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:
+ `& ?1 I* p2 UPreface to the first edition page viii
4 t! a6 ^* \& G( kPreface to the second edition xi
& }9 n5 M' v6 _6 _1 Introduction 1
, j" Q% W% g0 [; @2 Parabolic equations in one space variable 7
4 @; U4 o% S$ @1 n: }2.1 Introduction 7
% `+ B/ [7 G) j2.2 A model problem 7
2.3 Series approximation 9
: n. K  v$ W6 Q( w/ Y2.4 An explicit scheme for the model problem 10
. R) s9 _' {6 _7 S2.5 Difference notation and truncation error 12
9 [0 N8 N1 y" k4 Z( n! \& M2.6 Convergence of the explicit scheme 16
( A1 ~7 q8 f' `; Z7 y0 I% q+ ]1 P5 W2.7 Fourier analysis of the error 19
2.8 An implicit method 22
( T) o) M* j% X1 k2.9 The Thomas algorithm 24
2.10 The weighted average or θ-method 26
2.11 A maximum principle and convergence
for μ(1−θ)≤ 1
+ M$ {# e3 t' p% m4 L5 I0 |2 33
' |: g0 v" ?: e& T2.12 A three-time-level scheme 38
& o+ ~3 c  V7 y  F( k( M2.13 More general boundary conditions 39
  y! F9 p; ~5 g( r2.14 Heat conservation properties 44
5 I% s* ?, `) x# e& L7 H" L2.15 More general linear problems 46
& @  K! J# o, V3 i: k6 S% J2.16 Polar co-ordinates 52
6 T8 R' |6 j* q* g, v! u7 P2.17 Nonlinear problems 54
Bibliographic notes 56
) f9 z' i$ L% V6 T# s1 F- ZExercises 56
v
# F! U, E  p4 Q/ s0 t. Pvi Contents
  H$ `7 Y: D0 ?5 q" U3 2-D and 3-D parabolic equations 62
3.1 The explicit method in a rectilinear box 62
3.2 An ADI method in two dimensions 64
3.3 ADI and LOD methods in three dimensions 70
0 O" z9 x( F$ o6 i6 |3.4 Curved boundaries 71
9 A( I  ~$ q5 k, |6 \. i2 x3.5 Application to general parabolic problems 80
; k! w4 l" I8 s7 a0 }  JBibliographic notes 83
2 t  }8 W' a2 l/ ?Exercises 83
4 Hyperbolic equations in one space dimension 86
4.1 Characteristics 86
4.2 The CFL condition 89
# M: t  v" \& n6 p1 `0 [( w: i4.3 Error analysis of the upwind scheme 94
4.4 Fourier analysis of the upwind scheme 97
4.5 The Lax–Wendroff scheme 100
4.6 The Lax–Wendroff method for conservation laws 103
" y' ^0 _1 X. M9 D3 O+ H, j4.7 Finite volume schemes 110
0 X9 H* J% {4 R; A4.8 The box scheme 116
4.9 The leap-frog scheme 123
* ?( p7 w% u" d8 q) t. p4.10 Hamiltonian systems and symplectic
integration schemes 128
4.11 Comparison of phase and amplitude errors 135
4.12 Boundary conditions and conservation properties 139
5 O0 x, p1 y9 n* }8 R; y4.13 Extensions to more space dimensions 143
4 f! Z1 \3 Q( P1 |; V- ]" jBibliographic notes 146
Exercises 146
5 Consistency, convergence and stability 151
. P) [7 Y+ I# v* i* D- z* Y5.1 Definition of the problems considered 151
5.2 The finite difference mesh and norms 152
* @5 L* r0 n) ]! r2 ?* K5.3 Finite difference approximations 154
8 n! Z7 R. }% K$ x, y5.4 Consistency, order of accuracy and convergence 156
/ V8 w: b- T' j- s5.5 Stability and the Lax Equivalence Theorem 157
7 y+ Z# t7 E+ h$ _5.6 Calculating stability conditions 160
5.7 Practical (strict or strong) stability 166
* V* c6 [* S5 ~) J0 F' w9 I: i5.8 Modified equation analysis 169
5.9 Conservation laws and the energy method of analysis 177
5.10 Summary of the theory 186
7 `3 d7 U( d3 sBibliographic notes 189
/ W( P1 G) ^1 N1 _5 g5 Q9 pExercises 190
. a( [) b3 k. h. p9 [& y, lContents vii
% X7 K* l& L; N" Y. |2 R6 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
6.4 Boundary conditions on a curved boundary 199
* T2 ~0 h- W5 k9 N6 j* t- a) f& b  l8 e6.5 Error analysis using a maximum principle 203
. t: C( s6 T( a1 |' A; ^6.6 Asymptotic error estimates 213
: y2 U  l% U% A6 Z7 I6.7 Variational formulation and the finite
$ E3 D* z; u5 Y. f0 {1 ?4 Welement method 218
1 j' e( k2 L) v! u; d9 O6.8 Convection–diffusion problems 224
6.9 An example 228
2 J% w1 w, G# @2 ~1 xBibliographic notes 231
! o; d9 ?$ M! B& F0 d9 L+ X" _) mExercises 232
$ @# b+ F$ N) J( s7 Iterative solution of linear algebraic equations 235
$ C* x; F! Z9 s7.1 Basic iterative schemes in explicit form 237
# }8 M% t2 s: a4 W. E7.2 Matrix form of iteration methods and
their convergence 239
! c) g8 t& f2 l1 W4 L  c# p4 O2 E7.3 Fourier analysis of convergence 244
7.4 Application to an example 248
6 g/ C$ O! G) w2 e7.5 Extensions and related iterative methods 250
+ F1 M! O) v* G! c' e7.6 The multigrid method 252
7.7 The conjugate gradient method 258
7.8 A numerical example: comparisons 261
6 d7 r5 j$ _% z2 g4 }Bibliographic notes 263
$ X! v: e% @9 L! _8 fExercises 263
  z9 c+ ]6 n9 W2 K" q" F( d/ [6 oReferences 267
Index 273

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