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
" r0 @6 [1 m! g) t6 `1 N0 \0 Bcontents:
0 D9 f: }! C6 L# m8 v+ ]Preface to the first edition page viii
' m5 u+ w- s5 Y7 T' xPreface to the second edition xi
. d6 S! Y3 b3 y# ~+ C1 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
7 w' t7 A6 D  O% {" r# f2.4 An explicit scheme for the model problem 10
8 y) F* @- r6 b+ p9 K' H' v; `2.5 Difference notation and truncation error 12
+ B. d/ i) J- |/ e" B2.6 Convergence of the explicit scheme 16
2.7 Fourier analysis of the error 19
* n* s3 A- T0 y( r6 }1 A. v8 g' f2.8 An implicit method 22
2.9 The Thomas algorithm 24
! `" A$ p, E0 J( }9 C2 h2.10 The weighted average or θ-method 26
/ h9 S8 i, K( ?2 ?  q3 ?2.11 A maximum principle and convergence
for μ(1−θ)≤ 1
5 `+ I0 L4 S; Q- J2 33
& A" ~. A/ w7 M1 v. p7 s" D2.12 A three-time-level scheme 38
' e' |7 `" W- J! \: t! Y0 }4 p2.13 More general boundary conditions 39
# k( @# A3 k' @) v2.14 Heat conservation properties 44
2.15 More general linear problems 46
2.16 Polar co-ordinates 52
( N- ], _* R; j2.17 Nonlinear problems 54
Bibliographic notes 56
Exercises 56
v
vi Contents
. K2 e# k" i! Q% f% B3 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
/ @  [7 a1 t" C3 g5 P3.4 Curved boundaries 71
3 c' N6 o( D9 C, W7 w( I3.5 Application to general parabolic problems 80
  Y- T- U) h5 P" {: |+ eBibliographic notes 83
Exercises 83
4 Hyperbolic equations in one space dimension 86
/ h( u) P+ n% E% S4.1 Characteristics 86
+ t! r7 ~- ^' m! H4.2 The CFL condition 89
) z& P  F4 a7 X' B) H! H& l4.3 Error analysis of the upwind scheme 94
' l! p$ o: t7 G1 P+ i9 c7 D3 h4.4 Fourier analysis of the upwind scheme 97
5 d8 T8 I" \& ?) v4.5 The Lax–Wendroff scheme 100
6 n- t2 u) X  H' R9 Q4.6 The Lax–Wendroff method for conservation laws 103
4.7 Finite volume schemes 110
: H  v' h, e  M) @$ K  V9 Q4.8 The box scheme 116
4.9 The leap-frog scheme 123
. g  b, U4 l. h. q! q4.10 Hamiltonian systems and symplectic
integration schemes 128
' k( V! v) ~5 ?3 _8 O3 S4.11 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
. `7 q2 U3 p0 ~7 wExercises 146
- D2 r( f; b/ _5 Consistency, convergence and stability 151
5.1 Definition of the problems considered 151
7 e' r) y* p7 L$ a/ v  E  P, Q+ f" ?5.2 The finite difference mesh and norms 152
5 H& h% f  g& l0 z) S5.3 Finite difference approximations 154
5.4 Consistency, order of accuracy and convergence 156
6 P' q$ P; `8 i, H& Q; H+ M! E2 L5.5 Stability and the Lax Equivalence Theorem 157
5.6 Calculating stability conditions 160
; u8 e5 s" @8 Q# X5.7 Practical (strict or strong) stability 166
/ ]1 n- @  f7 q! m$ H5.8 Modified equation analysis 169
5.9 Conservation laws and the energy method of analysis 177
/ F' J! u1 ?1 X/ }$ W) s) R5.10 Summary of the theory 186
. |7 ]/ ]" i  w, GBibliographic notes 189
Exercises 190
Contents vii
6 Linear second order elliptic equations in
+ k6 y) _0 h2 H( Qtwo dimensions 194
3 Q: Q+ ~  P: R6 X( t- |6.1 A model problem 194
. c9 E$ |5 _3 x2 F3 h5 N; i6 B6.2 Error analysis of the model problem 195
6.3 The general diffusion equation 197
6 a3 V* n) d0 h% s! @5 {6.4 Boundary conditions on a curved boundary 199
6.5 Error analysis using a maximum principle 203
+ W2 G; d$ M# m; z# X; s6.6 Asymptotic error estimates 213
) K/ R  b1 r; Y6 v$ q* M8 ]4 y6.7 Variational formulation and the finite
, i1 l8 n9 q2 \$ `4 q$ M9 ~element method 218
6.8 Convection–diffusion problems 224
7 P: A2 `7 ^5 G. |; p* \+ G6.9 An example 228
6 Z. C; l0 n( K, [% MBibliographic notes 231
Exercises 232
7 Iterative solution of linear algebraic equations 235
7 I$ I0 n( d- _7 r7.1 Basic iterative schemes in explicit form 237
- c8 Y+ c/ j& D; r0 F7.2 Matrix form of iteration methods and
& N% p- `$ N1 \: ]! U" H) l: Ctheir convergence 239
2 c. [( A4 H9 p1 o/ Q9 |/ ^7.3 Fourier analysis of convergence 244
! V; ]% l8 d4 {/ n1 m5 x4 ~7.4 Application to an example 248
7.5 Extensions and related iterative methods 250
2 J1 f. ?$ \0 T/ E/ d. W7.6 The multigrid method 252
7.7 The conjugate gradient method 258
7.8 A numerical example: comparisons 261
& w; P# I6 I5 V) CBibliographic notes 263
Exercises 263
References 267
7 V$ b" d) Y9 i, h) gIndex 273

! j- f: ^* Y3 |, X3 [

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