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
Preface to the second edition xi
& e5 f4 N" [. B2 g8 a2 A1 Introduction 1
; E# h# M4 _- t0 j7 T$ ]* d* e3 z' r2 Parabolic equations in one space variable 7
1 ?/ g9 B  U# I: s$ L2.1 Introduction 7
2.2 A model problem 7
2.3 Series approximation 9
/ Y" h  W3 ]# j% S- {2.4 An explicit scheme for the model problem 10
2.5 Difference notation and truncation error 12
2.6 Convergence of the explicit scheme 16
2.7 Fourier analysis of the error 19
2 @6 i2 G# Z4 ]! z0 n) s' \  S2.8 An implicit method 22
' y" m! A" J9 \" ~2.9 The Thomas algorithm 24
% W9 i* v: _  L: j2.10 The weighted average or θ-method 26
4 ~5 @* G! u8 r' K. L7 o. ~2.11 A maximum principle and convergence
/ ^1 m* F# O9 _7 E6 zfor μ(1−θ)≤ 1
2 33
2.12 A three-time-level scheme 38
" Q- M- h* \, j# s, s. H- ]1 C2.13 More general boundary conditions 39
' G2 f' M" l$ _! A% d2.14 Heat conservation properties 44
# K% a& g; u+ i! _" `0 y2.15 More general linear problems 46
1 v1 R1 i) ^+ w/ q) k3 L2.16 Polar co-ordinates 52
2.17 Nonlinear problems 54
Bibliographic notes 56
Exercises 56
v
7 M. q: l8 t7 e/ E5 V7 {3 I, H1 @1 ]$ avi Contents
" S  Z* m5 A' W: e3 2-D and 3-D parabolic equations 62
3.1 The explicit method in a rectilinear box 62
4 Z/ S4 \% `7 H; X! l! w5 O$ Q( T4 ]3.2 An ADI method in two dimensions 64
" S2 }: ~' U& A8 Y3 |; {( C3.3 ADI and LOD methods in three dimensions 70
3.4 Curved boundaries 71
3.5 Application to general parabolic problems 80
7 {7 I/ D' c, x  t, C" F1 gBibliographic notes 83
. ~+ B9 U, V& d- U8 f0 rExercises 83
4 Hyperbolic equations in one space dimension 86
4.1 Characteristics 86
& m% F2 ?7 X' t3 X. N8 U4.2 The CFL condition 89
3 X. e! K& i+ q4.3 Error analysis of the upwind scheme 94
5 X( p0 S& A$ G/ X+ M4.4 Fourier analysis of the upwind scheme 97
4.5 The Lax–Wendroff scheme 100
4.6 The Lax–Wendroff method for conservation laws 103
$ c. v; }5 |: x5 l4 K& g, E, t7 P4.7 Finite volume schemes 110
( ?/ C8 C5 e3 D4.8 The box scheme 116
" }6 d" Y" J! T! W4.9 The leap-frog scheme 123
% J( e3 \( A  B* @4.10 Hamiltonian systems and symplectic
; Y+ o& {. R! Y/ ^, r* j  tintegration schemes 128
* Z; W% @8 R. ?+ ~' |) T$ {4.11 Comparison of phase and amplitude errors 135
4.12 Boundary conditions and conservation properties 139
4.13 Extensions to more space dimensions 143
Bibliographic notes 146
Exercises 146
5 Consistency, convergence and stability 151
& }1 M. e9 G* Z2 U) }4 U6 L. m* n5.1 Definition of the problems considered 151
3 \2 d7 X% }) }' n6 e: n' i; F2 j" h5.2 The finite difference mesh and norms 152
9 W& o# q' v: [$ Z2 g& f% L1 J5.3 Finite difference approximations 154
8 c) e6 C- Z+ k' R" U6 |# r: `5.4 Consistency, order of accuracy and convergence 156
; \, S, }' h' m# @0 v: B5 V5.5 Stability and the Lax Equivalence Theorem 157
5.6 Calculating stability conditions 160
5.7 Practical (strict or strong) stability 166
) y  [" n# ]  [) Y/ X( [% M5.8 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
Exercises 190
6 H; B) B0 C6 n, t- U& l$ XContents vii
; P9 F, c8 I/ S* J# O9 k6 Linear second order elliptic equations in
two dimensions 194
6.1 A model problem 194
6.2 Error analysis of the model problem 195
6.3 The general diffusion equation 197
2 Z% a. {! P" q4 ^4 k+ c7 Q6.4 Boundary conditions on a curved boundary 199
6.5 Error analysis using a maximum principle 203
# Q! w) U! c8 F9 e* T# U6.6 Asymptotic error estimates 213
2 H4 K0 c' K$ p' d9 L% U& x: q( Q4 D6.7 Variational formulation and the finite
% u2 s  y( x! s. Z# e- Nelement method 218
3 X! J6 Z0 Z0 S; q1 z5 O% r1 q6.8 Convection–diffusion problems 224
1 r0 K% T- T/ U2 @4 C9 V6.9 An example 228
; y; t7 Z0 d; VBibliographic notes 231
& I8 Y8 Q& ^' M, ~9 F- \: U9 {# kExercises 232
0 V5 X; C. @" `6 n# L" R* @* I# V7 Iterative solution of linear algebraic equations 235
$ Z* d/ e. A; E4 h! `2 Y: r7.1 Basic iterative schemes in explicit form 237
1 u7 e7 n: l4 E6 _0 y7.2 Matrix form of iteration methods and
their convergence 239
7.3 Fourier analysis of convergence 244
7.4 Application to an example 248
7.5 Extensions and related iterative methods 250
1 D- m6 d* {. c6 [! P7.6 The multigrid method 252
7.7 The conjugate gradient method 258
+ ~+ C$ b- m% ?2 p  v7 a7.8 A numerical example: comparisons 261
* ]" M# l9 |) a. [& U/ m1 bBibliographic notes 263
Exercises 263
/ h  r/ z; U) G. [/ H; U& {  [References 267
0 X. i/ E% b0 t& a' tIndex 273

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