Travelling wave solutions of multisymplectic discretizations of semi-linear wave equations

TL;DR

This paper investigates how well multisymplectic discretizations preserve travelling wave solutions of semi-linear wave equations, using backward error analysis to compare numerical solutions with the PDE's solutions and revealing grid-size dependent resonances.

Contribution

It introduces a framework applying backward error analysis to study travelling wave preservation in multisymplectic discretizations of semi-linear wave equations.

Findings

Exact solutions for discontinuous nonlinearities.

Resonances depending on grid size for smooth nonlinearities.

Backward error analysis links numerical and continuous solutions.

Abstract

How well do multisymplectic discretisations preserve travelling wave solutions? To answer this question, the 5-point central difference scheme is applied to the semi-linear wave equation. A travelling wave ansatz leads to an ordinary difference equation, whose solutions correspond to the numerical scheme and can be compared to travelling wave solutions of the corresponding PDE. For a discontinuous nonlinearity the difference equation is solved exactly. For continuous nonlinearities the difference equation is solved using a Fourier series, and resonances that depend on the grid-size are revealed for a smooth nonlinearity. In general, the infinite dimensional functional equation, which must be solved to get the travelling wave solutions, is intractable, but backward error analysis proves to be a powerful tool, as it provides a way to study the solutions of the equation through a simple ODE that describes the behavior to arbitrarily high order. A general framework for using backward error analysis to analyze preservation of travelling waves for other equations and discretisations is presented. Then, the advantages that multisymplectic methods have over other methods are briefly highlighted.