Numerical error analysis for Evans function computations: a numerical gap lemma, centered-coordinate methods, and the unreasonable effectiveness of continuous orthogonalization

TL;DR

This paper analyzes numerical errors in Evans function computations, revealing why certain methods like centered coordinates and continuous orthogonalization are more stable than expected, through a novel error analysis based on a numerical gap lemma.

Contribution

It introduces a numerical gap lemma for error estimates and explains the unexpected stability of continuous orthogonalization in Evans function calculations.

Findings

Centered coordinates improve stability in exterior product methods.

Continuous orthogonalization is neutrally stable when approximating unstable subspaces.

A simple nonlinear boundary-value method is proposed for large-scale systems.

Abstract

We perform error analyses explaining some previously mysterious phenomena arising in numerical computation of the Evans function, in particular (i) the advantage of centered coordinates for exterior product and related methods, and (ii) the unexpected stability of the (notoriously unstable) continuous orthogonalization method of Drury in the context of Evans function applications. The analysis in both cases centers around a numerical version of the gap lemma of Gardner--Zumbrun and Kapitula--Sandstede, giving uniform error estimates for apparently ill-posed projective boundary-value problems with asymptotically constant coefficients, so long as the rate of convergence of coefficients is greater than the "badness" of the boundary projections as measured by negative spectral gap. In the second case, we use also the simple but apparently previously unremarked observation that the Drury method is in fact (neutrally) stable when used to approximate an unstable subspace, so that continuous orthogonalization and the centered exterior product method are roughly equally well-conditioned as methods for Evans function approximation. The latter observation makes possible an extremely simple nonlinear boundary-value method for possible use in large-scale systems, extending ideas suggested by Sandstede. We suggest also a related linear method based on the conjugation lemma of M\'etivier--Zumbrun, an extension of the gap lemma mentioned above.