Numerical solution of partial differential equations

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface to the first edition......Page 10
Preface to the second edition......Page 13
1 Introduction......Page 17
2.2 A model problem......Page 23
2.3 Series approximation......Page 25
2.4 An explicit scheme for the model problem......Page 26
2.5 Difference notation and truncation error......Page 28
2.6 Convergence of the explicit scheme......Page 32
2.7 Fourier analysis of the error......Page 35
2.8 An implicit method......Page 38
2.9 The Thomas algorithm......Page 40
2.10 The weighted average or theta-method......Page 42
2.11 A maximum principle and convergence for…......Page 49
2.12 A three-time-level scheme......Page 54
2.13 More general boundary conditions......Page 55
2.14 Heat conservation properties......Page 60
2.15 More general linear problems......Page 62
2.16 Polar co-ordinates......Page 68
2.17 Nonlinear problems......Page 70
Exercises......Page 72
3.1 The explicit method in a rectilinear box......Page 78
3.2 An ADI method in two dimensions......Page 80
3.3 ADI and LOD methods in three dimensions......Page 86
3.4 Curved boundaries......Page 87
3.5 Application to general parabolic problems......Page 96
Exercises......Page 99
4.1 Characteristics......Page 102
4.2 The CFL condition......Page 105
4.3 Error analysis of the upwind scheme......Page 110
4.4 Fourier analysis of the upwind scheme......Page 113
4.5 The Lax–Wendroff scheme......Page 116
4.6 The Lax–Wendroff method for conservation laws......Page 119
4.7 Finite volume schemes......Page 126
4.8 The box scheme......Page 132
4.9 The leap-frog scheme......Page 139
4.10 Hamiltonian systems and symplectic integration schemes......Page 144
4.11 Comparison of phase and amplitude errors......Page 151
4.12 Boundary conditions and conservation properties......Page 155
4.13 Extensions to more space dimensions......Page 159
Exercises......Page 162
5.1 Definition of the problems considered......Page 167
5.2 The finite difference mesh and norms......Page 168
5.3 Finite difference approximations......Page 170
5.4 Consistency, order of accuracy and convergence......Page 172
5.5 Stability and the Lax Equivalence Theorem......Page 173
5.6 Calculating stability conditions......Page 176
5.7 Practical (strict or strong) stability......Page 182
5.8 Modified equation analysis......Page 185
5.9 Conservation laws and the energy method of analysis......Page 193
5.10 Summary of the theory......Page 202
Bibliographic notes and recommended reading......Page 205
Exercises......Page 206
6.1 A model problem......Page 210
6.2 Error analysis of the model problem......Page 211
6.3 The general diffusion equation......Page 213
6.4 Boundary conditions on a curved boundary......Page 215
6.5 Error analysis using a maximum principle......Page 219
6.6 Asymptotic error estimates......Page 229
6.7 Variational formulation and the finite element method......Page 234
6.8 Convection–diffusion problems......Page 240
6.9 An example......Page 244
Bibliographic notes and recommended reading......Page 247
Exercises......Page 248
7 Iterative solution of linear algebraic equations......Page 251
7.1 Basic iterative schemes in explicit form......Page 253
7.2 Matrix form of iteration methods and their convergence......Page 255
7.3 Fourier analysis of convergence......Page 260
7.4 Application to an example......Page 264
7.5 Extensions and related iterative methods......Page 266
7.6 The multigrid method......Page 268
7.7 The conjugate gradient method......Page 274
7.8 A numerical example: comparisons......Page 277
Exercises......Page 279
References......Page 283
Index......Page 289