Trefethen L.N. Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations

Cornell University, 1996. — 299 p. OCRPreface: Many books have been written on numerical methods for partial differential equations, ranging from the mathematical classic by Richtmyer and Morton to recent texts on computational fluid dynamics. But this is not an easy field to set forth in a book. It is too active and too large, touching a bewildering variety of issues in mathematics, physics, engineering, and computer science.
My goal has been to write an advanced textbook focused on basic principles. Two remarkably fruitful ideas pervade this field: numerical stability and Fourier analysis. ! want the reader to think at length about these and other fundamentals, beginning with simple examples, and build in the process a solid foundation for subsequent work on more realistic problems. Numerical methods for partial differential equations are a subject of subtlety and beauty, but the appreciation of them may be lost if one views them too narrowly as tools to be hurried to application.
This is not a book of pure mathematics. A number of theorems are proved, but my purpose is to present a set of ideas rather than a sequence of theorems. The book should be accessible to mathematically inclined graduate students and practitioners in various fields of science and engineering, so long as they are comfortable with Fourier analysis, basic partial differential equations, some numerical methods, the elements of complex variables, and linear algebra including vector and matrix norms. At MIT and Cornell, successive drafts of this text have formed the basis of a large graduate course taught annually since 1985.