TY - JOUR

T1 - An efficient Newton-Krylov implementation of the constrained runs scheme for initializing on a slow manifold

AU - Vandekerckhove, Christophe

AU - Kevrekidis, Ioannis

AU - Roose, Dirk

N1 - Funding Information:
Acknowledgements It is a pleasure to acknowledge Wim Vanroose, Giovanni Samaey and Pieter Van Leemput for several interesting discussions about various aspects of this work. This article presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The research was also funded by the Research Council of the K.U.Leuven (Project OT/03/34). The work of IGK was partially supported by NSF and by the US DOE (CMPD).

PY - 2009/5

Y1 - 2009/5

N2 - The long-term dynamic behavior 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) is often extremely difficult or practically impossible. For this class of problems, the equation-free framework has been developed to enable performing coarse-grained computations, based on short full model simulations. Each full model simulation should be initialized so that the full model state is consistent with the values of the observables and close to the slow manifold. To compute such an initial full model state, a class of constrained runs functional iterations was proposed (Gear and Kevrekidis, J. Sci. Comput. 25(1), 17-28, 2005; Gear et al., SIAM J. Appl. Dyn. Syst. 4(3), 711-732, 2005). 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 article, we develop an implementation of the constrained runs scheme that is based on a (preconditioned) Newton-Krylov method rather than on a simple functional iteration. The functional iteration and the Newton-Krylov method are compared in detail using a lattice Boltzmann model for one-dimensional reaction-diffusion as the full model simulator. Depending on the parameters of the lattice Boltzmann model, the functional iteration may converge slowly or even diverge. We show that both issues are largely resolved by using the Newton-Krylov method, especially when a coarse grid correction preconditioner is incorporated.

AB - The long-term dynamic behavior 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) is often extremely difficult or practically impossible. For this class of problems, the equation-free framework has been developed to enable performing coarse-grained computations, based on short full model simulations. Each full model simulation should be initialized so that the full model state is consistent with the values of the observables and close to the slow manifold. To compute such an initial full model state, a class of constrained runs functional iterations was proposed (Gear and Kevrekidis, J. Sci. Comput. 25(1), 17-28, 2005; Gear et al., SIAM J. Appl. Dyn. Syst. 4(3), 711-732, 2005). 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 article, we develop an implementation of the constrained runs scheme that is based on a (preconditioned) Newton-Krylov method rather than on a simple functional iteration. The functional iteration and the Newton-Krylov method are compared in detail using a lattice Boltzmann model for one-dimensional reaction-diffusion as the full model simulator. Depending on the parameters of the lattice Boltzmann model, the functional iteration may converge slowly or even diverge. We show that both issues are largely resolved by using the Newton-Krylov method, especially when a coarse grid correction preconditioner is incorporated.

KW - Constrained runs

KW - Equation-free computing

KW - Initialization

KW - Lattice Boltzmann model

KW - Newton-Krylov method

KW - Slow manifold

UR - http://www.scopus.com/inward/record.url?scp=67349084990&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67349084990&partnerID=8YFLogxK

U2 - 10.1007/s10915-008-9256-y

DO - 10.1007/s10915-008-9256-y

M3 - Article

AN - SCOPUS:67349084990

VL - 39

SP - 167

EP - 188

JO - Journal of Scientific Computing

JF - Journal of Scientific Computing

SN - 0885-7474

IS - 2

ER -