Strikwerda J.C. Finite Difference Schemes and Partial Differential Equations

Second Edition. - SIAM, 2004. - 448 p.The book presents the basic theory of finite difference schemes applied to the numerical solution of partial differential equations. It is designed to be used as an introductory graduate text for students in applied mathematics, engineering, and the sciences, and with that in mind, presents the theory of finite difference schemes in a way that is both rigorous and accessible to the typical graduate student in the course. The two aims of the text are to present the basic material necessary to do scientific computation with finite difference schemes and to present the basic theory for understanding these methods.The text was developed for two courses: a basic introduction to finite difference schemes for partial differential equations and an upper level graduate course on the theory related to initial value problems. Because students in these courses have diverse backgrounds in mathematics, the text presumes knowledge only through advanced calculus, although some mathematical maturity is required for the more advanced topics. Students taking an introduction to finite difference schemes are often acquainted with partial differential equations, but many have not had a formal course on the subject. For this reason, much of the necessary theory of partial differential equations is developed in the text.The chief motivation for this text was the desire to present the material on timedependent equations, Chapters 1 through 11, in a unified way that was accessible to students who would use the material in scientific and engineering studies. Chapters 1 through 11 contain much that is not in any other textbook, but more important, the unified treatment, using Fourier analysis, emphasizes that one can study finite difference schemes using a few powerful ideas to understand most of their properties. The material on elliptic partial differential equations, Chapters, 12, 13, and 14, is intended to be only an introduction; it should enable students to progress to more advanced texts and implement the basic methods knowledgably.