Spline Collocation Methods for Partial Differential Equations

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.

Auteur

WILLIAM E. SCHIESSER, PhD, ScD (hon.), is Emeritus McCann Professor of Biomolecular and Chemical Engineering and Professor of Mathematics at Lehigh University. He is the author, coauthor or coeditor of 17 books, including Method of Lines PDE Analysis in Biomedical Science and Engineering; Differential Equation Analysis in Biomedical Science and Engineering: Ordinary Differential Equation Applications with R and Differential Equation Analysis in Biomedical Science and Engineering: Partial Differential Equation Applications with R, published by Wiley.

Résumé
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.

Contenu

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 NavierStokes, 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 Kortewegde 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 PoissonNernstPlanck 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 FokkerPlanck 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 FisherKolmogorov 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 KleinGordon Equation 317

10.1 PDE Model, Linear Case 317

10.2 Ma…