Grassmannian spectral shooting
For researchers studying linear stability of coherent structures, this method offers a stable and robust alternative to existing approaches, though it is incremental in nature.
The paper introduces a new numerical method for computing the pure-point spectrum of linear stability of coherent structures using Grassmann manifold projection, avoiding representation singularities and achieving cubic complexity. It demonstrates competitiveness with continuous orthogonalization methods in three applications.
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.