Computational Partial Differential Equations Using MATLAB

by Li; Jichun

• ISBN13: 9781420089042
• ISBN10: 1420089048
• Edition: 1st
• Format: Hardcover
• Copyright: 2008-10-20
• Publisher: Chapman & Hall/

Summary

A Concise yet Solid Introduction to Advanced Numerical MethodsThis textbook introduces several major numerical methods for solving various partial differential equations (PDEs) in science and engineering, including elliptic, parabolic, and hyperbolic equations. It covers traditional techniques that include the classic finite difference method and the finite element method as well as state-of-the-art numerical methods, such as the high-order compact difference method and the radial basis function meshless method.Helps Students Better Understand the Numerical Methods through the Use of MATLAB ®The authors uniquely emphasize both theoretical numerical analysis and practical implementation of the algorithms in MATLAB, making the book useful for students in computational science and engineering. They provide students with simple, clear implementations instead of sophisticated usages of MATLAB functions.All the Material Neededfor a Numerical Analysis CourseBased on the authors' own courses, the text only requires some knowledge of computer programming, advanced calculus, and difference equations. It includes practical examples, exercises, references, and problems, along with a solutions manual for qualifying instructors. The accompanying CD-ROM contains MATLAB source code, enabling students to easily modify or improve the codes to solve their own problems.

Table of Contents

Preface	p. xi
Acknowledgments	p. xiii
Brief Overview of Partial Differential Equations	p. 1
The parabolic equations	p. 1
The wave equations	p. 2
The elliptic equations	p. 3
Differential equations in broader areas	p. 3
Electromagnetics	p. 3
Fluid mechanics	p. 4
Ground water contamination	p. 5
Petroleum reservoir simulation	p. 6
Finance modeling	p. 7
Image processing	p. 7
A quick review of numerical methods for PDEs	p. 8
References	p. 10
Finite Difference Methods for Parabolic Equations	p. 13
Introduction	p. 13
Theoretical issues: stability, consistence, and convergence	p. 15
1-D parabolic equations	p. 16
The [theta]-method	p. 16
Some extensions	p. 19
2-D and 3-D parabolic equations	p. 23
Standard explicit and implicit methods	p. 23
The ADI methods for 2-D problems	p. 25
The ADI methods for 3-D problems	p. 28
Numerical examples with MATLAB codes	p. 30
Bibliographical remarks	p. 33
Exercises	p. 33
References	p. 36
Finite Difference Methods for Hyperbolic Equations	p. 39
Introduction	p. 39
Some basic difference schemes	p. 40
Dissipation and dispersion errors	p. 42
Extensions to conservation laws	p. 44
The second-order hyperbolic PDEs	p. 45
Numerical examples with MATLAB codes	p. 49
Bibliographical remarks	p. 52
Exercises	p. 52
References	p. 54
Finite Difference Methods for Elliptic Equations	p. 57
Introduction	p. 57
Numerical solution of linear systems	p. 59
Direct methods	p. 59
Simple iterative methods	p. 61
Modern iterative methods	p. 64
Error analysis with a maximum principle	p. 66
Some extensions	p. 68
Mixed boundary conditions	p. 68
Self-adjoint problems	p. 69
A fourth-order scheme	p. 70
Numerical examples with MATLAB codes	p. 73
Bibliographical remarks	p. 75
Exercises	p. 76
References	p. 78
High-Order Compact Difference Methods	p. 79
One-dimensional problems	p. 79
Spatial discretization	p. 79
Approximations of high-order derivatives	p. 83
Temporal discretization	p. 92
Low-pass spatial filter	p. 92
Numerical examples with MATLAB codes	p. 93
High-dimensional problems	p. 110
Temporal discretization for 2-D problems	p. 110
Stability analysis	p. 112
Extensions to 3-D compact ADI schemes	p. 113
Numerical examples with MATLAB codes	p. 114
Other high-order compact schemes	p. 122
One-dimensional problems	p. 122
Two-dimensional problems	p. 124
Bibliographical remarks	p. 127
Exercises	p. 127
References	p. 130
Finite Element Methods: Basic Theory	p. 133
Introduction to one-dimensional problems	p. 133
The second-order equation	p. 133
The fourth-order equation	p. 136
Introduction to two-dimensional problems	p. 140
The Poisson's equation	p. 140
The biharmonic problem	p. 142
Abstract finite element theory	p. 143
Existence and uniqueness	p. 143
Stability and convergence	p. 145
Examples of conforming finite element spaces	p. 146
Triangular finite elements	p. 147
Rectangular finite elements	p. 149
Examples of nonconforming finite elements	p. 150
Nonconforming triangular elements	p. 150
Nonconforming rectangular elements	p. 151
Finite element interpolation theory	p. 153
Sobolev spaces	p. 154
Interpolation theory	p. 155
Finite element analysis of elliptic problems	p. 159
Analysis of conforming finite elements	p. 159
Analysis of nonconforming finite elements	p. 161
Finite element analysis of time-dependent problems	p. 163
Introduction	p. 163
FEM for parabolic equations	p. 164
Bibliographical remarks	p. 167
Exercises	p. 167
References	p. 169
Finite Element Methods: Programming	p. 173
FEM mesh generation	p. 173
Forming FEM equations	p. 178
Calculation of element matrices	p. 179
Assembly and implementation of boundary conditions	p. 184
The MATLAB code for P[subscript 1] element	p. 185
The MATLAB code for the Q[subscript 1] element	p. 188
Bibliographical remarks	p. 193
Exercises	p. 194
References	p. 197
Mixed Finite Element Methods	p. 199
An abstract formulation	p. 199
Mixed methods for elliptic problems	p. 203
The mixed variational formulation	p. 203
The mixed finite element spaces	p. 205
The error estimates	p. 208
Mixed methods for the Stokes problem	p. 211
The mixed variational formulation	p. 211
Mixed finite element spaces	p. 212
An example MATLAB code for the Stokes problem	p. 217
Mixed methods for viscous incompressible flows	p. 231
The steady Navier-Stokes problem	p. 231
The unsteady Navier-Stokes problem	p. 233
Bibliographical remarks	p. 234
Exercises	p. 235
References	p. 237
Finite Element Methods for Electromagnetics	p. 241
Introduction to Maxwell's equations	p. 241
The time-domain finite element method	p. 243
The mixed method	p. 243
The standard Galerkin method	p. 248
The discontinuous Galerkin method	p. 251
The frequency-domain finite element method	p. 256
The standard Galerkin method	p. 256
The discontinuous Galerkin method	p. 257
The mixed DG method	p. 261
The Maxwell's equations in dispersive media	p. 263
Isotropic cold plasma	p. 264
Debye medium	p. 268
Lorentz medium	p. 270
Double-negative metamaterials	p. 273
Bibliographical remarks	p. 281
Exercises	p. 281
References	p. 283
Meshless Methods with Radial Basis Functions	p. 287
Introduction	p. 287
The radial basis functions	p. 288
The MFS-DRM	p. 291
The fundamental solution of PDEs	p. 291
The MFS for Laplace's equation	p. 294
The MFS-DRM for elliptic equations	p. 297
Computing particular solutions using RBFs	p. 300
The RBF-MFS	p. 302
The MFS-DRM for the parabolic equations	p. 302
Kansa's method	p. 304
Kansa's method for elliptic problems	p. 304
Kansa's method for parabolic equations	p. 305
The Hermite-Birkhoff collocation method	p. 306
Numerical examples with MATLAB codes	p. 308
Elliptic problems	p. 308
Biharmonic problems	p. 315
Coupling RBF meshless methods with DDM	p. 322
Overlapping DDM	p. 323
Non-overlapping DDM	p. 324
One numerical example	p. 325
Bibliographical remarks	p. 327
Exercises	p. 328
References	p. 329
Other Meshless Methods	p. 335
Construction of meshless shape functions	p. 335
The smooth particle hydrodynamics method	p. 335
The moving least-square approximation	p. 337
The partition of unity method	p. 338
The element-free Galerkin method	p. 340
The meshless local Petrov-Galerkin method	p. 342
Bibliographical remarks	p. 345
Exercises	p. 345
References	p. 346
Answers to Selected Problems	p. 349
Index	p. 361
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