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原版英文书 第二版9 A! q! [* d7 K$ {" C
contents:
+ y, a" J$ }! PPreface to the first edition page viii
4 @( ]* K3 U4 t6 P$ ?Preface to the second edition xi* e+ n" g5 l( `" J
1 Introduction 1
+ ^ L# i& n& d. V2 Parabolic equations in one space variable 7
" @8 R; y0 E+ z. g1 E& u2.1 Introduction 77 K" O/ l) I! t; e w' i
2.2 A model problem 7+ ^. S9 x( T) ?2 D
2.3 Series approximation 9
% T |6 A- I3 G0 b2.4 An explicit scheme for the model problem 106 n/ s* Q) f% E6 g3 ?$ ~
2.5 Difference notation and truncation error 12, F/ s# \/ {1 c$ O; ]6 H) `
2.6 Convergence of the explicit scheme 16
" P( U& |4 k: R6 @2.7 Fourier analysis of the error 197 A0 Q+ ^3 @5 t) C
2.8 An implicit method 22/ O$ }4 H* s; u, D% j
2.9 The Thomas algorithm 24# C, g& M3 H* T' @
2.10 The weighted average or θ-method 26
6 B% m9 q# q+ }( A2.11 A maximum principle and convergence
4 m# U3 Z u% _5 l. s1 c: G6 Efor μ(1−θ)≤ 1
, ]+ \) \! Z5 c2 33
1 w! D3 D' b9 w6 o3 t2.12 A three-time-level scheme 38
# m9 H, J, i/ w2.13 More general boundary conditions 396 c& R H2 ]2 Q3 H% |' p
2.14 Heat conservation properties 446 y+ A( K2 T5 z( X+ Q# h) v
2.15 More general linear problems 466 R9 A: F7 O, P) \
2.16 Polar co-ordinates 522 ~* b' |" R/ Z$ l; X3 }5 o
2.17 Nonlinear problems 54
+ b& ~, P' E$ |# u4 a+ xBibliographic notes 56
# J' W8 \3 [! G# _' jExercises 569 t& m0 z K U/ F7 N* {/ r
v0 m2 m4 L8 W' L! m: a
vi Contents
5 w/ z3 S4 c& U* U/ P; l3 2-D and 3-D parabolic equations 62
0 F. i6 ~' m) z3.1 The explicit method in a rectilinear box 62# r! ]* I7 ]/ V* L
3.2 An ADI method in two dimensions 643 G5 H; Y: [' X4 ?* w e
3.3 ADI and LOD methods in three dimensions 70
5 k+ l( _; Y8 q. y8 V+ ~% y8 p3.4 Curved boundaries 714 x" B6 }3 V2 g! m/ l% G& |
3.5 Application to general parabolic problems 80+ g, h7 T/ G5 v/ Z% H
Bibliographic notes 831 h, ]! z) r6 l; M( F& F
Exercises 83 I" Y* w) e* O: G" N
4 Hyperbolic equations in one space dimension 86
0 ]8 _: ~1 \! c! ]( ]0 ]; U1 N o; V4.1 Characteristics 861 I# o. b8 [ k& q9 ]8 P
4.2 The CFL condition 89
3 A. Z" Y ^7 \# R5 U3 t4.3 Error analysis of the upwind scheme 94. h# @( G# R; v- J
4.4 Fourier analysis of the upwind scheme 97& I% H: \/ t! U% M) X4 g. b* H
4.5 The Lax–Wendroff scheme 100+ Y+ c) d1 u4 M/ q/ _- z( h
4.6 The Lax–Wendroff method for conservation laws 103
4 n# h& J4 P2 W9 P6 d; w; C( V4.7 Finite volume schemes 1108 F5 N" F0 Q1 }7 Z u* r0 e
4.8 The box scheme 116
5 G/ E, M( T6 ^7 R) s4.9 The leap-frog scheme 123
1 W4 b4 ^2 [; q9 i/ i7 M4.10 Hamiltonian systems and symplectic7 L5 O, I$ B; C$ |
integration schemes 1283 K, u: B* B, G/ G# ^# o" {
4.11 Comparison of phase and amplitude errors 135" U3 [" @: R O F
4.12 Boundary conditions and conservation properties 139# A( l- ^; M8 ]
4.13 Extensions to more space dimensions 143
+ Y1 c! v. U/ _" y9 m3 }4 wBibliographic notes 146( c6 Q; |5 r# z6 G2 G4 \
Exercises 146
5 Y; L& P) I7 ]/ K9 u# K3 ~! _$ c t! |5 Consistency, convergence and stability 1515 z) w7 T0 i* P* p6 i
5.1 Definition of the problems considered 151
* c/ U0 E2 c6 v6 [8 _& j8 x! a7 \. U5.2 The finite difference mesh and norms 152. c/ e0 A* Y9 S A+ E
5.3 Finite difference approximations 154) t* c: T9 r6 D2 I' V3 G
5.4 Consistency, order of accuracy and convergence 156
/ P% l) t1 E5 g- W. u5.5 Stability and the Lax Equivalence Theorem 157* \+ L3 e9 U1 L; n+ |2 ?/ d" N
5.6 Calculating stability conditions 160, R, ~$ ?1 i. L, _/ ^9 W& F( s& A
5.7 Practical (strict or strong) stability 166
2 S! H# `) F% O F2 W9 B3 h+ B5.8 Modified equation analysis 169
: Z; O4 U; b$ K! t: P. G5.9 Conservation laws and the energy method of analysis 177/ O) T. R+ ~, S
5.10 Summary of the theory 1861 w- [+ R% y: @0 x" V' M1 v
Bibliographic notes 1891 I$ N, |+ }) ~* K$ E4 q
Exercises 190! q/ c5 i& M* U1 f0 ~1 N
Contents vii
- B8 l1 C3 R/ ^, m4 k+ b6 Linear second order elliptic equations in
) b' K5 r0 t6 g6 X. S Q' k- dtwo dimensions 194) z% S% q! ]# k1 _0 @( ?
6.1 A model problem 1941 G/ W5 g" |( z
6.2 Error analysis of the model problem 195
. W4 f, E. G4 {, f6.3 The general diffusion equation 197. X2 [) u0 ]8 T
6.4 Boundary conditions on a curved boundary 199
- X8 @0 x. D; V# z6.5 Error analysis using a maximum principle 203' e5 z# Q( M" K$ ~
6.6 Asymptotic error estimates 213: h9 ?3 B l% P! j
6.7 Variational formulation and the finite
; \) ~. e. A7 ] k! b6 m: nelement method 2182 l! n( S) Z- F# v# O/ o
6.8 Convection–diffusion problems 224
4 m* x2 S; d+ D+ T4 \! p- S$ X6.9 An example 228
' ^: ?( Z/ ]) Q# O! k% }/ ?4 Q; kBibliographic notes 231/ b x n6 q, l
Exercises 232
- m0 P7 r! A& G1 P7 n. T7 Iterative solution of linear algebraic equations 235
: b: a1 G$ _: |) m. m/ [8 o# W7.1 Basic iterative schemes in explicit form 237" v! ~$ }# h/ g) q3 e$ s9 i, j* F
7.2 Matrix form of iteration methods and
: ^# h$ ^/ X3 d' o9 a$ vtheir convergence 239- W( V. R; p- C9 M ~
7.3 Fourier analysis of convergence 244
* X# R; I% H& b3 d) x T7.4 Application to an example 248
/ F6 S7 [# }. h3 ~7 {2 v7.5 Extensions and related iterative methods 2504 a1 d$ H: o- s0 b
7.6 The multigrid method 252$ F: ^! o! l$ Z1 V6 A, |9 Q
7.7 The conjugate gradient method 258
2 M+ L: j: y. D0 \& A, S7.8 A numerical example: comparisons 261
2 }9 v% J# z# c) O9 x5 s! pBibliographic notes 263
2 {6 l" |9 n6 |Exercises 263
/ w% b7 T# W* I# p7 `3 Z9 s. qReferences 267/ k* q7 K0 ~" W2 r8 Z z
Index 273
7 t3 {$ F2 J. ~! r' Y/ h& g2 ~$ }. k: p% D( Z. }4 Q: t, V
# Y8 p; |+ A: m( U
' @# d2 h: T( x# Y3 @: D' P: D6 J+ A$ K M# {/ t6 M
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