Partial differential equations arise in almost all areas of science, engineering, modeling, and forecasting. Finite difference and finite element methods have long histories as particularly flexible and powerful general-purpose numerical solution methods. In the last two decades, spectral and in particular pseudospectral (PS) methods have emerged as intriguing alternatives in many situations – and as superior ones in several areas.

The aim of this Practical Guide is to describe when, how, and why the PS approach works, in a style that makes the transition to actual numerical implementations as straightforward as possible. For this reason, the book focuses on illustrations, examples, and heuristic explanations, and includes key code segments and references, but contains only a few rigorous theorems or technical proofs. It is written primarily for scientists and engineers who are interested in applying PS methods to real problems. However, I also hope that it will prove suitable for graduate-level study, conveying to students that PS methods form an important and rapidly developing field in which elaborate mathematical preliminaries are unnecessary. Material that is normally included in undergraduate-level mathematics and numerical analysis courses is mentioned only if customary viewpoints need to be complemented.

The paper entitled “A Review of Pseudospectral Methods”, which was co-authored by Professor David Sloan, appeared in Acta Numerica 1994. The encouragement offered by Dr. Arieh Iserles (the principal editor for Acta Numerica) was critical both for the original article and for its subsequent extension into this monograph.