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原版英文书 第二版
0 u* l" {+ a% U- _contents:6 t- C$ S! f8 e. m2 @
Preface to the first edition page viii
a. l8 \& n/ d2 x- _Preface to the second edition xi' ]3 ?4 M. \3 p4 X5 ?+ ^
1 Introduction 1( @. x: ^% x, D R- J7 R# }/ k
2 Parabolic equations in one space variable 7; I0 l( p7 m$ y) K4 i! `- q# I) ?
2.1 Introduction 7
% }/ @# B A7 ?; E+ o# K i2.2 A model problem 7
4 K: e/ D, |! h. K, ]+ Z- _ _2.3 Series approximation 9
, q7 v$ I1 r! g( W% |# p. g2.4 An explicit scheme for the model problem 107 k/ A* [+ |0 w/ t, M
2.5 Difference notation and truncation error 12: j; k1 c% q5 j9 W, `& z* Q1 p B8 B
2.6 Convergence of the explicit scheme 16& c; V: h9 p5 B
2.7 Fourier analysis of the error 194 W% U; ]3 Q: h: s% z/ r# V
2.8 An implicit method 22' P5 T4 D8 S4 Z
2.9 The Thomas algorithm 24- L8 c c$ U p7 O; U! ~% Q
2.10 The weighted average or θ-method 267 K0 U) z) X2 _& l# r5 O" Y
2.11 A maximum principle and convergence
2 K! _0 j& J q2 B0 C5 W3 ?& p# ^for μ(1−θ)≤ 1: F. k- C z, g2 Z5 E& _1 k
2 333 U1 y! t3 v9 E& {* d* C
2.12 A three-time-level scheme 38
0 z* n8 {5 V; D& K2.13 More general boundary conditions 39. {( ~5 k/ n7 o' _' P9 Y1 C% v
2.14 Heat conservation properties 44
5 T, O, Q, G2 V2.15 More general linear problems 46
9 y1 n2 W; L+ z: Q* |2 {& }* l2.16 Polar co-ordinates 52
) j. {( @( E/ ]7 n* b! o2.17 Nonlinear problems 54
5 Y* g" d0 f4 M3 ^Bibliographic notes 56
6 @2 ]0 n8 `2 b: TExercises 56
) M$ L$ g9 J+ ^& `! [v0 K5 k: R2 g8 a% ?+ k1 {: ?$ d
vi Contents
9 `1 W2 s( P, n2 Z( x# c. L3 2-D and 3-D parabolic equations 62
% s1 y6 k6 F v6 o( A3.1 The explicit method in a rectilinear box 62+ A8 [6 T3 Z7 g; @' x1 F
3.2 An ADI method in two dimensions 64
( {; U' I# [( l; G" p; Q3.3 ADI and LOD methods in three dimensions 70
9 q7 x; W0 \" t1 `3.4 Curved boundaries 718 x; v$ ]# E) J2 B
3.5 Application to general parabolic problems 80
5 }) E1 L3 w8 TBibliographic notes 835 D1 F" I3 j( |. T5 o4 V
Exercises 83$ O: \ }- ?3 K! L: @+ m! V* `
4 Hyperbolic equations in one space dimension 86
/ q* G2 Q+ _( i" g" U4.1 Characteristics 86
( x$ K" X- I; e5 k4.2 The CFL condition 89
6 L6 H+ i' v0 x) U) Y4.3 Error analysis of the upwind scheme 94
* |$ G1 S5 z0 d1 ^; j4.4 Fourier analysis of the upwind scheme 97
: p7 X0 j; {1 c/ \; ~+ w5 _& |. l! W( O4.5 The Lax–Wendroff scheme 1003 m& Q+ G* l. t
4.6 The Lax–Wendroff method for conservation laws 103
4 f. R0 z3 h( K5 c" h' @4.7 Finite volume schemes 1104 ^4 n0 |" T3 w7 X/ {9 k
4.8 The box scheme 116# p! l; R$ Y7 J+ b: O' V1 |
4.9 The leap-frog scheme 123, P5 h3 ?0 Z6 B. a$ i1 X
4.10 Hamiltonian systems and symplectic
% d% d' L2 x$ O( [integration schemes 128" c% J" S- a" w6 m3 r
4.11 Comparison of phase and amplitude errors 135* U1 J9 t7 {2 N+ w4 P) p- g
4.12 Boundary conditions and conservation properties 139
' O- _/ @' [1 q; h2 g4 L4.13 Extensions to more space dimensions 143
H3 R3 M% i% L* n6 ^6 jBibliographic notes 1464 Q3 E+ [8 I5 u# G5 m
Exercises 146
; s2 S) Z4 z4 J3 i: ]5 Consistency, convergence and stability 151
4 y1 Y, U( h# ]( C- K5.1 Definition of the problems considered 151
4 \3 i* I8 i6 s, W. C3 B5.2 The finite difference mesh and norms 152
0 t: f! ]8 k3 N$ C" ]" N/ \' E" i% A5.3 Finite difference approximations 1548 O/ c$ q" b' q# W
5.4 Consistency, order of accuracy and convergence 156; H7 ~4 j/ v/ a! V# q
5.5 Stability and the Lax Equivalence Theorem 157
! P* P W$ Y5 M2 Z) E+ U5.6 Calculating stability conditions 160
6 B9 W# G Q' D. x }5.7 Practical (strict or strong) stability 166
! t7 e3 U3 a0 P: }# W5.8 Modified equation analysis 169" a$ F; Q f$ v3 m# t+ R: f0 t
5.9 Conservation laws and the energy method of analysis 1770 c5 }7 I+ N: {% B( m; I
5.10 Summary of the theory 186
% A' }+ y5 J, V4 ^' GBibliographic notes 189
" O( F/ X& S9 e2 V* c: {Exercises 190' ~1 z! P4 v2 \1 ~
Contents vii
9 f4 p& X# M$ o4 Y% b; s. p6 Linear second order elliptic equations in( V) v$ t% E0 F( A0 T; F3 E& ^
two dimensions 194
; Z: }, y6 ~/ z- j) @7 y6.1 A model problem 194' P3 K: G& J$ `% |: \) p$ ^
6.2 Error analysis of the model problem 195
1 V* ^8 Q, }% S/ |* j6.3 The general diffusion equation 197
1 Y% V7 q; M4 O7 r! |# Q6.4 Boundary conditions on a curved boundary 1998 k& e ^" {( L& P, y" C
6.5 Error analysis using a maximum principle 203
' L2 n p1 l$ i& `6.6 Asymptotic error estimates 213: o: N* P5 ~+ `# {. Z t
6.7 Variational formulation and the finite
0 e* j+ B: T# p& I1 R/ e+ \- yelement method 2184 a9 j! O$ a; n
6.8 Convection–diffusion problems 224
; s7 J1 a% L& X) Y# w& r6.9 An example 228
' \$ S" I. G* Y& ZBibliographic notes 231 `- _" T- H& d* a. ]& ]1 a
Exercises 2325 l3 R( B" D! ? O8 ?6 @$ g' L5 }
7 Iterative solution of linear algebraic equations 235
& y0 F- j3 q# @6 ~% q+ L9 P3 T7.1 Basic iterative schemes in explicit form 237
8 V1 _, \% t7 N/ a# }& b# z7.2 Matrix form of iteration methods and3 q9 s; p, D) s8 R3 Z
their convergence 239$ ]3 l0 e9 y5 |4 {# J- b; h
7.3 Fourier analysis of convergence 2448 I& N0 N8 U7 _
7.4 Application to an example 2482 `& h* }3 g1 F I- L
7.5 Extensions and related iterative methods 250
# ^- [1 u$ }) h' B9 L1 q7.6 The multigrid method 252
. M2 h; ~4 ~% ], w6 ~7.7 The conjugate gradient method 258& e" J0 o" J: T0 a! A! R* ~
7.8 A numerical example: comparisons 261( z, m4 j9 H) @3 Y& Q& W
Bibliographic notes 263
( I, t1 R3 c C6 IExercises 2630 Y8 ~7 i5 ]% _, ^! s
References 267% O/ S: ^. j- v& l: L5 q4 s
Index 273
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