This book provides a unified and accessible introduction to the basic theory of finite difference schemes applied to the numerical solution of partial differential equations. Originally published in 1989, its objective remains to clearly present the basic methods necessary to perform finite difference schemes and to understand the theory underlying these schemes. This is one of the few texts in the field to not only present the theory of stability in a rigorous and clear manner but also to discuss the theory of initial-boundary value problems in relation to finite difference schemes. In this updated edition the notion of a stability domain is now included in the definition of stability and is more prevalent throughout the book. The author has also added many new figures and tables to clarify important concepts and illustrate the properties of finite difference schemes.
This text/reference discusses the concepts of convergence, consistency, and stability for time-dependent equations. Methods of Fourier analysis are used throughout the book. Annotation c. Book News, Inc., Portland, OR (booknews.com)
Preface to the second edition; Preface to the first edition; 1. Hyperbolic partial differential equations; 2. Analysis of finite difference Schemes; 3. Order of accuracy of finite difference schemes; 4. Stability for multistep schemes; 5. Dissipation and dispersion; 6. Parabolic partial differential equations; 7. Systems of partial differential equations in higher dimensions; 8. Second-order equations; 9. Analysis of well-posed and stable problems; 10. Convergence estimates for initial value problems; 11. Well-posed and stable initial-boundary value problems; 12. Elliptic partial differential equations and difference schemes; 13. Linear iterative methods; 14. The method of steepest descent and the conjugate gradient method; Appendix A. Matrix and vectoranalysis; Appendix B. A survey of real analysis; Appendix C. A Survey of results from complex analysis; References; Index.