原版英文书 第二版# {0 Z e& v1 K8 P; D2 \# K
contents:
4 e3 g" u; y0 x" h6 sPreface to the first edition page viii
$ _) G. |6 R' _9 E1 O. T! B0 qPreface to the second edition xi8 V T/ X Z# r' z) S$ i) y/ I0 ^
1 Introduction 1+ m1 b/ h0 A q- b! ^; \
2 Parabolic equations in one space variable 7! c7 g Z; G2 M: H- d# S; U
2.1 Introduction 7
6 L+ S* a7 M, D0 H- W- k2.2 A model problem 7/ ~% D3 K/ V0 U/ W& z" A3 P% {4 v
2.3 Series approximation 93 j) {1 [/ e) O/ ?
2.4 An explicit scheme for the model problem 109 ?- i3 L3 b9 t% v! C- b0 B! q |# ?
2.5 Difference notation and truncation error 123 d# }8 @9 I( v
2.6 Convergence of the explicit scheme 16% C. W8 e$ c0 |
2.7 Fourier analysis of the error 19+ A2 L8 B+ Z3 [, @% Z7 H
2.8 An implicit method 22# L# I9 h( M: u6 h0 |4 m2 e! D
2.9 The Thomas algorithm 24* b& f* N; x2 w: m# X6 r$ ~
2.10 The weighted average or θ-method 26
9 ^3 `3 i, T8 e: O- _1 ~2.11 A maximum principle and convergence
: {" M5 v+ s. h" }7 wfor μ(1−θ)≤ 1
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, ]" z# M! C- J3 ~4 Q$ E& \2.12 A three-time-level scheme 38
+ d- N, u/ \5 U# ~4 x2.13 More general boundary conditions 39- n5 w# }% Z" ~8 w6 w+ y
2.14 Heat conservation properties 444 W: I8 f; H4 G9 H
2.15 More general linear problems 46- ~; n3 g7 l4 e# C
2.16 Polar co-ordinates 52
9 x% h" e4 ^6 H' X+ s9 |2.17 Nonlinear problems 54' Q0 z& d5 z+ G& e' U& [
Bibliographic notes 56. r' V4 Z( K# J# [1 v% B' t! U
Exercises 56: A. O$ d) x% d- y, C9 |
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vi Contents
, q9 t0 y' R- D9 V& T- N3 2-D and 3-D parabolic equations 62
$ t9 V; `- B# `1 K3.1 The explicit method in a rectilinear box 62
% I+ r4 w- }$ g6 b# ]" i3.2 An ADI method in two dimensions 64
% j" v, X! t) K3.3 ADI and LOD methods in three dimensions 70
9 L, Y6 ^7 C& t1 g; F5 ?3.4 Curved boundaries 71
' U" \2 V3 w$ ^' A8 {3.5 Application to general parabolic problems 80
+ j& W" f* `" |! K4 S! j: {/ hBibliographic notes 837 R+ n; i, [: [) S
Exercises 83! {5 o5 t6 s1 a
4 Hyperbolic equations in one space dimension 86
# N6 c1 x4 R$ h$ w4.1 Characteristics 865 R! @/ v( r) d( J- `& K4 n3 H
4.2 The CFL condition 89
" i$ X% }0 c9 c4.3 Error analysis of the upwind scheme 94* o' [: L) t; v( n4 N; o# B- S# q$ F. I
4.4 Fourier analysis of the upwind scheme 977 `" E! x. i: w" G( H) x( I5 K
4.5 The Lax–Wendroff scheme 100
8 n6 }/ E; \$ q8 t" u4.6 The Lax–Wendroff method for conservation laws 103 p0 M$ j2 L# g, ^
4.7 Finite volume schemes 1104 [& m8 V7 T. i. o( `3 G
4.8 The box scheme 116
# x" _, s- `4 g* ^0 @4.9 The leap-frog scheme 123
* B6 c4 S' ~1 ~) M4.10 Hamiltonian systems and symplectic
R b+ T2 o, ~( X, gintegration schemes 128+ v' O! \' r: y- T$ H
4.11 Comparison of phase and amplitude errors 135
2 r* P: i9 i, u; f: T/ M# o4.12 Boundary conditions and conservation properties 139- O* V+ {( p! `+ L
4.13 Extensions to more space dimensions 1436 J; U$ u& w7 Q. e# o+ A
Bibliographic notes 146. ]- H% |) U2 c) s6 ~
Exercises 146
/ P# j/ F5 j) D) h( I8 S5 Consistency, convergence and stability 151- I5 D- X% f5 A; Z1 E; [
5.1 Definition of the problems considered 151- a, S' b0 _2 {4 p7 q
5.2 The finite difference mesh and norms 152
1 p9 l& o. w6 }6 e" l3 \% O5.3 Finite difference approximations 154
) r% y& v) v3 `+ n: J% d$ z5.4 Consistency, order of accuracy and convergence 156
/ z1 p! ^1 h" }# F; j- u& T5.5 Stability and the Lax Equivalence Theorem 157
$ A9 Q9 j% ^/ v' j' T) Y0 m% p3 O' G( u$ R5.6 Calculating stability conditions 160
0 R1 [6 f8 x$ o H! t0 P5.7 Practical (strict or strong) stability 166# {) f- O! X8 |4 H* J$ ~
5.8 Modified equation analysis 169
/ H# C6 N D" c1 R& R% a. p5.9 Conservation laws and the energy method of analysis 1775 k1 I, G: @( @2 M6 I
5.10 Summary of the theory 186! ?) L! r8 C! }5 s; q( l
Bibliographic notes 189
n. ^9 X; o3 S; g& B' PExercises 190
, j& z, p" J. lContents vii4 S3 b9 F4 ?" x3 r
6 Linear second order elliptic equations in0 p `8 p0 n5 v. H, p- ], s" F( J
two dimensions 1941 h4 b0 z# v; @
6.1 A model problem 194
$ {, V; Y6 K R( ^8 a8 ]6.2 Error analysis of the model problem 195
% A1 n% u: q& d9 G9 o6 A( c6.3 The general diffusion equation 1973 s2 Z. L+ @& W! a! w
6.4 Boundary conditions on a curved boundary 199
! g7 B) M' w% D) y0 r6.5 Error analysis using a maximum principle 203
( J3 A/ }1 U! v. w/ L6.6 Asymptotic error estimates 213" W; P! O% O- h( r& m4 U
6.7 Variational formulation and the finite
6 ~8 r5 h r9 Qelement method 2185 ^2 Z! v9 H1 K9 ?3 Y
6.8 Convection–diffusion problems 224$ s; Y& q, ]( w* o5 q% P1 i
6.9 An example 228
, I1 g, B. `" w8 r0 @Bibliographic notes 231
$ n) W' g) g/ l& m' q" jExercises 232
( [4 Y# x; z) O2 }% C7 Iterative solution of linear algebraic equations 235
/ z v- t+ L! \# W, n% Q2 T+ t7.1 Basic iterative schemes in explicit form 237% |- e$ P9 @( U, t# A+ F% I
7.2 Matrix form of iteration methods and
# t. d! [2 o! j3 @" p) Q4 W/ _9 N7 atheir convergence 239
$ ^: r" u) x* N5 i: `; ^7.3 Fourier analysis of convergence 244, e/ q& s4 O# ]4 ]7 n: S
7.4 Application to an example 248
9 J* s) K0 J3 I% H8 h" t+ y7.5 Extensions and related iterative methods 250- }, o7 o; g9 k5 r4 R- ?
7.6 The multigrid method 252$ R/ j) Z' W6 `# ^. @
7.7 The conjugate gradient method 258
4 x+ F: o8 d( O; {7.8 A numerical example: comparisons 261) T$ t1 s; z/ h& E
Bibliographic notes 2632 D) u( d R( t; ~
Exercises 2638 Z9 c0 A$ g) J1 L
References 267
k7 j$ s! K; N& Y6 c2 `! ~6 RIndex 273
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