剑桥出版社---偏微分方程数值解法

数学中国—罂粟

原版英文书 第二版
contents:
Preface to the first edition page viii
3 s3 N) P6 o* pPreface to the second edition xi
/ A( m# O  G" f* s1 i8 b# g! k1 Introduction 1
2 Parabolic equations in one space variable 7
2.1 Introduction 7
2.2 A model problem 7
2.3 Series approximation 9
2.4 An explicit scheme for the model problem 10
. f- ]1 C! x) p! q, S2.5 Difference notation and truncation error 12
2.6 Convergence of the explicit scheme 16
2.7 Fourier analysis of the error 19
2.8 An implicit method 22
2.9 The Thomas algorithm 24
1 C& l# N4 [; M2 e2.10 The weighted average or θ-method 26
9 v1 q$ e2 e  F+ h; {' @7 t* Z3 ?2.11 A maximum principle and convergence
for μ(1−θ)≤ 1
2 33
, [" r$ e7 n. y2.12 A three-time-level scheme 38
+ s8 l8 X  E0 r. D# T2.13 More general boundary conditions 39
2.14 Heat conservation properties 44
; [, T8 P4 z: W) g8 r' a. }0 q2.15 More general linear problems 46
2.16 Polar co-ordinates 52
2.17 Nonlinear problems 54
, r* p  l1 r, a; ]0 rBibliographic notes 56
Exercises 56
v
3 G. w2 y$ z6 ]' j; @& }vi Contents
; \: h! i# m! e$ f3 2-D and 3-D parabolic equations 62
3.1 The explicit method in a rectilinear box 62
4 _4 ^2 \7 c0 [8 w3.2 An ADI method in two dimensions 64
7 M" C, j+ ]) q+ Y3.3 ADI and LOD methods in three dimensions 70
3.4 Curved boundaries 71
3.5 Application to general parabolic problems 80
3 U  i" ~' T3 C0 K& }( MBibliographic notes 83
2 h- z6 H* p( E9 i8 C) @: C  k. AExercises 83
4 Hyperbolic equations in one space dimension 86
, e7 L% z3 f2 ~) P4 V4.1 Characteristics 86
0 ^- @4 U" |" T# Z( C# v; u4.2 The CFL condition 89
, X. W9 Q5 ^; V: I+ D2 o' E4.3 Error analysis of the upwind scheme 94
4.4 Fourier analysis of the upwind scheme 97
4.5 The Lax–Wendroff scheme 100
0 q1 i2 j2 d5 `  I; t4.6 The Lax–Wendroff method for conservation laws 103
9 J# t  ]9 g6 C6 I8 d( a) c$ m" N- T4.7 Finite volume schemes 110
4.8 The box scheme 116
4.9 The leap-frog scheme 123
4.10 Hamiltonian systems and symplectic
integration schemes 128
2 G3 I2 i0 b- X: ?. J4.11 Comparison of phase and amplitude errors 135
& m2 {7 O$ R9 S: I/ ]4.12 Boundary conditions and conservation properties 139
' c9 n, u$ _" ]; W1 K3 S4.13 Extensions to more space dimensions 143
+ W. G7 \& N6 L, }. p  H1 GBibliographic notes 146
6 M: ^! ]% t1 q" ^$ j. C% iExercises 146
9 {! ~2 l+ a" k, i' I! v5 z5 Consistency, convergence and stability 151
$ p) [- G' C( ?9 ?0 f( \, p1 E5.1 Definition of the problems considered 151
$ E7 J* M9 p. [9 K& R6 I5.2 The finite difference mesh and norms 152
5 @8 T8 \9 Y* _: H8 c( a1 I5.3 Finite difference approximations 154
5.4 Consistency, order of accuracy and convergence 156
0 c- F9 @/ ]' m+ ]: m/ N5.5 Stability and the Lax Equivalence Theorem 157
5.6 Calculating stability conditions 160
2 B: q$ U' k7 a8 t/ o0 x5.7 Practical (strict or strong) stability 166
5.8 Modified equation analysis 169
5.9 Conservation laws and the energy method of analysis 177
5.10 Summary of the theory 186
Bibliographic notes 189
5 \4 G3 @3 {$ y& \Exercises 190
Contents vii
6 Linear second order elliptic equations in
two dimensions 194
6.1 A model problem 194
6.2 Error analysis of the model problem 195
' q3 t$ q2 _0 a1 M" e, j6.3 The general diffusion equation 197
! s6 K2 ?: T+ j' k+ w% s6.4 Boundary conditions on a curved boundary 199
7 |; s3 g: Q: j1 L& D) O! [6.5 Error analysis using a maximum principle 203
( p. q& H/ q, Y* r# X, L6 W0 q/ ]7 C+ M6.6 Asymptotic error estimates 213
3 u- K4 U, ]. P# D6 H. y5 S6.7 Variational formulation and the finite
element method 218
6.8 Convection–diffusion problems 224
6.9 An example 228
6 \) `, ]$ ?* @! v8 uBibliographic notes 231
) u( p2 I4 w. t, a+ o; n+ c' ]Exercises 232
4 T1 a1 {3 r* K# }8 A% s7 Iterative solution of linear algebraic equations 235
7 q) `3 Q, g1 W" _; p  G  h+ d7.1 Basic iterative schemes in explicit form 237
7.2 Matrix form of iteration methods and
. {( B" S  \) ^2 ]0 t$ Ytheir convergence 239
7.3 Fourier analysis of convergence 244
7.4 Application to an example 248
7.5 Extensions and related iterative methods 250
2 k$ \+ F4 s! X- W( ?# c5 ^* W7.6 The multigrid method 252
7.7 The conjugate gradient method 258
7.8 A numerical example: comparisons 261
" J$ P8 n; X. [# _% vBibliographic notes 263
! F$ ^$ r/ G6 t$ W& r& `  AExercises 263
, Q/ i' T& x& J0 v& m9 xReferences 267
Index 273

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