Computing Evans functions numerically via boundary-value problems
For researchers studying spectral stability of travelling waves, this provides a linear and scalable alternative to existing shooting methods.
The authors present a new boundary-value problem formulation for computing Evans functions numerically, proving convergence and demonstrating scalability to large problems including multi-dimensional eigenvalue computations.
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.