A comprehensive approach to numerical partial differential equations
Spline Collocation Methods for Partial Differential Equations combines the collocation analysis of partial differential equations (PDEs) with the method of lines (MOL) in order to simplify the solution process. Using a series of example applications, the author delineates the main features of the approach in detail, including an established mathematical framework. The book also clearly demonstrates that spline collocation can offer a comprehensive method for numerical integration of PDEs when it is used with the MOL in which spatial (boundary value) derivatives are approximated with splines, including the boundary conditions.
R, an open-source scientific programming system, is used throughout for programming the PDEs and numerical algorithms, and each section of code is clearly explained. As a result, readers gain a complete picture of the model and its computer implementation without having to fill in the details of the numerical analysis, algorithms, or programming. The presentation is not heavily mathematical, and in place of theorems and proofs, detailed example applications are provided.
Appropriate for scientists, engineers, and applied mathematicians, Spline Collocation Methods for Partial Differential Equations:
- Introduces numerical methods by first presenting basic examples followed by more complicated applications
- Employs R to illustrate accurate and efficient solutions of the PDE models
- Presents spline collocation as a comprehensive approach to the numerical integration of PDEs and an effective alternative to other, well established methods
- Discusses how to reproduce and extend the presented numerical solutions
- Identifies the use of selected algorithms, such as the solution of nonlinear equations and banded or sparse matrix processing
- Features a companion website that provides the related R routines
Spline Collocation Methods for Partial Differential Equations is a valuable reference and/or self-study guide for academics, researchers, and practitioners in applied mathematics and engineering, as well as for advanced undergraduates and graduate-level students.
Table of Contents:
Preface xiii
About the CompanionWebsite xv
1 Introduction 1
1.1 Uniform Grids 2
1.2 Variable Grids 18
1.3 Stagewise Differentiation 24
Appendix A1 – Online Documentation for splinefun 27
Reference 30
2 One-Dimensional PDEs 31
2.1 Constant Coefficient 31
2.1.1 Dirichlet BCs 32
2.1.1.1 Main Program 33
2.1.1.2 ODE Routine 40
2.1.2 Neumann BCs 43
2.1.2.1 Main Program 44
2.1.2.2 ODE Routine 46
2.1.3 Robin BCs 49
2.1.3.1 Main Program 50
2.1.3.2 ODE Routine 55
2.1.4 Nonlinear BCs 60
2.1.4.1 Main Program 61
2.1.4.2 ODE Routine 63
2.2 Variable Coefficient 64
2.2.1 Main Program 67
2.2.2 ODE Routine 71
2.3 Inhomogeneous, Simultaneous, Nonlinear 76
2.3.1 Main Program 78
2.3.2 ODE routine 85
2.3.3 Subordinate Routines 88
2.4 First Order in Space and Time 94
2.4.1 Main Program 96
2.4.2 ODE Routine 101
2.4.3 Subordinate Routines 105
2.5 Second Order in Time 107
2.5.1 Main Program 109
2.5.2 ODE Routine 114
2.5.3 Subordinate Routine 117
2.6 Fourth Order in Space 120
2.6.1 First Order in Time 120
2.6.1.1 Main Program 121
2.6.1.2 ODE Routine 125
2.6.2 Second Order in Time 138
2.6.2.1 Main Program 140
2.6.2.2 ODE Routine 143
References 155
3 Multidimensional PDEs 157
3.1 2D in Space 157
3.1.1 Main Program 158
3.1.2 ODE Routine 163
3.2 3D in Space 170
3.2.1 Main Program, Case 1 170
3.2.2 ODE Routine 174
3.2.3 Main Program, Case 2 183
3.2.4 ODE Routine 187
3.3 Summary and Conclusions 193
4 Navier–Stokes, Burgers’ Equations 197
4.1 PDE Model 197
4.2 Main Program 198
4.3 ODE Routine 203
4.4 Subordinate Routine 205
4.5 Model Output 206
4.6 Summary and Conclusions 208
Reference 209
5 Korteweg–de Vries Equation 211
5.1 PDE Model 211
5.2 Main Program 212
5.3 ODE Routine 225
Contents ix
5.4 Subordinate Routines 228
5.5 Model Output 234
5.6 Summary and Conclusions 238
References 239
6 Maxwell Equations 241
6.1 PDE Model 241
6.2 Main Program 243
6.3 ODE Routine 248
6.4 Model Output 252
6.5 Summary and Conclusions 252
Appendix A6.1. Derivation of the Analytical Solution 257
Reference 259
7 Poisson–Nernst–Planck Equations 261
7.1 PDE Model 261
7.2 Main Program 265
7.3 ODE Routine 271
7.4 Model Output 276
7.5 Summary and Conclusions 284
References 286
8 Fokker–Planck Equation 287
8.1 PDE Model 287
8.2 Main Program 288
8.3 ODE Routine 293
8.4 Model Output 295
8.5 Summary and Conclusions 301
References 303
9 Fisher–Kolmogorov Equation 305
9.1 PDE Model 305
9.2 Main Program 306
9.3 ODE Routine 311
9.4 Subordinate Routine 313
9.5 Model Output 314
9.6 Summary and Conclusions 316
Reference 316
10 Klein–Gordon Equation 317
10.1 PDE Model, Linear Case 317
10.2 Main Program 318
10.3 ODE Routine 323
10.4 Model Output 326
10.5 PDE Model, Nonlinear Case 328
10.6 Main Program 330
10.7 ODE Routine 335
10.8 Subordinate Routines 338
10.9 Model Output 339
10.10 Summary and Conclusions 342
Reference 342
11 Boussinesq Equation 343
11.1 PDE Model 343
11.2 Main Program 344
11.3 ODE Routine 350
11.4 Subordinate Routines 354
11.5 Model Output 355
11.6 Summary and Conclusions 358
References 358
12 Cahn–Hilliard Equation 359
12.1 PDE Model 359
12.2 Main Program 360
12.3 ODE Routine 366
12.4 Model Output 369
12.5 Summary and Conclusions 379
References 379
13 Camassa–Holm Equation 381
13.1 PDE Model 381
13.2 Main Program 382
13.3 ODE Routine 388
13.4 Model Output 391
13.5 Summary and Conclusions 394
13.6 Appendix A13.1: Second Example of a PDE with a Mixed Partial Derivative 395
13.7 Main Program 395
13.8 ODE Routine 398
13.9 Model Output 400
Reference 403
14 Burgers–Huxley Equation 405
14.1 PDE Model 405
14.2 Main Program 406
14.3 ODE Routine 411
14.4 Subordinate Routine 416
14.5 Model Output 417
14.6 Summary and Conclusions 422
References 422
15 Gierer–Meinhardt Equations 423
15.1 PDE Model 423
15.2 Main Program 424
15.3 ODE Routine 429
15.4 Model Output 432
15.5 Summary and Conclusions 437
Reference 440
16 Keller–Segel Equations 441
16.1 PDE Model 441
16.2 Main Program 443
16.3 ODE Routine 449
16.4 Subordinate Routines 453
16.5 Model Output 453
16.6 Summary and Conclusions 458
Appendix A16.1. Diffusion Models 458
References 459
17 Fitzhugh–Nagumo Equations 461
17.1 PDE Model 461
17.2 Main Program 462
17.3 ODE Routine 467
17.4 Model Output 470
17.5 Summary and Conclusions 475
Reference 475
18 Euler–Poisson–Darboux Equation 477
18.1 PDE Model 477
18.2 Main Program 478
18.3 ODE Routine 483
18.4 Model Output 488
18.5 Summary and Conclusions 493
References 493
19 Kuramoto–Sivashinsky Equation 495
19.1 PDE Model 495
19.2 Main Program 496
19.3 ODE Routine 503
19.4 Subordinate Routines 506
19.5 Model Output 508
19.6 Summary and Conclusions 513
References 514
20 Einstein–Maxwell Equations 515
20.1 PDE Model 515
20.2 Main Program 516
20.3 ODE Routine 521
20.4 Model Output 526
20.5 Summary and Conclusions 533
Reference 536
A Differential Operators in Three Orthogonal Coordinate Systems 537
References 539
Index 541
