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
8 S$ N( G( G7 {contents:
Preface to the first edition page viii
Preface to the second edition xi
1 Introduction 1
2 Parabolic equations in one space variable 7
5 u. k' E4 M6 F! e" v# A- b  P; w2.1 Introduction 7
2.2 A model problem 7
6 M. N' o8 s  A2 P2.3 Series approximation 9
  q7 s7 d7 h# [, ]. L) q5 M  m% c2.4 An explicit scheme for the model problem 10
+ ~7 D, Q: H* e! {2.5 Difference notation and truncation error 12
; p: y' Q# W3 u2.6 Convergence of the explicit scheme 16
2.7 Fourier analysis of the error 19
+ U+ L- B% O( [  z, g+ J- w3 b2.8 An implicit method 22
2.9 The Thomas algorithm 24
2.10 The weighted average or θ-method 26
0 T6 n* i' v7 ?. v2.11 A maximum principle and convergence
5 w" b* z/ a! f5 K* wfor μ(1−θ)≤ 1
2 33
2.12 A three-time-level scheme 38
- u: I# Z8 a6 X3 W2.13 More general boundary conditions 39
2.14 Heat conservation properties 44
2 D- Z- Z+ T5 j, d2.15 More general linear problems 46
2.16 Polar co-ordinates 52
$ i0 j2 V4 r- T5 D3 G! O' w) i  A2.17 Nonlinear problems 54
# {2 [7 K! ?# H; n: A# LBibliographic notes 56
: i* {  Q' P2 M: l9 f% y  [Exercises 56
v
vi Contents
3 2-D and 3-D parabolic equations 62
3.1 The explicit method in a rectilinear box 62
/ q; f$ J6 B7 |5 J8 ?$ Y' E# G3.2 An ADI method in two dimensions 64
2 v/ T8 c6 J8 Y# a& G3.3 ADI and LOD methods in three dimensions 70
5 }+ T* M/ G1 \$ K3.4 Curved boundaries 71
: y" H' n& k; ?- n3.5 Application to general parabolic problems 80
# i# O3 s2 f( Q* n$ TBibliographic notes 83
- o- t% t# w: J& m) TExercises 83
4 Hyperbolic equations in one space dimension 86
4.1 Characteristics 86
- I' |' O* @7 X7 C3 ?  q$ Q4.2 The CFL condition 89
( h/ W& c9 n, K7 c4.3 Error analysis of the upwind scheme 94
9 j1 y3 ?" d% o% `3 z3 ~! O4.4 Fourier analysis of the upwind scheme 97
' L! C/ N* W7 c1 f+ X, \8 i4.5 The Lax–Wendroff scheme 100
4 M; w8 M0 K" }' G8 c4 s1 G6 i4.6 The Lax–Wendroff method for conservation laws 103
4.7 Finite volume schemes 110
' z9 E5 @) ^6 W  l* O9 B  n( S  ~% i4.8 The box scheme 116
4.9 The leap-frog scheme 123
$ A$ ^* i1 f0 g4 O8 a4.10 Hamiltonian systems and symplectic
; A% n! V2 o( f1 R# B% wintegration schemes 128
% L2 s4 T) l( A0 C, N4.11 Comparison of phase and amplitude errors 135
4.12 Boundary conditions and conservation properties 139
# C8 j2 ?+ T* j- y: N. ?- D4.13 Extensions to more space dimensions 143
Bibliographic notes 146
& S9 s, t* E* B  ?Exercises 146
5 Consistency, convergence and stability 151
5.1 Definition of the problems considered 151
5.2 The finite difference mesh and norms 152
4 Y) s- {" U) x( j9 T5.3 Finite difference approximations 154
5.4 Consistency, order of accuracy and convergence 156
- t1 D  }4 s( k5.5 Stability and the Lax Equivalence Theorem 157
, n& O- m" V- Z5.6 Calculating stability conditions 160
5.7 Practical (strict or strong) stability 166
4 G/ k  q3 D$ W6 R( n0 B$ o: d- W5.8 Modified equation analysis 169
# V& `" J) q3 p7 `, \* o1 d5.9 Conservation laws and the energy method of analysis 177
5.10 Summary of the theory 186
Bibliographic notes 189
( }' [9 G7 z5 L7 b# I) ]$ X8 [Exercises 190
# G' W, X9 ]; v4 d& x  CContents 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
0 X7 {1 L- G  l9 o7 w$ w/ o6.3 The general diffusion equation 197
6.4 Boundary conditions on a curved boundary 199
6 G* D4 F% r- g$ S6.5 Error analysis using a maximum principle 203
0 D; d7 z9 H! D5 J* |6.6 Asymptotic error estimates 213
6.7 Variational formulation and the finite
element method 218
4 Y5 i4 S) d. i9 l  H6.8 Convection–diffusion problems 224
6.9 An example 228
Bibliographic notes 231
Exercises 232
7 Iterative solution of linear algebraic equations 235
7.1 Basic iterative schemes in explicit form 237
" v9 n) q$ f& Q+ O. d! J7.2 Matrix form of iteration methods and
their convergence 239
7.3 Fourier analysis of convergence 244
7.4 Application to an example 248
- H0 O4 ?" N6 h0 ~+ o) N; t8 k5 i7.5 Extensions and related iterative methods 250
7.6 The multigrid method 252
3 b. x! _7 c0 l1 i! J9 l* q7.7 The conjugate gradient method 258
7.8 A numerical example: comparisons 261
- U5 y: I% j# O0 V& B# C' i3 z$ ~Bibliographic notes 263
6 [* {/ g! a6 w3 A; l1 ?/ [, QExercises 263
0 D' N* h+ ^( D, z9 SReferences 267
; Y& ]7 p5 D  UIndex 273

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