On matrix-free computation of 2D unstable manifolds

arXiv:1003.4463v1 [math.DS] 23 Mar 2010 ON MATRIX-FREE COMPUTATION OF 2D UNSTABLE MANIFOLDS∗ L. VAN VEEN†, GENTA KAWAHARA‡, AND MATSUMURA ATSUSHI‡ Abstract. Recently, a flexible and stable algorithm was introduced for the computation of 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. The boundary value problem is under determined 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 boundary value problem is discretised, 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 Newton- Krylov continuation. We show that the number of time integrations necessary for each continuation step scales only 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 boundary value problem 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. Key words. Unstable manifold, orbit continuation, Newton-Krylov continuation, shear turbu- lence. AMS subject classifications. 65P99, 37N10, 34K19, 65L10 1. Introduction. In recent years, an increasing number of algorithms from nu- merical dynamical systems theory have become available for systems with many de- grees of freedom. These algorithms are designed for the computation and continuation of equilibrium states, time-periodic solutions, invariant tori and connecting orbits and are usually of the prediction-correction variety. The prediction can be based simply on data filtered from simulations or on extrapolation of a previously computed part of the continuation curve. The correction step, however, involves solving a large set of coupled nonlinear equations by an iterative method. Because of its quadratic conver- gence, the most desirable method here is Newton-Raphson iteration. This method, in turn, requires the repeated solution of a large linear system. It is not surprising, then, that the most significant step forward in this field was the introduction of Krylov sub- space methods for solving the linear systems. The combination of Newton-Raphson iteration with a Krylov subspace method for prediction-correction methods is now referred to as Newton-Krylov continuation. Newton-Krylov continuation has been used extensively in the context of fluid dynamics, where a large system of ordinary differential equations (ODEs) results from discretisation of the Navier-Stokes equation and possibly the continuity and energy equations. Early examples include the computation of equilibrium states in Taylor vortex flow by Edwards et al. [3] and the continuation of time-periodic solutions by S´anchez et al. [14]. More recently, the algorithm has also been used for the computation of quasi-periodic solutions [15]. ∗This work was supported by Le Fonds Qu´eb´ecois de la Recherche sur la Nature et les Technologies grant nr. 2009-NC-125259 and by a Grant-in-Aid for Scientific Research of the Japan Society for Promotion of Science †Faculty of Science, University of Ontario Institute of Technology, 2000 Simcoe St. N., Oshawa, ON L1H 7K4, Canada (lennaert.vanveen@uoit.ca) ‡Department of Mechanical Science and Bioengineering, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan (kawahara@me.es.osaka-u.ac.jp) 1 2 L. van Veen et al. Now that algorithms for the study of equilibria and (quasi) periodic solutions are available, it is natural to consider their stable and unstable manifolds and the global dynamical structures they represent. It is a well-known fact from dynamical systems theory that these manifolds, and the way they intersect in phase space, play an essential role in such phenomena as the generation of chaos, boundary crises and bursting behaviour. The tractability of manifold computation depends critically on the manifold dimension. A one-dimensional (un)stable manifold is just an integral curve of the ODEs and its computation is simple. The computation of manifolds of dimension three and up would be a formidable task. Even if we would design an algorithm which works regardless of the dimension of the ambient space, the repre- sentation of the manifold in a discrete data set and its interpretation would present great difficulties. In the current paper we focus on the computation of two-dimensional invari- ant manifolds. A number of algorithms has been designed for this end, mostly for low-dimensional systems. An overview of methods can be found in Krauskopf et al. [9]. One method presented there is par

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