Description and Philosophy of Spectral Methods

Abstract

We describe the use of spectral methods in computational fluid dynamics. Spectral methods are generally more accurate and often faster than finite-differences. For example, the ∇² operator in 2 or 3 dimensions is easier to invert with spectral techniques because the spatial dependence of the operator separates in a more natural way. We warn against the use of some of the more common spectral expansions. Bessel series expansions of functions in cylindrical geometries converge poorly. However, other series expansions of the same functions converge quickly. We show how to choose basis functions that give fast convergence and outline the differences between Galerkin, tau, modal, collocation, and pseudo-spectral methods.