Grassmannian spectral shooting

Veerle Ledoux, S. J A Malham, Vera Thümmler

Research output: Contribution to journalArticle

18 Citations (Scopus)

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 the Ekman boundary layer. © 2010 American Mathematical Society.

Original languageEnglish
Pages (from-to)1585-1619
Number of pages35
JournalMathematics of Computation
Volume79
Issue number271
DOIs
Publication statusPublished - Jul 2010

Keywords

  • Grassmann manifolds
  • Numerical shooting
  • Spectral theory

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