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Grassmannian spectral shooting

Veerle Ledoux, Simon JA Malham and Vera Thümmler (2010) MATHEMATICS OF COMPUTATION. 79(271). p.1585-1619
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.
Please use this url to cite or link to this publication:
author
organization
year
type
journalArticle (original)
publication status
published
subject
keyword
spectral theory, numerical shooting, Grassmann manifolds, FREDHOLM DETERMINANTS, SCHRODINGER-EQUATION, SYMPLECTIC INTEGRATORS, DIFFERENTIAL-EQUATIONS, EFFICIENT COMPUTATION, MATRIX RICCATI-EQUATIONS, GENERALIZED POLAR DECOMPOSITIONS, UNEQUAL DIFFUSION RATES, TRAVELING-WAVES, EVANS FUNCTION
journal title
MATHEMATICS OF COMPUTATION
Math. Comp.
volume
79
issue
271
pages
1585 - 1619
Web of Science type
Article
Web of Science id
000279776900013
JCR category
MATHEMATICS, APPLIED
JCR impact factor
1.382 (2010)
JCR rank
43/235 (2010)
JCR quartile
1 (2010)
ISSN
0025-5718
DOI
10.1090/S0025-5718-10-02323-9
language
English
UGent publication?
yes
classification
A1
copyright statement
I have transferred the copyright for this publication to the publisher
id
1013246
handle
http://hdl.handle.net/1854/LU-1013246
date created
2010-07-16 10:45:15
date last changed
2016-12-19 15:46:20
@article{1013246,
  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.},
  author       = {Ledoux, Veerle and Malham, Simon JA and Th{\"u}mmler, Vera},
  issn         = {0025-5718},
  journal      = {MATHEMATICS OF COMPUTATION},
  keyword      = {spectral theory,numerical shooting,Grassmann manifolds,FREDHOLM DETERMINANTS,SCHRODINGER-EQUATION,SYMPLECTIC INTEGRATORS,DIFFERENTIAL-EQUATIONS,EFFICIENT COMPUTATION,MATRIX RICCATI-EQUATIONS,GENERALIZED POLAR DECOMPOSITIONS,UNEQUAL DIFFUSION RATES,TRAVELING-WAVES,EVANS FUNCTION},
  language     = {eng},
  number       = {271},
  pages        = {1585--1619},
  title        = {Grassmannian spectral shooting},
  url          = {http://dx.doi.org/10.1090/S0025-5718-10-02323-9},
  volume       = {79},
  year         = {2010},
}

Chicago
Ledoux, Veerle, Simon JA Malham, and Vera Thümmler. 2010. “Grassmannian Spectral Shooting.” Mathematics of Computation 79 (271): 1585–1619.
APA
Ledoux, Veerle, Malham, S. J., & Thümmler, V. (2010). Grassmannian spectral shooting. MATHEMATICS OF COMPUTATION, 79(271), 1585–1619.
Vancouver
1.
Ledoux V, Malham SJ, Thümmler V. Grassmannian spectral shooting. MATHEMATICS OF COMPUTATION. 2010;79(271):1585–619.
MLA
Ledoux, Veerle, Simon JA Malham, and Vera Thümmler. “Grassmannian Spectral Shooting.” MATHEMATICS OF COMPUTATION 79.271 (2010): 1585–1619. Print.