Basic Approaches to Constructing Spectral Methods

Abstract

For the remainder of this book we shall be concerned with the use of spectral methods to obtain approximate solutions to ordinary differential equations (ODEs) and, especially, partial differential equations (PDEs). With very few exceptions spectral methods have only been applied to the approximation of spatial, and not temporal, derivatives. Our focus in this book is on the spatial approximations. The reader is referred to Appendix D for a review of time-discretization methods, including some brief comments on spectral approximations to time derivatives. When all or part of the time discretization is implicit, then the solution of implicit equations is required to advance in time. This topic is covered in the following chapter.