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Full Description
This book introduces finite difference methods for both ordinary differential equations (ODEs) and partial differential equations (PDEs) and discusses the similarities and differences between algorithm design and stability analysis for different types of equations. A unified view of stability theory for ODEs and PDEs is presented, and the interplay between ODE and PDE analysis is stressed. The text emphasizes standard classical methods, but several newer approaches also are introduced and are described in the context of simple motivating examples. Exercises and student projects are available on the book's webpage, along with Matlab mfiles for implementing methods. Readers will gain an understanding of the essential ideas that underlie the development, analysis, and practical use of finite difference methods as well as the key concepts of stability theory, their relation to one another, and their practical implications. The author provides a foundation from which students can approach more advanced topics.
Contents
Preface
Part I: Boundary Value Problems and Iterative Methods.
Chapter 1: Finite Difference Approximations
Chapter 2: Steady States and Boundary Value Problems
Chapter 3: Elliptic Equations
Chapter 4: Iterative Methods for Sparse Linear Systems
Part II: Initial Value Problems.
Chapter 5: The Initial Value Problem for Ordinary Differential Equations
Chapter 6: Zero-Stability and Convergence for Initial Value Problems
Chapter 7: Absolute Stability for Ordinary Differential Equations
Chapter 8: Stiff Ordinary Differential Equations
Chapter 9: Diffusion Equations and Parabolic Problems
Chapter 10: Advection Equations and Hyperbolic Systems
Chapter 11: Mixed Equations
Appendix A: Measuring Errors
Appendix B: Polynomial Interpolation and Orthogonal Polynomials
Appendix C: Eigenvalues and Inner-Product Norms
Appendix D: Matrix Powers and Exponentials
Appendix E: Partial Differential Equations
Bibliography
Index.
