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原版英文书 第二版1 ?% i7 o- ]# i' _: w
contents:2 r; D4 ^2 v9 ]$ _
Preface to the first edition page viii/ J9 s$ w/ D8 P0 y4 {; T* h; w5 y0 a
Preface to the second edition xi
2 M+ j" `2 W9 S4 K u/ k' i$ e; D1 Introduction 1
9 }! ] _4 o: e% C: D6 F ]3 r4 b0 `2 Parabolic equations in one space variable 7! }4 u; a2 _1 u- I5 E" z
2.1 Introduction 7
* b6 z9 @3 Y/ ?2.2 A model problem 7
1 e& a0 b: ^* v. b S2.3 Series approximation 9$ X; u9 m, G( j4 E# V
2.4 An explicit scheme for the model problem 10
$ M2 E5 K0 b+ I2.5 Difference notation and truncation error 12
$ W2 l/ S$ }3 L# a$ X% n4 W2.6 Convergence of the explicit scheme 163 V0 U q+ `# v% t; l3 j# o0 _) ~/ N
2.7 Fourier analysis of the error 19' |3 F% `6 R; U1 b& W4 Z6 o6 F
2.8 An implicit method 22
* A; ~2 r" o7 e1 O2 d' j- n$ j2.9 The Thomas algorithm 24
: p, }' _9 b$ u; L& I2.10 The weighted average or θ-method 26
( f! A0 n! V# {( A" @0 W* K2.11 A maximum principle and convergence
8 \8 O# U) V4 Wfor μ(1−θ)≤ 1
0 Z, |' x0 P- d: e5 \8 F6 f8 u2 33( \, { l: G. Q- a5 [" j, h
2.12 A three-time-level scheme 38
& r' [4 R& V) U3 R# V8 @* x2.13 More general boundary conditions 39
) A7 A+ Y. H8 ^: T$ G2.14 Heat conservation properties 44
+ x9 }! c/ l, F! o r2.15 More general linear problems 469 M+ J8 H7 S" k+ e8 Z! O
2.16 Polar co-ordinates 52
9 b$ j' e6 M3 @; {2.17 Nonlinear problems 54
. w) {! u. m5 `* gBibliographic notes 56. W. V* g" {6 Q) M }; S* ]
Exercises 56
1 h/ G+ x4 C1 Q& J: Tv/ K9 T8 S% m1 z o p
vi Contents1 c% |) o, u+ M# S
3 2-D and 3-D parabolic equations 621 D( V6 \4 C+ l3 D/ h8 M* m
3.1 The explicit method in a rectilinear box 62
& H7 C, y' \% ?- h: p5 U3.2 An ADI method in two dimensions 64
7 a+ T3 A1 t7 |* j) A$ s3.3 ADI and LOD methods in three dimensions 70
8 P. Z& X4 S h$ ~0 R3.4 Curved boundaries 71
9 L7 C7 U3 G' |8 [3.5 Application to general parabolic problems 802 k# r% U: K7 M# i' C
Bibliographic notes 83
^1 n4 J# |: ~! WExercises 83
5 C9 b. m+ _; S6 N- d, H4 Hyperbolic equations in one space dimension 86
" b7 }/ U; T }* O4.1 Characteristics 86
" b; u7 V6 J" _( e) @4.2 The CFL condition 89; p. C: G) t4 r* T# ~' t4 V4 I( P
4.3 Error analysis of the upwind scheme 94
* R& W" p5 t8 U5 C4.4 Fourier analysis of the upwind scheme 97
) v2 z% p6 r) D8 B! Q& _4.5 The Lax–Wendroff scheme 100* v! ?! P+ u$ V3 W
4.6 The Lax–Wendroff method for conservation laws 103$ u2 T2 }" H+ r1 P/ k4 Z; F
4.7 Finite volume schemes 110: U" i' F4 |8 g8 D- }" H/ [
4.8 The box scheme 116. H P# U- g2 G9 j7 G5 I- Z
4.9 The leap-frog scheme 123. Y# d& K+ M5 c- u/ w3 |% n. Q* [. z3 ]; ^
4.10 Hamiltonian systems and symplectic+ L* v1 T4 ]0 W5 B% }
integration schemes 1280 G4 }% c1 M' |/ v
4.11 Comparison of phase and amplitude errors 135% y2 J0 z0 A) {9 s
4.12 Boundary conditions and conservation properties 139
( @9 ?' t: Y5 H, K. t6 F( n* p0 h4.13 Extensions to more space dimensions 143
/ v2 f) Y, x$ \% i! G* fBibliographic notes 146% G9 l; J( r# ] X% }: y0 b
Exercises 146. l0 D1 ^: W, h1 x6 Z6 J2 W+ w! ?( ]
5 Consistency, convergence and stability 151
- j- N3 B4 h& ?6 ^- w5 T5.1 Definition of the problems considered 151
' M( B4 }: F6 {$ ^% @3 a5.2 The finite difference mesh and norms 152$ E, W$ m+ }6 B2 ^' W7 J+ Q
5.3 Finite difference approximations 154; o& y1 [# [# J3 j" p( B+ J
5.4 Consistency, order of accuracy and convergence 156, M8 k, |4 H4 P+ q* D; ?) p
5.5 Stability and the Lax Equivalence Theorem 157, [; p1 z: m+ A" P/ c1 l1 `( A
5.6 Calculating stability conditions 160
! j9 i: X! w3 M* i5.7 Practical (strict or strong) stability 166
: F/ }3 P2 @( U5.8 Modified equation analysis 169! I+ i* Y- I1 ]& s t4 H: i+ u
5.9 Conservation laws and the energy method of analysis 177
' [) d% P2 |% |8 l5.10 Summary of the theory 186% L4 U: C- J/ M- ^0 F4 G- c% F8 K& p
Bibliographic notes 189
- W8 r+ Y a9 k5 e0 L8 G" f& `Exercises 190& K/ m1 B# S9 Q
Contents vii4 x% r" y) @$ O" {& J. T
6 Linear second order elliptic equations in
9 `! C; {+ z( G) [4 j% wtwo dimensions 194
( a8 c" s2 y: O6.1 A model problem 194
3 ^- F4 O& A' m8 L2 u5 g6.2 Error analysis of the model problem 195
8 r. {0 j; D" j: D6.3 The general diffusion equation 197
8 @ y- {& h# E; o6.4 Boundary conditions on a curved boundary 199
* O. v' g+ [3 A' u6.5 Error analysis using a maximum principle 203
( V# N+ R1 L+ H& @- V2 O/ l6.6 Asymptotic error estimates 213( |1 ?, {" v. v: P+ q9 O( a
6.7 Variational formulation and the finite
6 V% ]# q' j" q( q3 \ [, relement method 218% @1 u8 F0 U' j5 i: K1 F! |
6.8 Convection–diffusion problems 224
$ n) t& a* b' ]" f$ K& f6.9 An example 228% a1 \% [7 a4 o b' ]4 |9 O
Bibliographic notes 231
1 ]+ ?- J& l) L8 zExercises 232
& T2 ?9 M2 M% N% y) }" R+ u( d, H! Q7 Iterative solution of linear algebraic equations 2354 M% ^+ l1 C' t; c! w8 @
7.1 Basic iterative schemes in explicit form 2374 Z4 H& d7 [& E* l# B( s3 D X; T
7.2 Matrix form of iteration methods and' \' z4 G: u# N
their convergence 239' I7 u9 J5 c4 T, E* W
7.3 Fourier analysis of convergence 244: n+ D# o/ [* [8 I @; x& W
7.4 Application to an example 248
, X. z' P8 N+ J7.5 Extensions and related iterative methods 250) U& [: N* G/ ]
7.6 The multigrid method 252
7 [( t p. V; e1 C7.7 The conjugate gradient method 258
1 W6 ]! `6 V9 u" S9 a7.8 A numerical example: comparisons 261$ E" R, u9 m1 }: v3 p# a1 o/ k
Bibliographic notes 263' F* h& ~1 E' [2 T
Exercises 263
) x9 I8 N- y" Q6 M* qReferences 267
+ L' P/ A" \, Q' G7 IIndex 273
K: s" G. z; }) Z3 b
4 t, O9 |% ~. N# _: _
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