Fundamentals of Spectral Methods for PDEs

Abstract

For the remainder of this book we shall be concerned with the use of spectral methods to approximate solutions to partial differential equations (PDEs). Our concern in this chapter is to illustrate how spectral methods are actually implemented for PDEs. We start by deriving the semi-discrete (continuous in time) ODE equations which are satisfied by various spectral approximations to Burgers equation. This will involve a discussion of non-linear terms, boundary conditions, projection operators, and different spectral discretizations. The second section provides a detailed discussion of transform methods for evaluating convolution sums. Next, we discuss Neumann, Robin and radiation boundary conditions. Finally, we remark on the treatment of coordinate singularities and the use of mapping techniques in two-dimensional problems.