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Grassmannian spectral shooting

(2010) MATHEMATICS OF COMPUTATION. 79(271). p.1585-1619
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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.
Keywords
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

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Citation

Please use this url to cite or link to this publication:

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.
@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},
}

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