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 such as the Korteweg-de Vries and nonlinear Schrodinger equations. For the beet computational efficiency, one needs also to use high-order methods in time while somehow bypassing the usual severe stability restrictions. We use linearly implicit multistep methods, with the innovation of choosing different methods for different ranges in Fourier space-high accuracy at low wavenumbers and A-stability at high wavenumbers. This new approach compares favorably to alternatives such as split-step and integrating factor (or linearly exact) methods. (C) 1999 Academic Press.