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This 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 Numerical Methods through Use of MATLAB(R) 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 Needed for a Numerical Analysis Course Based 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.
Students can download MATLAB code from www.crcpress.com, enabling them to easily modify or improve the codes to solve their own problems.
University of California at Los Angeles, USA University of Nevada, Las Vegas, USA
Brief Overview of Partial Differential Equations The parabolic equations The wave equations The elliptic equations Differential equations in broader areas A quick review of numerical methods for PDEs Finite Difference Methods for Parabolic Equations Introduction Theoretical issues: stability, consistence, and convergence 1-D parabolic equations 2-D and 3-D parabolic equations Numerical examples with MATLAB codes Finite Difference Methods for Hyperbolic Equations Introduction Some basic difference schemes Dissipation and dispersion errors Extensions to conservation laws The second-order hyperbolic PDEs Numerical examples with MATLAB codes Finite Difference Methods for Elliptic Equations Introduction Numerical solution of linear systems Error analysis with a maximum principle Some extensions Numerical examples with MATLAB codes High-Order Compact Difference Methods 1-D problems High-dimensional problems Other high-order compact schemes Finite Element Methods: Basic Theory Introduction to 1-D problems Introduction to 2-D problems Abstract finite element theory Examples of conforming finite element spaces Examples of nonconforming finite elements Finite element interpolation theory Finite element analysis of elliptic problems Finite element analysis of time-dependent problems Finite Element Methods: Programming Finite element method mesh generation Forming finite element method equations Calculation of element matrices Assembly and implementation of boundary conditions The MATLAB code for P1 element The MATLAB code for the Q1 element Mixed Finite Element Methods An abstract formulation Mixed methods for elliptic problems Mixed methods for the Stokes problem An example MATLAB code for the Stokes problem Mixed methods for viscous incompressible flows Finite Element Methods for Electromagnetics Introduction to Maxwell's equations The time-domain finite element method The frequency-domain finite element method Maxwell's equations in dispersive media Meshless Methods with Radial Basis Functions Introduction The radial basis functions The MFS-DRM Kansa's method Numerical examples with MATLAB codes Coupling RBF meshless methods with DDM Other Meshless Methods Construction of meshless shape functions The element-free Galerkin method The meshless local Petrov-Galerkin method Answers to Selected Problems Index Bibliographical remarks, Exercises, and References appear at the end of each chapter.
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- ID: 9781420089042
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