Approximating travelling waves by equilibria of non local equations

TL;DR

This paper introduces a method to approximate traveling wave solutions of parabolic equations by transforming them into non-local equations, enabling stable computation of wave profiles and speeds without moving meshes.

Contribution

The authors develop a novel approach transforming parabolic equations with traveling waves into non-local equations, facilitating efficient numerical approximation of wave profiles and speeds.

Findings

The non-local equation has a unique, asymptotically stable stationary solution.

The method accurately approximates traveling wave profiles and speeds.

Numerical examples demonstrate the effectiveness of the approach.

Abstract

We consider an evolution equation of parabolic type in R having a travelling wave solution. We perform an appropriate change of variables which transforms the equation into a non local evolution one having a travelling wave solution with zero speed of propagation with exactly the same profile as the original one. We analyze the relation of the new equation with the original one in the entire real line. We also analyze the behavior of the non local problem in a bounded interval with appropriate boundary conditions and show that it has a unique stationary solution which is asymptotically stable for large enough intervals and that converges to the travelling wave as the interval approaches the entire real line. This procedure allows to compute simultaneously the travelling wave profile and its propagation speed avoiding moving meshes, as we illustrate with several numerical examples.