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“Coarse” stability and bifurcation analysis using time-steppers: a reaction-diffusion example. (English) Zbl 1064.65121
Summary: 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 [G. M. Shroff and H. B. Keller, SIAM J. Numer. Anal. 30, No. 4, 1099–1120 (1993; Zbl 0789.65037)] 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.

MSC:
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
76M28 Particle methods and lattice-gas methods
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