The Initial Value Problem for Weakly Nonlinear PDE

TL;DR

This paper extends a pseudospectral method for numerically solving weakly nonlinear PDEs, including soliton equations like KdV, SGE, and NLS, emphasizing its speed, stability, and convergence properties.

Contribution

It generalizes an existing pseudospectral algorithm to a broader class of weakly nonlinear evolution equations, with proven convergence and enhanced stability.

Findings

The method is simple to implement and fast.

It exhibits remarkable stability during simulations.

Convergence of the algorithm is proven via a fixed point theorem.

Abstract

We will discuss an extension of the pseudospectral method developed by Wineberg, McGrath, Gabl, and Scott for the numerical integration of the 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 that we will make precise. This class includes in particular the other classical soliton equations (SGE and 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 will 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.