Computational Partial Differential Equations Using MATLAB® (Textbooks in Mathematics)

✍ Scribed by Jichun Li, Yi-Tung Chen

• Publisher: CRC Press
• Year: 2019
• Tongue: English
• Leaves: 423
• Edition: 2
• Category: Library

✦ Synopsis

In this popular text for an Numerical Analysis course, the authors introduce several major methods of solving various partial differential equations (PDEs) including elliptic, parabolic, and hyperbolic equations. It covers traditional techniques including the classic finite difference method, finite element method, and state-of-the-art numercial methods.The text uniquely emphasizes both theoretical numerical analysis and practical implementation of the algorithms in MATLAB. This new edition includes a new chapter, Finite Value Method, the presentation has been tightened, new exercises and applications are included, and the text refers now to the latest release of MATLAB.

Key Selling Points:

• A successful textbook for an undergraduate text on numerical analysis or methods taught in mathematics and computer engineering.
• This course is taught in every university throughout the world with an engineering department or school.
• Competitive advantage broader numerical methods (including finite difference, finite element, meshless method, and finite volume method), provides the MATLAB source code for most popular PDEs with detailed explanation about the implementation and theoretical analysis. No other existing textbook in the market offers a good combination of theoretical depth and practical source codes.

✦ Table of Contents

Cover
Half Title
Series Page
Title Page
Copyright Page
Contents
Preface
Acknowledgments
1. Brief Overview of Partial Differential Equations
1.1 The parabolic equations
1.2 The wave equations
1.3 The elliptic equations
1.4 Differential equations in broader areas
1.4.1 Electromagnetics
1.4.2 Fluid mechanics
1.4.3 Groundwater contamination
1.4.4 Petroleum reservoir simulation
1.4.5 Finance modeling
1.4.6 Image processing
1.5 A quick review of numerical methods for PDEs
References
2. Finite Difference Methods for Parabolic Equations
2.1 Introduction
2.2 Theoretical issues: stability, consistence, and convergence
2.3 1-D parabolic equations
2.3.1 The θ-method and its analysis
2.3.2 Some extensions
2.4 2-D and 3-D parabolic equations
2.4.1 Standard explicit and implicit methods
2.4.2 The ADI methods for 2-D problems
2.4.3 The ADI methods for 3-D problems
2.5 Numerical examples with MATLAB codes
2.6 Bibliographical remarks
2.7 Exercises
References
3. Finite Difference Methods for Hyperbolic Equations
3.1 Introduction
3.2 Some basic difference schemes
3.3 Dissipation and dispersion errors
3.4 Extensions to conservation laws
3.5 The second-order hyperbolic equations
3.5.1 The 1-D case
3.5.2 The 2-D case
3.6 Numerical examples with MATLAB codes
3.7 Bibliographical remarks
3.8 Exercises
References
4. Finite Difference Methods for Elliptic Equations
4.1 Introduction
4.2 Numerical solution of linear systems
4.2.1 Direct methods
4.2.2 Simple iterative methods
4.2.3 Modern iterative methods
4.3 Error analysis with a maximum principle
4.4 Some extensions
4.4.1 Mixed boundary conditions
4.4.2 Self-adjoint problems
4.4.3 A fourth-order scheme
4.5 Numerical examples with MATLAB codes
4.6 Bibliographical remarks
4.7 Exercises
References
5. High-Order Compact Difference Methods
5.1 One-dimensional problems
5.1.1 Spatial discretization
5.1.2 Dispersive error analysis
5.1.3 Temporal discretization
5.1.4 Low-pass spatial filter
5.1.5 Numerical examples with MATLAB codes
5.2 High-dimensional problems
5.2.1 Temporal discretization for 2-D problems
5.2.2 Stability analysis
5.2.3 Extensions to 3-D compact ADI schemes
5.2.4 Numerical examples with MATLAB codes
5.3 Other high-order compact schemes
5.3.1 One-dimensional problems
5.3.2 Two-dimensional problems
5.4 Bibliographical remarks
5.5 Exercises
References
6. Finite Element Methods: Basic Theory
6.1 Introduction to one-dimensional problems
6.1.1 The second-order equation
6.1.2 The fourth-order equation
6.2 Introduction to two-dimensional problems
6.2.1 The Poisson equation
6.2.2 The biharmonic problem
6.3 Abstract finite element theory
6.3.1 Existence and uniqueness
6.3.2 Stability and convergence
6.4 Examples of conforming finite element spaces
6.4.1 Triangular finite elements
6.4.2 Rectangular finite elements
6.5 Examples of nonconforming finite elements
6.5.1 Nonconforming triangular elements
6.5.2 Nonconforming rectangular elements
6.6 Finite element interpolation theory
6.6.1 Sobolev spaces
6.6.2 Interpolation theory
6.7 Finite element analysis of elliptic problems
6.7.1 Analysis of conforming finite elements
6.7.2 Analysis of nonconforming finite elements
6.8 Finite element analysis of time-dependent problems
6.8.1 Introduction
6.8.2 FEM for parabolic equations
6.9 Bibliographical remarks
6.10 Exercises
References
7. Finite Element Methods: Programming
7.1 FEM mesh generation
7.2 Forming FEM equations
7.3 Calculation of element matrices
7.4 Assembly and implementation of boundary conditions
7.5 The MATLAB code for P1 element
7.6 The MATLAB code for the Q1 element
7.7 Bibliographical remarks
7.8 Exercises
References
8. Mixed Finite Element Methods
8.1 An abstract formulation
8.2 Mixed methods for elliptic problems
8.2.1 The mixed variational formulation
8.2.2 The mixed finite element spaces
8.2.3 The error estimates
8.3 Mixed methods for the Stokes problem
8.3.1 The mixed variational formulation
8.3.2 Mixed finite element spaces
8.4 An example MATLAB code for the Stokes problem
8.5 Mixed methods for viscous incompressible flows
8.5.1 The steady Navier-Stokes problem
8.5.2 The unsteady Navier-Stokes problem
8.6 Bibliographical remarks
8.7 Exercises
References
9. Finite Element Methods for Electromagnetics
9.1 Introduction to Maxwell’s equations
9.2 The time-domain finite difference method
9.2.1 The semi-discrete scheme
9.2.2 The fully discrete scheme
9.3 The time-domain finite element method
9.3.1 The mixed method
9.3.2 The standard Galerkin method
9.3.3 The discontinuous Galerkin method
9.4 The frequency-domain finite element method
9.4.1 The standard Galerkin method
9.4.2 The discontinuous Galerkin method
9.4.3 The mixed DG method
9.5 Maxwell’s equations in dispersive media
9.5.1 Isotropic cold plasma
9.5.2 Debye medium
9.5.3 Lorentz medium
9.5.4 Double-negative metamaterials
9.6 Bibliographical remarks
9.7 Exercises
References
10. Meshless Methods with Radial Basis Functions
10.1 Introduction
10.2 The radial basis functions
10.3 The MFS-DRM
10.3.1 The fundamental solution of PDEs
10.3.2 The MFS for Laplace’s equation
10.3.3 The MFS-DRM for elliptic equations
10.3.4 Computing particular solutions using RBFs
10.3.5 The RBF-MFS
10.3.6 The MFS-DRM for the parabolic equations
10.4 Kansa’s method
10.4.1 Kansa’s method for elliptic problems
10.4.2 Kansa’s method for parabolic equations
10.4.3 The Hermite-Birkhoff collocation method
10.5 Numerical examples with MATLAB codes
10.5.1 Elliptic problems
10.5.2 Biharmonic problems
10.6 Coupling RBF meshless methods with DDM
10.6.1 Overlapping DDM
10.6.2 Non-overlapping DDM
10.6.3 One numerical example
10.7 Bibliographical remarks
10.8 Exercises
References
11. Other Meshless Methods
11.1 Construction of meshless shape functions
11.1.1 The smooth particle hydrodynamics method
11.1.2 The moving least-square approximation
11.1.3 The partition of unity method
11.2 The element-free Galerkin method
11.3 The meshless local Petrov-Galerkin method
11.4 Bibliographical remarks
11.5 Exercises
References
Appendix A: Answers to Selected Problems
Index