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
$ o; U  K8 l' s& uPreface to the second edition xi
; Y. L. Y- i5 S5 o  g3 N( }' B) S1 Introduction 1
7 v) e' d( d1 x* x) L2 Parabolic equations in one space variable 7
+ B0 Y, f" f3 C* y2.1 Introduction 7
2.2 A model problem 7
7 s* R: C3 R# ~8 r2 S: Y2.3 Series approximation 9
2.4 An explicit scheme for the model problem 10
2.5 Difference notation and truncation error 12
) ~3 A8 Q$ C- z' o0 u& @' X- R2.6 Convergence of the explicit scheme 16
( `) N& _5 ^. e9 [" }% ]  T2.7 Fourier analysis of the error 19
! j/ V! f) S; F* `5 I1 r2.8 An implicit method 22
2.9 The Thomas algorithm 24
2.10 The weighted average or θ-method 26
0 B" G# @) x$ L+ K8 {, W! T2.11 A maximum principle and convergence
for μ(1−θ)≤ 1
) _$ d: y- `' N+ A4 S. v2 33
+ u: c# d5 d) y! z3 T. _9 `5 [2.12 A three-time-level scheme 38
/ M+ A; O9 V: N5 b0 B3 [2.13 More general boundary conditions 39
- c( T( g" `( V* F" n2.14 Heat conservation properties 44
2.15 More general linear problems 46
2.16 Polar co-ordinates 52
( p- f- `2 }; o" a& g2.17 Nonlinear problems 54
Bibliographic notes 56
Exercises 56
v
/ H0 Y$ `- ~1 ^  q; m. Mvi Contents
7 @8 ?- l; j7 z' G! z3 2-D and 3-D parabolic equations 62
3.1 The explicit method in a rectilinear box 62
7 }) q' `+ i8 v  x) Q1 p7 Q9 d6 D4 k3.2 An ADI method in two dimensions 64
3.3 ADI and LOD methods in three dimensions 70
5 g. D  V) e7 ?! g+ A3 ]% `# ^+ [( j- h3.4 Curved boundaries 71
4 |0 S6 I5 I5 t$ L2 N3.5 Application to general parabolic problems 80
8 d4 Z1 a3 C& [Bibliographic notes 83
. @4 c0 ~  d- q. k, ~4 w2 S" M* l3 sExercises 83
( I# V' D7 |& w4 Hyperbolic equations in one space dimension 86
  ~6 K& \# ?7 [/ C9 E  s4.1 Characteristics 86
4.2 The CFL condition 89
7 a% [* u4 |  b/ l4.3 Error analysis of the upwind scheme 94
4.4 Fourier analysis of the upwind scheme 97
4 h+ J  \( G( h. s' z4.5 The Lax–Wendroff scheme 100
* Z9 ~$ Y/ s9 x' ^, ?$ J4.6 The Lax–Wendroff method for conservation laws 103
4.7 Finite volume schemes 110
/ {) Y2 K  ~+ c4.8 The box scheme 116
4.9 The leap-frog scheme 123
4.10 Hamiltonian systems and symplectic
integration schemes 128
4.11 Comparison of phase and amplitude errors 135
4.12 Boundary conditions and conservation properties 139
+ k! ?$ D7 A1 k2 t) Y4.13 Extensions to more space dimensions 143
Bibliographic notes 146
Exercises 146
  T! u( E& s  _& f  S5 Consistency, convergence and stability 151
. `& u( i# i. N( N( f3 q6 {5.1 Definition of the problems considered 151
. A3 }& f; F) R( B0 ~' t5.2 The finite difference mesh and norms 152
0 A2 n0 A  n7 X+ b7 o7 ^5.3 Finite difference approximations 154
* z, n# Q: x5 v, F: r3 N8 |* i& m5.4 Consistency, order of accuracy and convergence 156
5.5 Stability and the Lax Equivalence Theorem 157
2 c$ z4 ^8 s6 E5.6 Calculating stability conditions 160
5.7 Practical (strict or strong) stability 166
5.8 Modified equation analysis 169
" p+ n& |. w  L( `  ^5.9 Conservation laws and the energy method of analysis 177
5.10 Summary of the theory 186
8 E3 x  O9 k1 l0 `% DBibliographic notes 189
2 L7 r3 R7 W! N/ G2 w+ o, G, N  BExercises 190
Contents vii
6 Linear second order elliptic equations in
/ j" q5 X& X0 q; Z) q9 dtwo dimensions 194
6.1 A model problem 194
6.2 Error analysis of the model problem 195
$ D% q; T" a% M6.3 The general diffusion equation 197
. G! L3 Y5 l& P" r3 b( a6.4 Boundary conditions on a curved boundary 199
! y* D; `5 z/ E6.5 Error analysis using a maximum principle 203
6.6 Asymptotic error estimates 213
6.7 Variational formulation and the finite
3 u3 v0 B0 S$ X) j  K6 P; m  helement method 218
6.8 Convection–diffusion problems 224
6.9 An example 228
Bibliographic notes 231
" `. l. a+ \7 p) eExercises 232
7 Iterative solution of linear algebraic equations 235
3 W1 w# s$ F( [  u0 o& e& u7.1 Basic iterative schemes in explicit form 237
, k- s8 i% a( R9 [) Z7.2 Matrix form of iteration methods and
their convergence 239
" r' x# o& e( B/ J! e  l7.3 Fourier analysis of convergence 244
3 Z7 w' ^+ z# ~7.4 Application to an example 248
7.5 Extensions and related iterative methods 250
7.6 The multigrid method 252
7.7 The conjugate gradient method 258
9 U( v! u5 e4 i! w7.8 A numerical example: comparisons 261
" q% R! D1 `+ ^( dBibliographic notes 263
Exercises 263
References 267
+ P# n- y" r" w/ @Index 273

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