.7 Relativestabilityregionsandorderstars . . . . . . . . . . . . . . . . 164
8 StiffOrdinaryDifferentialEquations 167
8.1 Numericaldifficulties . . . . . . . . . . . . . . . . . . . . . . . . . . 168
8.2 Characterizationsofstiffness . . . . . . . . . . . . . . . . . . . . . . 169
8.3 Numericalmethodsforstiffproblems . . . . . . . . . . . . . . . . . . 170
i i
i i
“rjlfdm”
2007/6/1
i ipagex
i i
x Contents
8.3.1 A-stabilityandA(˛)-stability . . . . . . . . . . . . . . . 171
8.3.2 L-stability . . . . . . . . . . . . . . . . . . . . . . . . . 171
8.4 BDFmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8.5 TheTR-BDF2method . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.6 Runge–Kutta–Chebyshevexplicitmethods . . . . . . . . . . . . . . . 175
9 DiffusionEquationsandParabolicProblems 181
9.1 Localtruncationerrorsandorderofaccuracy . . . . . . . . . . . . . . 183
9.2 Methodoflinesdiscretizations . . . . . . . . . . . . . . . . . . . . . 184
9.3 Stabilitytheory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
9.4 Stiffnessoftheheatequation . . . . . . . . . . . . . . . . . . . . . . 186
9.5 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
9.5.1 PDEversusODEstabilitytheory . . . . . . . . . . . . . 191
9.6 VonNeumannanalysis . . . . . . . . . . . . . . . . . . . . . . . . . 192
9.7 Multidimensionalproblems . . . . . . . . . . . . . . . . . . . . . . . 195
9.8 Thelocallyone-dimensionalmethod . . . . . . . . . . . . . . . . . . 197
9.8.1 Boundaryconditions . . . . . . . . . . . . . . . . . . . . 198
9.8.2 Thealternatingdirectionimplicitmethod . . . . . . . . . 199
9.9 Otherdiscretizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
10 AdvectionEquationsandHyperbolicSystems 201
10.1 Advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
10.2 Methodoflinesdiscretization . . . . . . . . . . . . . . . . . . . . . . 203
10.2.1 ForwardEulertimediscretization . . . . . . . . . . . . . 204
10.2.2 Leapfrog . . . . . . . . . . . . . . . . . . . . . . . . . . 205
10.2.3 Lax–Friedrichs . . . . . . . . . . . . . . . . . . . . . . . 206
10.3 TheLax–Wendroffmethod . . . . . . . . . . . . . . . . . . . . . . . 207
10.3.1 Stabilityanalysis . . . . . . . . . . . . . . . . . . . . . . 209
10.4 Upwindmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
10.4.1 Stabilityanalysis . . . . . . . . . . . . . . . . . . . . . . 211
10.4.2 TheBeam–Warmingmethod . . . . . . . . . . . . . . . 212
10.5 VonNeumannanalysis . . . . . . . . . . . . . . . . . . . . . . . . . 212
10.6 Characteristictracingandinterpolation . . . . . . . . . . . . . . . . . 214
10.7 TheCourant–Friedrichs–Lewycondition . . . . . . . . . . . . . . . . 215
10.8 Somenumericalresults . . . . . . . . . . . . . . . . . . . . . . . . . 218
10.9 Modifiedequations . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
10.10 Hyperbolicsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
10.10.1 Characteristicvariables . . . . . . . . . . . . . . . . . . 224
10.11 Numericalmethodsforhyperbolicsystems . . . . . . . . . . . . . . . 225
10.12 Initialboundaryvalueproblems . . . . . . . . . . . . . . . . . . . . . 226
10.12.1 Analysisofupwindontheinitialboundaryvalueproblem 226
10.12.2 Outflowboundaryconditions . . . . . . . . . . . . . . . 228
10.13 Otherdiscretizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
11 MixedEquations 233
11.1 Someexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
i i
i i
“rjlfdm”
2007/6/1
i ipagexi
i i
Contents xi
11.2 Fullycoupledmethodoflines . . . . . . . . . . . . . . . . . . . . . . 235
11.3 FullycoupledTaylorseriesmethods . . . . . . . . . . . . . . . . . . 236
11.4 Fractionalstepmethods . . . . . . . . . . . . . . . . . . . . . . . . . 237
11.5 Implicit-explicitmethods . . . . . . . . . . . . . . . . . . . . . . . . 239
11.6 Exponentialtimedifferencingmethods . . . . . . . . . . . . . . . . . 240
11.6.1 Implementingexponentialtimedifferencingmethods . . 241
III Appendices 243
A MeasuringErrors 245
A.1 Errorsinascalarvalue. . . . . . . . . . . . . . . . . . . . . . . . . . 245
A.1.1 Absoluteerror . . . . . . . . . . . . . . . . . . . . . . . 245
A.1.2 Relativeerror. . . . . . . . . . . . . . . . . . . . . . . . 246
A.2 “Big-oh”and“little-oh”notation . . . . . . . . . . . . . . . . . . . . 247
A.3 Errorsinvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
A.3.1 Normequivalence . . . . . . . . . . . . . . . . . . . . . 249
A.3.2 Matrixnorms. . . . . . . . . . . . . . . . . . . . . . . . 250
A.4 Errorsinfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
A.5 Errorsingridfunctions . . . . . . . . . . . . . . . . . . . . . . . . . 251
A.5.1 Normequivalence . . . . . . . . . . . . . . . . . . . . . 252
A.6 Estimatingerrorsinnumericalsolutions . . . . . . . . . . . . . . . . 254
A.6.1 Estimatesfromthetruesolution . . . . . . . . . . . . . . 255
A.6.2 Estimatesfromafine-gridsolution . . . . . . . . . . . . 256
A.6.3 Estimatesfromcoarsersolutions . . . . . . . . . . . . . 256
B PolynomialInterpolationandOrthogonalPolynomials 259
B.1 Thegeneralinterpolationproblem. . . . . . . . . . . . . . . . . . . . 259
B.2 Polynomialinterpolation . . . . . . . . . . . . . . . . . . . . . . . . 260
B.2.1 Monomialbasis . . . . . . . . . . . . . . . . . . . . . . 260
B.2.2 Lagrangebasis . . . . . . . . . . . . . . . . . . . . . . . 260
B.2.3 Newtonform . . . . . . . . . . . . . . . . . . . . . . . . 260
B.2.4 Errorinpolynomialinterpolation . . . . . . . . . . . . . 262
B.3 Orthogonalpolynomials . . . . . . . . . . . . . . . . . . . . . . . . . 262
B.3.1 Legendrepolynomials . . . . . . . . . . . . . . . . . . . 264
B.3.2 Chebyshevpolynomials . . . . . . . . . . . . . . . . . . 265
C EigenvaluesandInner-Product Norms 269
C.1 Similaritytransformations . . . . . . . . . . . . . . . . . . . . . . . . 270
C.2 Diagonalizablematrices . . . . . . . . . . . . . . . . . . . . . . . . . 271
C.3 TheJordancanonicalform . . . . . . . . . . . . . . . . . . . . . . . 271
C.4 SymmetricandHermitianmatrices . . . . . . . . . . . . . . . . . . . 273
C.5 Skew-symmetricandskew-Hermitianmatrices . . . . . . . . . . . . . 274
C.6 Normalmatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
C.7 Toeplitzandcirculantmatrices . . . . . . . . . . . . . . . . . . . . . 275
C.8 TheGershgorintheorem . . . . . . . . . . . . . . . . . . . . . . . . . 277
i i
i i