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:
9 ~0 P) K% G1 RPreface to the first edition page viii
3 h) Q5 H8 h+ K( VPreface to the second edition xi
2 e9 S* Q& F% @" J, \/ A1 A% k1 Introduction 1
2 Parabolic equations in one space variable 7
2.1 Introduction 7
2.2 A model problem 7
9 r0 T2 ~% d# m2.3 Series approximation 9
2.4 An explicit scheme for the model problem 10
0 I" o) |$ r! G5 M3 S; X2.5 Difference notation and truncation error 12
5 K8 |6 `, X3 r  M% Z4 _( D2.6 Convergence of the explicit scheme 16
2.7 Fourier analysis of the error 19
7 _3 n, k- e/ T  r, F2.8 An implicit method 22
2.9 The Thomas algorithm 24
: ?4 Y9 q4 ^2 J2.10 The weighted average or θ-method 26
2.11 A maximum principle and convergence
for μ(1−θ)≤ 1
, g" i) D  u1 y* x2 33
3 B) o! ]% `& i7 m( C' s% l" l2.12 A three-time-level scheme 38
2.13 More general boundary conditions 39
2 w: T  K) o; _( k9 M2.14 Heat conservation properties 44
2.15 More general linear problems 46
2.16 Polar co-ordinates 52
2.17 Nonlinear problems 54
0 X! l6 _- I. B1 H3 a) S$ wBibliographic notes 56
Exercises 56
v
vi Contents
3 2-D and 3-D parabolic equations 62
9 i+ i7 k" F0 V3.1 The explicit method in a rectilinear box 62
" }4 i( S7 Y$ g4 ~3.2 An ADI method in two dimensions 64
" S& l4 R% o0 ]+ @3.3 ADI and LOD methods in three dimensions 70
3.4 Curved boundaries 71
7 @5 j( O4 H1 t4 z) ]3.5 Application to general parabolic problems 80
9 o( I3 U. c7 ]* M0 i. R& MBibliographic notes 83
" E, o* Y5 i/ ^; `4 W' JExercises 83
4 Hyperbolic equations in one space dimension 86
4.1 Characteristics 86
% Z  \( A8 o! }! f* B2 \# p4.2 The CFL condition 89
4.3 Error analysis of the upwind scheme 94
' d& l( _6 h  ?7 H2 a! b4.4 Fourier analysis of the upwind scheme 97
  F6 e8 ^# T0 @: f4 L) ]# B- R5 {* Z4.5 The Lax–Wendroff scheme 100
4.6 The Lax–Wendroff method for conservation laws 103
4.7 Finite volume schemes 110
4.8 The box scheme 116
- S, z3 L1 j. G7 }) e* O" W8 v4.9 The leap-frog scheme 123
4.10 Hamiltonian systems and symplectic
integration schemes 128
( |" a- l9 U  Q2 f4.11 Comparison of phase and amplitude errors 135
& K! Y# l' D- k7 _4.12 Boundary conditions and conservation properties 139
4.13 Extensions to more space dimensions 143
7 v' u+ F/ D  A9 S0 _& ?Bibliographic notes 146
+ k5 X- b- R. I! h! L! \5 JExercises 146
* w! e  j+ @" ~- }8 A3 ?! x: u5 Consistency, convergence and stability 151
5.1 Definition of the problems considered 151
( f5 ^: W& T" |" @( s, j5.2 The finite difference mesh and norms 152
3 U8 l; n3 Y/ {: q" E: j. T3 }; y5.3 Finite difference approximations 154
6 g. t0 M' _2 G, N+ Z5.4 Consistency, order of accuracy and convergence 156
5.5 Stability and the Lax Equivalence Theorem 157
% V2 v6 g* U3 w& L' I5.6 Calculating stability conditions 160
8 [/ p. S6 B. ]& I& _0 k5.7 Practical (strict or strong) stability 166
5.8 Modified equation analysis 169
5.9 Conservation laws and the energy method of analysis 177
5.10 Summary of the theory 186
; j2 d; R; Q; DBibliographic notes 189
: m7 G: C% Y3 J5 R! `6 l+ N* aExercises 190
Contents vii
6 Linear second order elliptic equations in
; c/ h' D& a  i* ~  _- b9 ttwo dimensions 194
6.1 A model problem 194
! \( L: Y7 {8 Q( _6.2 Error analysis of the model problem 195
' Z# s  P' c$ q+ `" N# Q6.3 The general diffusion equation 197
- N* E/ n" t0 l1 b6 U7 P* }6.4 Boundary conditions on a curved boundary 199
6.5 Error analysis using a maximum principle 203
6.6 Asymptotic error estimates 213
6.7 Variational formulation and the finite
element method 218
, x- g8 h' ^$ C6.8 Convection–diffusion problems 224
/ [& q) e0 ~# ^- V7 x+ K3 G6.9 An example 228
. J( j  }5 A& D% r, r3 EBibliographic notes 231
* \4 V, j4 e+ g4 N5 S3 PExercises 232
7 Iterative solution of linear algebraic equations 235
- c! @! c9 ~  k7 g0 w3 ^2 o7.1 Basic iterative schemes in explicit form 237
7.2 Matrix form of iteration methods and
their convergence 239
7.3 Fourier analysis of convergence 244
& a; a; W) j' @# {' I+ g0 [  x7.4 Application to an example 248
7.5 Extensions and related iterative methods 250
( a" d& ]- f- O$ A$ q7.6 The multigrid method 252
  h, g/ w$ T4 Y/ Z$ V2 x' Y7.7 The conjugate gradient method 258
# E! k$ ?$ }9 C5 u7.8 A numerical example: comparisons 261
Bibliographic notes 263
6 Y  y1 a# Z9 }3 x5 gExercises 263
References 267
  Y, K  C2 m. ]! |Index 273

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