The purpose of this book is to collect, in one volume, all the ingredients necessary for the understanding of spectral methods for time-dependent problems, and, in particular, hyperbolic partial differential equations. It is intended as a graduate-level text, covering not only the basic concepts in spectral methods, but some of the modern developments as well. There are already several excellent books on spectral methods by authors who are well-known and active researchers in this field. This book is distinguished by the exclusive treatment of time-dependent problems, and so the derivation of spectral methods is influenced primarily by the research on finite-difference schemes, and less so by the finite-element methodology. Furthermore, this book is unique in its focus on the stability analysis of spectral methods, both for the semi-discrete and fully discrete cases. In the book we address advanced topics such as spectral methods for discontinuous problems and spectral methods on arbitrary grids, which are necessary for the implementation of pseudo-spectral methods on complex multi-dimensional domains.

In Chapter 1, we demonstrate the benefits of high order methods using phase error analysis. Typical finite difference methods use a local stencil to compute the derivative at a given point; higher order methods are then obtained by using a wider stencil, i.e., more points. The Fourier spectral method is obtained by using all the points in the domain. In Chapter 2, we discuss the trigonometric polynomial approximations to smooth functions, and the associated approximation theory for both the continuous and the discrete case. In Chapter 3, we present Fourier spectral methods, using both the Galerkin and collocation approaches, and discuss their stability for both hyperbolic and parabolic equations.