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数学中国—罂粟

原版英文书 第二版
' Q1 i1 ]* w+ K- G8 S% }' Ncontents:
% t1 M' e0 ~6 Z; u  WPreface to the first edition page viii
Preface to the second edition xi
1 Introduction 1
2 Parabolic equations in one space variable 7
( G9 Z3 ?$ V/ U$ n. t/ Z. [; @( Q$ K2.1 Introduction 7
6 C$ }$ V4 k  N) {% }3 t2.2 A model problem 7
& P2 E, U" M( A9 W0 K" L0 X# n2.3 Series approximation 9
2.4 An explicit scheme for the model problem 10
9 T. E' {8 q& Q: e+ C% k# F9 d# }" g2.5 Difference notation and truncation error 12
5 j9 A& H* k0 }/ ?2.6 Convergence of the explicit scheme 16
2.7 Fourier analysis of the error 19
2.8 An implicit method 22
2.9 The Thomas algorithm 24
( P) A7 B' i4 p2.10 The weighted average or θ-method 26
2.11 A maximum principle and convergence
for μ(1−θ)≤ 1
2 33
2.12 A three-time-level scheme 38
2.13 More general boundary conditions 39
  t' j5 M, h" o$ [& l4 O2.14 Heat conservation properties 44
5 U$ H; U0 j; \% f5 ^! g  @2.15 More general linear problems 46
" [9 E; f7 q7 G9 s2.16 Polar co-ordinates 52
2.17 Nonlinear problems 54
Bibliographic notes 56
3 W' L9 i, h/ T! g( X2 aExercises 56
/ F) Q4 W: C7 m6 l6 Dv
vi Contents
0 M) K. v8 |9 `0 h% K! W3 2-D and 3-D parabolic equations 62
4 _/ h+ l+ ]3 c, g3.1 The explicit method in a rectilinear box 62
3.2 An ADI method in two dimensions 64
+ E. |, s! x( J3 C4 ~7 w9 e3.3 ADI and LOD methods in three dimensions 70
3.4 Curved boundaries 71
3.5 Application to general parabolic problems 80
Bibliographic notes 83
Exercises 83
# d# Y: q8 v+ O4 Z% N4 Hyperbolic equations in one space dimension 86
4 q! ^" f+ W& d) ?# S+ ?2 t4.1 Characteristics 86
4.2 The CFL condition 89
4.3 Error analysis of the upwind scheme 94
4.4 Fourier analysis of the upwind scheme 97
4.5 The Lax–Wendroff scheme 100
6 p' i* g2 g7 Y: a% n( r4.6 The Lax–Wendroff method for conservation laws 103
4.7 Finite volume schemes 110
( s/ O4 i& \' N/ _4 B! J' S0 O4.8 The box scheme 116
4.9 The leap-frog scheme 123
4.10 Hamiltonian systems and symplectic
integration schemes 128
+ D8 T8 |3 E( p8 p# \4.11 Comparison of phase and amplitude errors 135
4.12 Boundary conditions and conservation properties 139
! ^# I$ K! G7 ~: w2 [5 F$ J4.13 Extensions to more space dimensions 143
Bibliographic notes 146
3 }- P& ~/ ]4 h, T0 ^) n, OExercises 146
+ r- ?/ C2 Z+ n, M  u5 Consistency, convergence and stability 151
5.1 Definition of the problems considered 151
5 _! q7 n; q- F" X6 \0 ?5.2 The finite difference mesh and norms 152
5.3 Finite difference approximations 154
1 ]8 p5 t" s; r7 }) K5 n4 g% _& Q5.4 Consistency, order of accuracy and convergence 156
! R% b- P' {/ Q! Z+ G7 J5.5 Stability and the Lax Equivalence Theorem 157
3 r, I: ~, v* y' q2 j& a8 h5.6 Calculating stability conditions 160
5.7 Practical (strict or strong) stability 166
/ n3 I! u& e: j* W7 P- z5.8 Modified equation analysis 169
) J% L* B, J5 i, L' A9 e5.9 Conservation laws and the energy method of analysis 177
5.10 Summary of the theory 186
Bibliographic notes 189
Exercises 190
* {, P& p9 V/ m1 p9 jContents vii
6 Linear second order elliptic equations in
two dimensions 194
6.1 A model problem 194
6.2 Error analysis of the model problem 195
6.3 The general diffusion equation 197
6.4 Boundary conditions on a curved boundary 199
6.5 Error analysis using a maximum principle 203
6.6 Asymptotic error estimates 213
+ D1 m* U: R" ^, U) ~8 }6.7 Variational formulation and the finite
element method 218
* I& [3 Y. _- w7 w: i: \$ ]6 s3 b6.8 Convection–diffusion problems 224
6.9 An example 228
# Y1 r: q8 b$ I9 f, I+ ?! `Bibliographic notes 231
9 F' e8 T) q8 q0 J0 ]Exercises 232
) r3 Q, ]. w' N' P+ s. E% f% ]7 Iterative solution of linear algebraic equations 235
3 G2 ?! U* r+ T; W7.1 Basic iterative schemes in explicit form 237
7.2 Matrix form of iteration methods and
their convergence 239
* z# E' @- l4 A7.3 Fourier analysis of convergence 244
7.4 Application to an example 248
7.5 Extensions and related iterative methods 250
7.6 The multigrid method 252
7.7 The conjugate gradient method 258
/ p2 q4 @4 L/ x$ e& H$ n" y! Y7.8 A numerical example: comparisons 261
( D6 `) i0 r# y/ w& FBibliographic notes 263
8 G, Y& i) p$ V+ S& |* f3 RExercises 263
7 @* G( d9 o* \/ V! gReferences 267
  t: L  s* R- O% |( EIndex 273

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