NDLI: On Matrix-Free Computation of 2D Unstable Manifolds

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Atsushi, Matsumura Kawahara, Genta van Veen, Lennaert
Copyright Year	2011
Abstract	Recently, a flexible and stable algorithm was introduced for the computation of two-dimensional (2D) unstable manifolds of periodic solutions to systems of ordinary differential equations. The main idea of this approach is to represent orbits in this manifold as the solutions of an appropriate boundary value problem (BVP). The BVP is underdetermined, and a one-parameter family of solutions can be found by means of arclength continuation. This family of orbits covers a piece of the manifold. The quality of this covering depends on the way the BVP is discretized, as do the tractability and accuracy of the computation. In this paper, we describe an implementation of the orbit continuation algorithm which relies on multiple shooting and NewtonKrylov continuation. We show that the number of time integrations necessary for each continuation step scales with the number of shooting intervals but not with the number of degrees of freedom of the dynamical system. The number of shooting intervals is chosen based on linear stability analysis to keep the conditioning of the BVP in check. We demonstrate our algorithm with two test systems: a low-order model of shear flow and a well-resolved simulation of turbulent plane Couette flow.
Starting Page	25
Ending Page	44
Page Count	20
File Format	PDF
ISSN	10648275
DOI	10.1137/100789804
e-ISSN	10957197
Issue Number	1
Volume Number	33
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2011-01-06
Access Restriction	Subscribed
Subject Keyword	shear turbulence Boundary value problems NewtonKrylov continuation Dynamical systems in fluid mechanics, oceanography and meteorology orbit continuation Invariant manifolds unstable manifold
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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