The initial value problem for weakly nonlinear PDE

Abstract

We discuss an extension of the pseudospectral method developed by Wineberg, McGrath, Gabl, Scott, and Southwell (1991) for the numerical integration of the Korteweg–de Vries (KdV) initial value problem. Our generalization of their algorithm can be used to solve initial value problems for a wide class of evolution equations that are “weakly nonlinear” in a sense we will make precise. This class includes in particular the other classical soliton equations, Sine-Gordon equation (SGE) and nonlinear Schr¨odinger equation (NLS). As well as being very simple to implement, this method exhibits remarkable speed and stability, making it ideal for use with visualization tools where it makes it possible to experiment in real time with soliton interactions and to see how a general solution decomposes into solitons. We analyze the structure of the algorithm, discuss some of the reasons behind its robust numerical behavior, and finally describe a fixed point theorem we have found that proves that the pseudospectral stepping algorithm converges.