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原版英文书 第二版8 S2 h* F( ^7 k/ E- i3 {
contents:
6 x, v. C! R7 ~- l4 TPreface to the first edition page viii- G5 G7 Y2 A$ I$ G- d6 o1 j) z$ `. c
Preface to the second edition xi" ^1 d* i3 G; q* |
1 Introduction 1; C/ ]# u' s, H
2 Parabolic equations in one space variable 7
+ g4 x7 `2 P L5 u0 w8 c3 E2.1 Introduction 7 Z- I+ D' V8 `1 e( l8 T
2.2 A model problem 7
' \6 L$ C2 [( X) d2.3 Series approximation 9
. i0 E, c3 D% e2 K0 E- e2.4 An explicit scheme for the model problem 10
! G2 l" ^: P- c" t) C8 [, G& Q7 F2.5 Difference notation and truncation error 12# s( P% |' e. ?1 N
2.6 Convergence of the explicit scheme 16& b7 m' H3 s5 m6 \0 \4 B# h; ~
2.7 Fourier analysis of the error 19) I) ^+ [( ^+ a0 S, l0 O2 D9 h) Q
2.8 An implicit method 22* C# o; f6 X3 M0 S* b# R8 ?
2.9 The Thomas algorithm 24! W! ~; @8 v! k: Y$ u9 D
2.10 The weighted average or θ-method 26- p$ [; ^ }/ }
2.11 A maximum principle and convergence
3 u% {/ {! E3 G( _% Y, Z/ afor μ(1−θ)≤ 1
* h4 i! [: _9 e2 33
& d a7 o1 }& N; N& [2.12 A three-time-level scheme 38
6 y( _4 X5 ^3 [6 e Z, |2.13 More general boundary conditions 39
. z' L2 r" |/ i6 @2.14 Heat conservation properties 449 y+ a8 T2 d3 h7 A1 q- `& R
2.15 More general linear problems 462 p" b- W2 V/ p7 V& L8 C
2.16 Polar co-ordinates 52+ b& f S* w2 ^8 Q5 E5 j1 u8 A
2.17 Nonlinear problems 54
! D5 F* H% k/ D pBibliographic notes 56/ I( O: R! ?& L
Exercises 56
1 s: \7 S/ h8 ^( B" n! R9 Zv. l V0 p9 v4 c: `/ P& p
vi Contents
3 H1 f1 g$ k4 O1 l y3 2-D and 3-D parabolic equations 62" E% \7 U j1 |! ^" r5 f
3.1 The explicit method in a rectilinear box 62
7 T' y" r* L- G* g; Z' \9 y: Z3.2 An ADI method in two dimensions 64
& D& V& L, S( ~, O; |: Y3.3 ADI and LOD methods in three dimensions 70
% ]. Q+ R* s! E4 S# l3.4 Curved boundaries 71 Q; ?0 x' O6 I
3.5 Application to general parabolic problems 80
: _+ s: K' t6 Z& @. f* H& ?Bibliographic notes 83 I8 r9 _+ z' i6 t$ Q
Exercises 83# V$ f+ J. @+ L) Y* v
4 Hyperbolic equations in one space dimension 86
3 S: Y& o' h8 y8 Y0 K% }) x: b4.1 Characteristics 86
) Q! a: [. `! Q/ T4.2 The CFL condition 89
$ k, v2 Q) @3 ~* w$ L( r' H4.3 Error analysis of the upwind scheme 948 ^1 C. M+ |. H* l* _
4.4 Fourier analysis of the upwind scheme 978 L; [ k2 c! v
4.5 The Lax–Wendroff scheme 1003 A: R) p7 v: J. m2 _* |' e" Z
4.6 The Lax–Wendroff method for conservation laws 1038 N. R t) m+ h: L7 b
4.7 Finite volume schemes 1109 f# @+ g4 L3 t, b2 X
4.8 The box scheme 1163 k4 M) \% t- A" v$ s
4.9 The leap-frog scheme 123, v3 `4 H7 L; |1 q
4.10 Hamiltonian systems and symplectic
+ i. @0 S- B& y4 Z; Pintegration schemes 128
+ d5 [2 w+ R% `" _4.11 Comparison of phase and amplitude errors 135
3 `( g9 o, @8 b' h' ?( ?4.12 Boundary conditions and conservation properties 139( F$ G7 b. v& ?: @0 @! |! M
4.13 Extensions to more space dimensions 1431 S2 B8 i, z2 a
Bibliographic notes 146
. Q9 [* ?# e1 _5 o4 UExercises 146
" x4 f9 ?: [; _- @5 Consistency, convergence and stability 151% z/ C' i1 d( ^# J# R3 @! ?
5.1 Definition of the problems considered 151/ @, ~- m5 s6 \
5.2 The finite difference mesh and norms 1528 {& }, J( z5 o
5.3 Finite difference approximations 154+ b: ]( M( N* O; \) i3 S; ?
5.4 Consistency, order of accuracy and convergence 156
' ], E; \$ K3 r5.5 Stability and the Lax Equivalence Theorem 157
4 S* L( V$ Q. Q+ h4 S8 E5.6 Calculating stability conditions 1602 q9 u9 g# E' o' j y
5.7 Practical (strict or strong) stability 166
# o3 {8 ^3 i/ m" R, s3 X5.8 Modified equation analysis 169
+ t2 Z7 D% T" Q, c9 ?3 r% I5.9 Conservation laws and the energy method of analysis 177) n7 K" O. N) J/ T, z, v8 C
5.10 Summary of the theory 186* i! {: R, o, R: n4 B% ?
Bibliographic notes 1891 y3 ^/ [( C, x4 w- [
Exercises 1907 y! E v, N7 q1 v3 V
Contents vii# L( o7 h: J- ?" Q" }9 T, t8 R5 w& ]: m
6 Linear second order elliptic equations in
- `! q0 ~. j! W( E; |8 @two dimensions 194% M$ o4 t) Q- `/ p. o
6.1 A model problem 1944 k; p; {& l9 M! m9 D$ |- }, }
6.2 Error analysis of the model problem 195( _% ?0 Q$ R' P: O
6.3 The general diffusion equation 197
6 M4 T2 H3 j+ m; O6.4 Boundary conditions on a curved boundary 199
3 n2 S! w/ Y! p3 H6.5 Error analysis using a maximum principle 203
+ U- N1 R4 N4 j% n" O6.6 Asymptotic error estimates 213% j! J/ e/ N+ c) Y% W
6.7 Variational formulation and the finite4 g4 d1 ~3 }2 K
element method 218* L. \ c/ L1 \. J" I D
6.8 Convection–diffusion problems 224
) ~* Y9 R1 ]8 j; y0 ^1 p( H6.9 An example 228/ t) V' q8 }8 O/ C: I
Bibliographic notes 231" k. u* F' a5 h8 U8 F4 K
Exercises 232
5 J) e, I) P9 w, K( h I, K" _! w( W7 Iterative solution of linear algebraic equations 235- t8 r6 l+ ~' B# Z h5 a- O) Z
7.1 Basic iterative schemes in explicit form 2376 r; v/ F8 b, U1 i1 b4 P$ n$ q, O
7.2 Matrix form of iteration methods and
- f3 n4 E q" L4 L! H4 ptheir convergence 239
2 `& @% ~5 N6 o/ A# o5 ~8 W4 L; G7.3 Fourier analysis of convergence 244; B l& [- N% ?! c5 K$ C
7.4 Application to an example 248
& _/ t5 h% f( Y; A3 Z4 g7.5 Extensions and related iterative methods 250 d2 z7 d/ M# C# I0 M# ~
7.6 The multigrid method 252
! u% E2 Y1 j) ? I. D C7.7 The conjugate gradient method 258
V! D! Q4 v1 J9 |. t% D" |7.8 A numerical example: comparisons 261
$ y! t1 p& v$ m/ e I; K. [3 nBibliographic notes 263( x4 ^, ]7 J% N S4 u, [
Exercises 2631 C- v& I5 Q; e" Q8 l
References 267
9 A3 B1 ?! D1 h6 F( o4 I/ DIndex 273
1 l' |/ D. `2 g& r. f! d# o2 h- f6 L B* ^9 p& Y/ g* q
" H2 |3 U- _2 L" @. K0 K5 n' G
4 K4 H4 O5 E3 K4 P9 L# ]3 r: h$ w6 S6 v, C9 V0 D
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