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原版英文书 第二版
4 m+ W; t$ W& F7 T/ C1 econtents:
0 f7 P( T6 y" b4 e) @9 w" a0 PPreface to the first edition page viii
- D0 d) Z/ H4 r0 t& x! w* t5 mPreface to the second edition xi
- ] J; z, q1 u( A7 Y y8 W2 c1 Introduction 19 _- D+ N2 o) O5 t
2 Parabolic equations in one space variable 7
( W$ [! i4 n1 D, j. }7 S5 X; Y2.1 Introduction 7
9 h& v( k) [1 }2.2 A model problem 7$ P8 w3 m0 ~/ Z* w/ }
2.3 Series approximation 9
' f0 x9 r6 j4 p7 M9 t2.4 An explicit scheme for the model problem 10! {9 o% G h: G4 A4 k
2.5 Difference notation and truncation error 12) F9 D: U" t1 A/ {# v) @0 {( i
2.6 Convergence of the explicit scheme 16* r8 i7 j3 F8 c+ O' ^
2.7 Fourier analysis of the error 19
/ C. r( u$ Z* p# a2 G; P+ Z2.8 An implicit method 22 O" ]5 e, p, E5 {9 d
2.9 The Thomas algorithm 24) P3 L: E# J! t. K5 B9 I
2.10 The weighted average or θ-method 268 X* {0 L( _2 B" a' b& k1 [7 P
2.11 A maximum principle and convergence
! O7 U1 j* x! i _+ Ifor μ(1−θ)≤ 1! h1 w+ X5 Q x/ f
2 33% j- g' A! q$ G5 {$ A, [8 X
2.12 A three-time-level scheme 38# c' m) Y5 b/ V3 O3 @+ P* I5 F
2.13 More general boundary conditions 39) Z$ X0 c- M) o& X/ H% Y6 }* w
2.14 Heat conservation properties 44
5 e; R3 _& n9 b( x3 W, q2.15 More general linear problems 46; F) b! L% b2 v: [% E
2.16 Polar co-ordinates 52
6 f4 _6 T- l" I3 M2.17 Nonlinear problems 54
! M* z) _$ {- c2 Y2 {8 zBibliographic notes 56/ e5 l: _, h' ^6 w2 p
Exercises 56
/ V) W3 L" L& v8 g1 gv
/ P0 _1 r2 d6 e4 Z/ x" ]vi Contents
+ I& m0 a$ _* ?9 {$ r3 2-D and 3-D parabolic equations 629 p; ]' U- ^7 g1 b% K) d
3.1 The explicit method in a rectilinear box 623 k+ ]( m/ c4 x- j# d* [
3.2 An ADI method in two dimensions 64
; d3 j% V7 `# _3.3 ADI and LOD methods in three dimensions 70
; V) M3 Y Z& p6 B" D9 L* o. k3.4 Curved boundaries 715 }$ r' L! n; f
3.5 Application to general parabolic problems 80
5 X4 D" m5 x% T# IBibliographic notes 839 U3 ~ ~- |- o$ a- g7 W0 b" a6 t
Exercises 83# |5 r% P/ G& R0 v: v4 A
4 Hyperbolic equations in one space dimension 86% I9 C1 r' N/ }8 I+ X, F, v
4.1 Characteristics 86
0 X$ I/ s' C+ M! w9 C1 s4.2 The CFL condition 89
; p A: A8 r. o& V Q4.3 Error analysis of the upwind scheme 94
1 a5 U f$ y# z' H I5 a4.4 Fourier analysis of the upwind scheme 97
! Q; @, H4 F( c1 Q) D' i4.5 The Lax–Wendroff scheme 100
8 k1 J, ^! Y3 x- \/ Q" H! L' a2 p4.6 The Lax–Wendroff method for conservation laws 1034 E! H% v, d9 i* j$ S: Y/ S3 f
4.7 Finite volume schemes 110; g. K; K: K) a/ }5 F% S; c& K- Z
4.8 The box scheme 116, o- `3 }7 h8 q
4.9 The leap-frog scheme 123
* S3 D" n; T2 _+ z# B4.10 Hamiltonian systems and symplectic z3 w! }, V+ J. j
integration schemes 128
N* O0 L6 ~$ z( b6 G# f4.11 Comparison of phase and amplitude errors 135
- n4 ?) o9 t) t, h# |* b4.12 Boundary conditions and conservation properties 139
- X! Q; P) Q, N4.13 Extensions to more space dimensions 143
( ~0 Q, j' O% D3 z7 v/ d' H8 b o! \Bibliographic notes 146
3 d6 |. h& ^# i( ~ j( O1 DExercises 146
% _& y5 k) x- z3 i" _8 @+ c$ F6 g5 Consistency, convergence and stability 151
# I; L9 ]2 w) J- k" b/ c+ T0 ~% R4 W5.1 Definition of the problems considered 151
( a4 g0 u$ M' ~2 @) l/ C+ }5.2 The finite difference mesh and norms 1522 J7 r/ r! i& _( \2 W/ `
5.3 Finite difference approximations 154
3 ]6 w, t! c0 D/ R5.4 Consistency, order of accuracy and convergence 156
7 I0 d9 k+ Y8 G3 M4 G& `$ P( s- ~5.5 Stability and the Lax Equivalence Theorem 157
3 T6 I/ p+ e4 i% m5.6 Calculating stability conditions 160% \) A# {/ k* v3 C- @' ?! }# i
5.7 Practical (strict or strong) stability 1665 |( T# u n% U9 ?
5.8 Modified equation analysis 169% P* q) h' }2 n" [3 f; J+ s
5.9 Conservation laws and the energy method of analysis 177 ~" k2 R) m: @9 i% V
5.10 Summary of the theory 186, M, p: I2 p$ r* M: J
Bibliographic notes 189. o a5 x4 V. o% i* O
Exercises 190
$ c$ p: k$ ^7 } y. U# QContents vii+ A) |7 ]9 o, B6 d& y
6 Linear second order elliptic equations in
# h: O$ }: l& x0 |6 z5 k5 `. V2 e5 P, [two dimensions 194
' `* x1 D! ] u7 H- }5 N6.1 A model problem 194: w# \' C( L, r! g0 c8 y9 P. S- q
6.2 Error analysis of the model problem 1952 `& j1 T5 ]; p2 ]# E
6.3 The general diffusion equation 197
6 v' H4 e4 l( e) b2 n6 @- e6.4 Boundary conditions on a curved boundary 199: m- G2 o4 y" `) I
6.5 Error analysis using a maximum principle 203/ ^/ |& `$ v6 \0 F' j+ A
6.6 Asymptotic error estimates 213# E8 t, e% _: u9 I0 k5 o
6.7 Variational formulation and the finite* i% b5 W# {9 u0 ^
element method 2183 ?: m: X9 H2 K$ }3 u8 T+ _
6.8 Convection–diffusion problems 224
( A. b0 g* P* n! X6.9 An example 228
' A3 x5 z, A1 n* mBibliographic notes 231
- @, \1 f/ q l% UExercises 232
; N, ]+ `8 \/ d. N7 Iterative solution of linear algebraic equations 2357 j7 c! [/ n! ^& K
7.1 Basic iterative schemes in explicit form 237 Z( h5 V% e* v+ F& g" H( i
7.2 Matrix form of iteration methods and
* F( `2 C: X0 m) a( C/ R; H" ^+ }5 L2 ttheir convergence 239
+ R8 m: Y$ x0 K5 K. t7.3 Fourier analysis of convergence 244' U6 n3 J2 b b2 }0 |% n
7.4 Application to an example 248, H6 U! }# p: A
7.5 Extensions and related iterative methods 250
7 v; p% N4 o# j2 {7.6 The multigrid method 2522 ~( Q, A. v9 H! a! Z
7.7 The conjugate gradient method 258! s! l5 M0 n- |0 d6 N8 W4 T
7.8 A numerical example: comparisons 261* }7 b) A% D) H R: K
Bibliographic notes 263+ s+ D8 i, P" [. k' V8 C
Exercises 2639 R5 |( R4 {' R
References 267
! X& x6 R g7 \7 } L/ Z- D. LIndex 273 ?- O6 o# P8 u4 P6 C: U' K7 N( b
* g5 i/ X1 p( }. P2 h5 f* j) }2 a8 L! k2 \
) G' |' f/ l8 H3 x
/ s, p6 l9 j% @: S4 Y, v- E; n
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