Grassmannian spectral shooting

TL;DR

This paper introduces a stable, robust numerical method for computing spectra related to the stability of coherent structures, leveraging Grassmannian projections to improve accuracy and avoid singularities.

Contribution

It develops a new Grassmannian spectral shooting method that enhances stability and robustness in spectral computations for stability analysis of coherent structures.

Findings

Method avoids representation singularities

Comparable or better performance than continuous orthogonalization methods

Demonstrated effectiveness on Boussinesq waves, autocatalytic waves, and Ekman layers

Abstract

We present a new numerical method for computing the pure-point spectrum associated with the linear stability of coherent structures. In the context of the Evans function shooting and matching approach, all the relevant information is carried by the flow projected onto the underlying Grassmann manifold. We show how to numerically construct this projected flow in a stable and robust manner. In particular, the method avoids representation singularities by, in practice, choosing the best coordinate patch representation for the flow as it evolves. The method is analytic in the spectral parameter and of complexity bounded by the order of the spectral problem cubed. For large systems it represents a competitive method to those recently developed that are based on continuous orthogonalization. We demonstrate this by comparing the two methods in three applications: Boussinesq solitary waves, autocatalytic travelling waves and Ekman boundary layer.