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"Coarse" Stability and Bifurcation Analysis Using Time-Steppers: A Reaction-Diffusion Example
Constantinos Theodoropoulos, Yue-Hong Qian and Ioannis G. Kevrekidis
Proceedings of the National Academy of Sciences of the United States of America
Vol. 97, No. 18 (Aug. 29, 2000), pp. 9840-9843
Published by: National Academy of Sciences
Stable URL: http://www.jstor.org/stable/123274
Page Count: 4
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Evolutionary, pattern forming partial differential equations (PDEs) are often derived as limiting descriptions of microscopic, kinetic theory-based models of molecular processes (e.g., reaction and diffusion). The PDE dynamic behavior can be probed through direct simulation (time integration) or, more systematically, through stability/bifurcation calculations; time-stepper-based approaches, like the Recursive Projection Method [Shroff, G. M. & Keller, H. B. (1993) SIAM J. Numer. Anal. 30, 1099-1120] provide an attractive framework for the latter. We demonstrate an adaptation of this approach that allows for a direct, effective ("coarse") bifurcation analysis of microscopic, kinetic-based models; this is illustrated through a comparative study of the FitzHugh-Nagumo PDE and of a corresponding Lattice-Boltzmann model.
Proceedings of the National Academy of Sciences of the United States of America © 2000 National Academy of Sciences