Grassmannian spectral shooting. (English) Zbl 1196.65132
Summary: 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.
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.
MSC:
| 65L15 | Numerical solution of eigenvalue problems involving ordinary differential equations |
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
Keywords:
Grassmann manifolds; spectral theory; numerical shooting; Stiefel manifold; Riccati flow; Drury-Oja flow; algorithms; numerical examples; pure-point spectrum; linear stability; coherent structures; Evans function shooting and matching; Boussinesq solitary waves; autocatalytic travelling waves; Ekman boundary layerSoftware:
MATCONTReferences:
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