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数学中国—罂粟

原版英文书 第二版
; |& x; c9 N8 zcontents:
$ f- e5 _% U$ r; c& lPreface to the first edition page viii
# A7 k( ~# |& I. s* K7 k' A% T$ BPreface to the second edition xi
1 Introduction 1
2 Parabolic equations in one space variable 7
1 C1 L# w3 j# V- e2.1 Introduction 7
2.2 A model problem 7
/ O1 d+ f" M# M1 s# Q$ R' `2.3 Series approximation 9
2.4 An explicit scheme for the model problem 10
2 G+ x3 p0 Q' l2 z5 x2.5 Difference notation and truncation error 12
2.6 Convergence of the explicit scheme 16
7 B( {# s1 U7 u+ v2.7 Fourier analysis of the error 19
8 D) X* a* u3 w% K2 p/ d2.8 An implicit method 22
( _' Y' q- h9 `1 v4 k2.9 The Thomas algorithm 24
2.10 The weighted average or θ-method 26
7 g- N: `2 r( F8 ?! l2.11 A maximum principle and convergence
) m/ b. L9 @6 ]9 U: c( U; j; Dfor μ(1−θ)≤ 1
2 33
2.12 A three-time-level scheme 38
2.13 More general boundary conditions 39
2.14 Heat conservation properties 44
- \; U! l! g- T# o; E' W2.15 More general linear problems 46
5 I, ~* i5 p) A2.16 Polar co-ordinates 52
2.17 Nonlinear problems 54
Bibliographic notes 56
Exercises 56
v
# P  Q# p3 P) y0 ovi Contents
8 e/ ?; ~/ G  w9 e. U3 2-D and 3-D parabolic equations 62
) R# a% H& S+ k& \/ }2 r. U4 N3.1 The explicit method in a rectilinear box 62
3.2 An ADI method in two dimensions 64
0 \# Y9 }* s' \3.3 ADI and LOD methods in three dimensions 70
3.4 Curved boundaries 71
3.5 Application to general parabolic problems 80
Bibliographic notes 83
" d3 q- G. ~$ k0 g4 {Exercises 83
4 Hyperbolic equations in one space dimension 86
# A9 ^2 P: }* x, X, i4.1 Characteristics 86
4.2 The CFL condition 89
$ m) c/ q; Z4 z3 O/ A$ E+ i4.3 Error analysis of the upwind scheme 94
4.4 Fourier analysis of the upwind scheme 97
4.5 The Lax–Wendroff scheme 100
4.6 The Lax–Wendroff method for conservation laws 103
1 Z* P0 P9 g' m( E0 h4.7 Finite volume schemes 110
4.8 The box scheme 116
4.9 The leap-frog scheme 123
4.10 Hamiltonian systems and symplectic
integration schemes 128
3 P& P6 g, ]) G" v6 x; {4 I! e4.11 Comparison of phase and amplitude errors 135
0 v/ L" |6 q3 q9 f7 {4.12 Boundary conditions and conservation properties 139
4.13 Extensions to more space dimensions 143
' Z; V% T9 P+ J5 }Bibliographic notes 146
Exercises 146
0 w3 x( Y4 ^! h  C+ y7 g5 Consistency, convergence and stability 151
5.1 Definition of the problems considered 151
; ?0 B0 m8 _9 H, q* B  j9 ^5.2 The finite difference mesh and norms 152
! R8 ~( R' I/ Z5.3 Finite difference approximations 154
9 E6 m0 H; o: ]5.4 Consistency, order of accuracy and convergence 156
  G5 z) q5 u+ b' O* D5.5 Stability and the Lax Equivalence Theorem 157
5.6 Calculating stability conditions 160
5.7 Practical (strict or strong) stability 166
& Z6 X+ M/ W- W" ~5.8 Modified equation analysis 169
5.9 Conservation laws and the energy method of analysis 177
5.10 Summary of the theory 186
Bibliographic notes 189
, |: ?1 [/ n: ^% |" sExercises 190
1 L8 H9 D6 G2 L! f* \Contents vii
6 Linear second order elliptic equations in
) l3 e, b6 K# p# A0 K/ stwo dimensions 194
6 m$ z6 g" J- R  A. ]/ J7 x6.1 A model problem 194
7 [4 Z- K* T" p! r7 W6.2 Error analysis of the model problem 195
6.3 The general diffusion equation 197
! K8 a& O# {8 D6.4 Boundary conditions on a curved boundary 199
6.5 Error analysis using a maximum principle 203
6.6 Asymptotic error estimates 213
6.7 Variational formulation and the finite
5 w2 T! b5 [7 ?+ Q7 o9 belement method 218
6.8 Convection–diffusion problems 224
6.9 An example 228
% C: X( ?2 q1 Y0 UBibliographic notes 231
/ b' i4 x, K+ [( c$ a# I- i% jExercises 232
$ j! J6 _" ^; K4 H0 J/ T, @7 Iterative solution of linear algebraic equations 235
8 |5 H6 |% G, D6 G) C7.1 Basic iterative schemes in explicit form 237
7.2 Matrix form of iteration methods and
: K, M" N* Y- a( E! `' atheir convergence 239
7.3 Fourier analysis of convergence 244
7.4 Application to an example 248
7.5 Extensions and related iterative methods 250
) A& q9 S$ @) }7 ^+ K7.6 The multigrid method 252
5 y' H9 Y& _' c) L2 @, _* }7.7 The conjugate gradient method 258
' J0 X9 Y2 J' d' v' g9 H! T( {7.8 A numerical example: comparisons 261
Bibliographic notes 263
Exercises 263
  D# E( s" `  M, H& x0 W+ zReferences 267
: }& r0 @: {. e; w$ k/ IIndex 273

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