A local greedy algorithm and higher order extensions for global numerical continuation of analytically varying subspaces

TL;DR

This paper introduces a simple, stable greedy algorithm and higher-order extensions for numerically continuing analytically varying invariant subspaces, with applications to stability analysis of PDE traveling waves.

Contribution

It presents a novel, easy-to-implement greedy algorithm and second-order extensions for numerical continuation of invariant subspaces, improving stability and simplicity over existing methods.

Findings

The first-order greedy algorithm is stable and easy to program.

Second-order extensions provide higher accuracy.

Applicable to numerical Evans function computations for PDE stability.

Abstract

We present a family of numerical implementations of Kato's ODE propagating global bases of analytically varying invariant subspaces, of which the first-order version is a surprising simple "greedy algorithm" that is both stable and easy to program and the second-order version a relaxation of a first-order scheme of Brin and Zumbrun. The method has application to numerical Evans function computations used to assess stability of traveling-wave solutions of time-evolutionary PDE.