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Grassmannian spectral shooting

Authors: Veerle Ledoux, Simon J. A. Malham and Vera Thümmler
Journal: Math. Comp. 79 (2010), 1585-1619
MSC (2010): Primary 65L15, 65L10
Published electronically: January 25, 2010
MathSciNet review: 2630004
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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.

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Additional Information

Veerle Ledoux
Affiliation: Vakgroep Toegepaste Wiskunde en Informatica, Ghent University, Krijgslaan, 281-S9, B-9000 Gent, Belgium

Simon J. A. Malham
Affiliation: Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom

Vera Thümmler
Affiliation: Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

Keywords: Grassmann manifolds, spectral theory, numerical shooting
Received by editor(s): September 3, 2008
Received by editor(s) in revised form: July 6, 2009
Published electronically: January 25, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.