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TW 502
Christophe Vandekerckhove, Ioannis G. Kevrekidis and Dirk Roose
An efficient Newton-Krylov implementation of the constrained runs scheme for initializing on a slow manifold
Abstract
The long-term dynamics of many dynamical systems evolves on a low-dimensional, attracting, invariant slow manifold, which can be parameterized by only a few variables (``observables''). The explicit derivation of such a slow manifold (and thus, the reduction of the long-term system dynamics) can be extremely difficult, or practically impossible. For this class of problems, the equation-free computational approach has been developed to perform numerical tasks with the unavailable reduced (coarse-grained) model based on short full model simulations. Each full model simulation should be initialized consistent with the values of the observables and close to the slow manifold. For this purpose, a class of constrained runs functional iterations was recently proposed. The schemes in this class only use the full model simulator and converge, under certain conditions, to an approximation of the desired state on the slow manifold. In this paper, we develop a constrained runs implementation that is based on a (preconditioned) Newton-Krylov method rather than on a functional iteration. We implement and compare both the functional iteration and Newton-Krylov method, using, as the full simulator, a lattice-Boltzmann model for a one-dimensional nonlinear reaction diffusion system. Depending on the parameters of the lattice Boltzmann model, the constrained runs functional iteration may converge slowly or even diverge. We show that both issues are largely resolved by using the constrained runs Newton-Krylov method, especially when a coarse grid correction preconditioner is incorporated.
report.pdf (392K) / mailto: C. Vandekerckhove