Spectral Methods for a Nonlinear Initial Value Problem Involving Pseudo Differential Operators

Abstract

Spectral methods (Fourier methods) for approximating the solution of a nonlinear initial value problem involving pseudo differential operators are defined and analysed. A semidiscrete approximation to the nonlinear equation based on an $L^2 $ projection is described. The semidiscrete $L^2 $ approximation is shown to be a priori stable and convergent under sufficient decay and smoothness assumptions on the initial data. It is shown that the semidiscrete method converges with infinite order, that is, higher order decay and smoothness assumptions imply higher order error bounds. Spectral schemes based on spacial collocation are also discussed.