Computing Evans functions numerically via boundary-value problems

TL;DR

This paper introduces a new boundary-value problem approach for numerically computing Evans functions, offering a linear, scalable alternative to existing shooting methods, with proven convergence and demonstrated effectiveness on complex problems.

Contribution

A novel boundary-value problem formulation for Evans function computation that is linear, scalable, and proven to converge, improving upon existing shooting methods.

Findings

The proposed method converges reliably.

It is scalable to multi-dimensional problems.

It outperforms shooting methods in efficiency.

Abstract

The Evans function has been used extensively to study spectral stability of travelling-wave solutions in spatially extended partial differential equations. To compute Evans functions numerically, several shooting methods have been developed. In this paper, an alternative scheme for the numerical computation of Evans functions is presented that relies on an appropriate boundary-value problem formulation. Convergence of the algorithm is proved, and several examples, including the computation of eigenvalues for a multi-dimensional problem, are given. The main advantage of the scheme proposed here compared with earlier methods is that the scheme is linear and scalable to large problems.