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原版英文书 第二版 [7 ^3 \8 p' f
contents:0 r3 W2 E' ?! v' P! F! ^& |7 L
Preface to the first edition page viii3 M& \& K0 y5 w; B: s
Preface to the second edition xi1 Q! e( D' {. q4 y3 Y5 O; U' |
1 Introduction 1
: c5 i1 s1 P& n N9 L& z2 Parabolic equations in one space variable 7! t+ j5 o% ]0 ]# d
2.1 Introduction 7
3 r! n, v) ]8 }$ M: M2.2 A model problem 7: e9 ~* V6 j& b0 v5 s1 N" w
2.3 Series approximation 9" F& j! [$ g" ]
2.4 An explicit scheme for the model problem 109 B6 ?; S9 X9 I1 }/ A! D
2.5 Difference notation and truncation error 12- e$ I/ G6 C: L& p5 @
2.6 Convergence of the explicit scheme 16' X- ?* ^- p9 e3 M8 L. H
2.7 Fourier analysis of the error 19# f* m& q5 D9 `- j2 t$ J$ }! ?
2.8 An implicit method 22
1 P7 ^3 n9 f l b' I& u) f6 v* v6 e2.9 The Thomas algorithm 24
- ^# t7 M: ^6 o* B& [0 a2.10 The weighted average or θ-method 26& Q) T$ m/ I$ ?3 |" C4 W3 Q
2.11 A maximum principle and convergence
& C; X& R4 W0 c% ]/ Q3 v, _for μ(1−θ)≤ 1
7 y1 ?! g. a$ o/ `5 [! @ \2 33" Q2 Y' ? Z6 W- \- j
2.12 A three-time-level scheme 38
/ K* ~& e! y M7 a5 v( j7 t2.13 More general boundary conditions 39
2 q6 d; Y4 t: {6 j0 J M+ u* b2.14 Heat conservation properties 44: v9 j! A' x* X3 @+ {
2.15 More general linear problems 460 A/ l1 u6 n' ~% `$ |4 I% I
2.16 Polar co-ordinates 52
. D, i# V+ ]# N0 U# H2.17 Nonlinear problems 54
3 F, y1 j4 Y/ A+ [1 r C1 ZBibliographic notes 56' q4 \( v% R8 I$ W+ F7 ~* [
Exercises 566 J, c7 k/ X3 X
v0 F' m$ J2 E. v
vi Contents# L8 m, W! g9 s! C/ a; X
3 2-D and 3-D parabolic equations 62 i0 T% N- P" @
3.1 The explicit method in a rectilinear box 62$ I7 @7 H' \6 r- T; S6 j7 K
3.2 An ADI method in two dimensions 64
" p, u+ ` j( i# o3.3 ADI and LOD methods in three dimensions 70
2 o9 @6 p- N# Y1 t3.4 Curved boundaries 71
5 [& w2 a2 v! U0 i8 W! m3.5 Application to general parabolic problems 80
F9 V! ] h+ p: u2 J* N9 r6 FBibliographic notes 83
5 A) C3 G) ]" _9 E/ Z, gExercises 83
: z" H( \" n* m9 Z" H/ @' }$ _8 H4 Hyperbolic equations in one space dimension 86& M" [' w/ J+ \
4.1 Characteristics 86+ Q; O4 w4 H; h l" H) }
4.2 The CFL condition 89
( v5 }3 l% R$ ~" Z8 z9 e4.3 Error analysis of the upwind scheme 94
, l. W5 w2 n; N9 ?1 b& b4.4 Fourier analysis of the upwind scheme 97
6 z7 d4 z. J4 U: S4.5 The Lax–Wendroff scheme 1004 r, s+ I/ f7 W I
4.6 The Lax–Wendroff method for conservation laws 103+ c' j$ c7 e" k7 s' ]
4.7 Finite volume schemes 110
( A, O( r) A; |, A1 x3 q4.8 The box scheme 116- @* J* C* B/ H8 v4 j
4.9 The leap-frog scheme 123
. r5 H, o/ Y! K" ]/ E/ A4.10 Hamiltonian systems and symplectic
. }/ t. [ Q& B# P! ~- Eintegration schemes 128
2 }9 g# d, D, x, j( d" X: G# B; m0 r4.11 Comparison of phase and amplitude errors 135
! r5 f9 d% Q0 z9 }: @& r. P+ ~' R4.12 Boundary conditions and conservation properties 139! \' A5 G5 A8 r1 ~9 B
4.13 Extensions to more space dimensions 143& v! a$ s0 f( t( ?* o+ w) y
Bibliographic notes 146
: c; U( ]! L _0 dExercises 1468 u- ]) {* R* i D
5 Consistency, convergence and stability 151
; R8 q' w" W# M5.1 Definition of the problems considered 151- M8 @, z9 ?5 Z9 g4 T
5.2 The finite difference mesh and norms 152
! K9 S" E4 h! `6 K' Q5.3 Finite difference approximations 154. y% E( s# M$ f
5.4 Consistency, order of accuracy and convergence 156
: L: m v+ A* p5 O. P5.5 Stability and the Lax Equivalence Theorem 1572 [$ K' x) i z$ {, l7 M
5.6 Calculating stability conditions 160 [2 F5 P o# A Y; p. E: w
5.7 Practical (strict or strong) stability 166" ]9 _' Z3 D9 I' @3 h2 }
5.8 Modified equation analysis 169 d; T4 `* X+ u7 y
5.9 Conservation laws and the energy method of analysis 177
; F+ F/ N" {0 E( U9 | w \! ?: u3 A- Z5.10 Summary of the theory 186$ y+ E% t& a% O- b) b$ B
Bibliographic notes 189: b6 B. I0 H0 M3 j
Exercises 190
- J- `- j+ K# y* E) zContents vii7 S. X) A. k' _& [
6 Linear second order elliptic equations in
, J9 n, b! K4 T( ^two dimensions 194# u4 e6 p: X2 J
6.1 A model problem 194
. U/ d* M# m* G4 h& M8 u6.2 Error analysis of the model problem 1950 ?9 w1 C. Q" ^1 l _
6.3 The general diffusion equation 197
& M) \) G) A( `; V6.4 Boundary conditions on a curved boundary 199# p7 i/ X6 Z+ ^) x0 c
6.5 Error analysis using a maximum principle 2031 k/ l' p" d' d# B* n( {$ T
6.6 Asymptotic error estimates 213
+ {" t' |2 k8 ^6.7 Variational formulation and the finite
+ k) _" K6 Q- E2 zelement method 2183 n* Y: ]1 |: D$ J0 B6 ^
6.8 Convection–diffusion problems 224# P/ }$ P1 n8 {2 p6 x% a
6.9 An example 2282 K8 L, l9 \2 O- ^
Bibliographic notes 2313 L! ~9 ]/ s+ C% W& u& v
Exercises 232
! w: W. ^, f1 {! b) _( K" f" N7 Iterative solution of linear algebraic equations 2353 r. ?1 K$ n; l
7.1 Basic iterative schemes in explicit form 237
w$ y% t0 P# i, q3 D7.2 Matrix form of iteration methods and' l! @+ I* d" W& y$ f( _4 [
their convergence 239( R1 h5 J. C# D) ~
7.3 Fourier analysis of convergence 244
1 S6 A' [) W5 H6 L% h. S6 ?7.4 Application to an example 248
' u1 x* D! j/ Q2 n' t3 ]( j7.5 Extensions and related iterative methods 250( H8 [/ T: L. m' {- `8 Q) L
7.6 The multigrid method 252" f( ?. e! c+ X, H3 G0 K
7.7 The conjugate gradient method 258' V' Q/ F7 R2 B% A& K
7.8 A numerical example: comparisons 261# D) L& E& d+ L% Y# a
Bibliographic notes 2631 o. ?0 c' k/ g7 }# H
Exercises 2636 w+ |! B/ \" f& K/ E
References 267' I; W0 ?: C' r% ]% ] J9 R
Index 273 : v- K1 _% {4 ] r6 O$ S: a
+ T: Y& [+ G# E2 v
% |9 B2 m# Q5 }- |' ]/ _% @
" J- ]/ `0 Y# ~- y
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