The Numerical Solution of Ordinary and Partial Differential Equations

The Numerical Solution of Ordinary and Partial Differential Equations......Page 2
Contents......Page 10
Preface......Page 14
0.1 General Linear Systems......Page 16
0.2 Systems Requiring No Pivoting......Page 20
0.3 The L U Decomposition......Page 23
0.4 Banded Linear Systems......Page 26
0.5 Sparse Direct Methods......Page 32
0.6 Problems......Page 39
1.0 Introduction......Page 42
1.1 Euler's Method......Page 43
1.2 Truncation Error, Stability. and Convergence......Page 45
1.3 Multistep Methods......Page 50
1.4 Adams Multistep Methods......Page 54
1.5 Backward Difference Methods for Stiff Problems......Page 61
1.6 Runge-Kutta Methods......Page 66
1.7 Problems......Page 73
2.0 Introduction......Page 77
2.1 An Explicit Method......Page 80
2.2 Implicit Methods......Page 85
2.3 A One-Dimensional Example......Page 89
2.4 Multidimensional Problems......Page 92
2.5 A Diffusion-Reaction Example......Page 98
2.6 Problems......Page 101
3.0 Introduction......Page 106
3.1 Explicit Methods for the Transport Problem......Page 112
3.2 The Method of Characteristics......Page 118
3.3 An Explicit Method for the Wave Equation......Page 123
3.4 A Damped Wave Example......Page 128
3.5 Problems......Page 131
4.0 Introduction......Page 135
4.1 Finite Difference Methods......Page 138
4.2 A Nonlinear Example......Page 140
4.3 A Singular Example......Page 142
4.4 Shooting Methods......Page 144
4.5 Multidimensional Problems......Page 148
4.6 Successive Overrelaxation......Page 152
4.7 Successive Overrelaxation Examples......Page 155
4.8 The Conjugate-Gradient Method......Page 165
4.9 Systems of Differential Equations......Page 171
4.10 The Eigenvalue Problem......Page 175
4.11 The Inverse Power Method......Page 179
4.12 Problems......Page 183
5.1 The Galerkin Method......Page 189
5.2 Example Using Piecewise Linear Trial Functions......Page 194
5.3 Example Using Cubic Hermite Trial Functions......Page 197
5.4 A Singular Example and The Collocation Method......Page 207
5.5 Linear Triangular Elements......Page 214
5.6 An Example Using Triangular Elements......Page 217
5.7 Time-Dependent Problems......Page 221
5.8 A One-Dimensional Example......Page 224
5.9 Time-Dependent Example Using Triangles......Page 228
5.10 The Eigenvalue Problem......Page 232
5.11 Eigenvalue Examples......Page 234
5.12 Problems......Page 242
A.1 History......Page 250
A.2 The PDE2D Interactive User Interface......Page 251
A.3 One-Dimensional Steady-State Problems......Page 254
A.4 Two-Dimensional Steady-State Problems......Page 256
A.5 Three-Dimensional Steady-State Problems......Page 263
A.6 Nonrectangular 3D Regions......Page 266
A.7 Time-Dependent Problems......Page 273
A.8 Eigenvalue Problems......Page 276
A.9 The PDE2D Parallel Linear System Solvers......Page 277
A.10 Examples......Page 283
A .11 Problems......Page 290
Appendix B - The Fourier Stability Method......Page 297
Appendix C - MATLAB Programs......Page 303
Appendix D - Can “ANYTHING” Happen in an Open System?......Page 331
Appendix E - Answers to Selected Exercises......Page 335
References......Page 342
Index......Page 346