A fast spectral algorithm for nonlinear wave equations with linear dispersion

Spectral algorithms offer very high spatial resolution for a wide range of nonlinear wave equations on periodic domains, including well-known cases su ch as the Korteweg-de Vries and nonlinear Schrodinger equations. For the be et computational efficiency, one needs also to use high-order methods in ti me while somehow bypassing the usual severe stability restrictions. We use linearly implicit multistep methods, with the innovation of choosing differ ent methods for different ranges in Fourier space-high accuracy at low wave numbers and A-stability at high wavenumbers. This new approach compares fav orably to alternatives such as split-step and integrating factor (or linear ly exact) methods. (C) 1999 Academic Press.