Newton–Krylov continuation of periodic orbits for Navier–Stokes flows

Efficient numerical algorithms for the continuation of periodic orbits of high-dimensional dissipative dynamical systems, and for analyzing their stability are presented. They are based on shooting, Newton–Krylov and Arnoldi methods. A thermal convection fluid dynamics problem, which has a rich bifu...

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Bibliographic Details
Published in:Journal of computational physics 2004-11, Vol.201 (1), p.13-33
Main Authors: Sánchez, J., Net, M., Garcı́a-Archilla, B., Simó, C.
Format: Article
Language:English
Subjects:
Arnoldi decomposition
Computational techniques
Continuation methods
Exact sciences and technology
Krylov methods
Mathematical methods in physics
Periodic orbits
Physics
Poincaré maps
Subspace iteration
Variational equations
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Summary:Efficient numerical algorithms for the continuation of periodic orbits of high-dimensional dissipative dynamical systems, and for analyzing their stability are presented. They are based on shooting, Newton–Krylov and Arnoldi methods. A thermal convection fluid dynamics problem, which has a rich bifurcation diagram due to symmetries, has been used as test. After a pseudo-spectral discretization of the equations a system of dimension O(10 4) has been obtained. The efficiency of the algorithms, which allows the unfolding of a complex diagram of periodic orbits, makes the methods suitable for the study of large nonlinear dissipative partial differential equations.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2004.04.018