0 Direct Solution of Linear Systems ............................ 1
0.0 Introduction ............................................ 1
0.1 General Linear Systems .................................. 1
0.2 Systems Requiring No Pivoting ........................... 5
0.3 The LU Decomposition .................................... 8
0.4 Banded Linear Systems .................................. 11
0.5 Sparse Direct Methods .................................. 17
0.6 Problems ............................................... 23
1 Initial Value Ordinary Differential Equations ............... 27
1.0 Introduction ........................................... 27
1.1 Euler's Method ......................................... 28
1.2 Truncation Error, Stability, and Convergence ........... 30
1.3 Multistep Methods ...................................... 35
1.4 Adams Multistep Methods ................................ 39
1.5 Backward Difference Methods for Stiff Problems ......... 46
1.6 Runge-Kutta Methods .................................... 51
1.7 Problems ............................................... 58
2 The Initial Value Diffusion Problem ......................... 62
2.0 Introduction ........................................... 62
2.1 An Explicit Method ..................................... 65
2.2 Implicit Methods ....................................... 70
2.3 A One-Dimensional Example .............................. 76
2.4 Multidimensional Problems .............................. 78
2.5 A Diffusion-Reaction Example ........................... 84
2.6 Problems .............................................. 87
3 The Initial Value Transport and Wave Problems .............. 92
3.0 Introduction ........................................... 92
3.1 Explicit Methods for the Transport Problem ............. 98
3.2 The Method of Characteristics ......................... 104
3.3 An Explicit Method for the Wave Equation .............. 109
3.4 A Damped Wave Example ................................. 114
3.5 Problems .............................................. 117
4 Boundary Value Problems .................................... 121
4.0 Introduction .......................................... 121
4.1 Finite Difference Methods ............................. 124
4.2 A Nonlinear Example ................................... 126
4.3 A Singular Example .................................... 128
4.4 Shooting Methods ...................................... 130
4.5 Multidimensional Problems ............................. 134
4.6 Successive Overrelaxation ............................. 137
4.7 Successive Overrelaxation Examples .................... 141
4.8 The Conjugate-Gradient Method ......................... 151
4.9 Systems of Differential Equations ..................... 157
4.10 The Eigenvalue Problem ................................ 161
4.11 The Inverse Power Method .............................. 165
4.12 Problems .............................................. 169
5 The Finite Element Method .................................. 174
5.0 Introduction .......................................... 174
5.1 The Galerkin Method ................................... 174
5.2 Example Using Piecewise Linear Trial Functions ........ 179
5.3 Example Using Cubic Hermite Trial Functions ........... 182
5.4 A Singular Example and The Collocation Method ......... 192
5.5 Linear Triangular Elements ............................ 199
5.6 An Example Using Triangular Elements .................. 202
5.7 Time-Dependent Problems ............................... 206
5.8 A One-Dimensional Example ............................. 209
5.9 Time-Dependent Example Using Triangles ................ 216
5.10 The Eigenvalue Problem ................................ 220
5.11 Eigenvalue Examples ................................... 222
5.12 Problems .............................................. 229
Appendix A - Solving PDEs with PDE2D .......................... 238
A.l History ............................................... 238
A.2 The PDE2D Interactive and Graphical User Interfaces ... 239
A.3 One-Dimensional Steady-State Problems ................. 243
A.4 Two-Dimensional Steady-State Problems ................. 245
A.5 Three-Dimensional Steady-State Problems ............... 252
A.6 Nonrectangular 3D Regions ............................. 255
A.7 Time-Dependent Problems ............................... 263
A.8 Eigenvalue Problems ................................... 266
A.9 The PDE2D Parallel Linear System Solvers .............. 268
A.10 Examples .............................................. 272
A.ll Problems .............................................. 281
Appendix В - The Fourier Stability Method ..................... 285
Appendix С - MATLAB Programs .................................. 291
Appendix D - Answers to Selected Exercises .................... 320
References .................................................... 327
Index ......................................................... 331
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