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Coarse-grained numerical bifurcation analysis of lattice Boltzmann models. (English) Zbl 1107.82061
Summary: In this paper we study the coarse-grained bifurcation analysis approach proposed by I. G. Kevrekidis and collaborators in [C. Theodoropoulos, Y. H. Qian and I. G. Kevrekidis, “Coarse” stability and bifurcation analysis using time-steppers: a reaction-diffusion example, Proc. Natl. Acad. Sci. 97, No. 18, 9840–9843 (2000; Zbl 1064.65121)]. We extend the results obtained in that paper for a one-dimensional FitzHugh-Nagumo lattice Boltzmann (LB) model in several ways. First, we extend the coarse-grained time stepper concept to enable the computation of periodic solutions and we use the more versatile Newton-Picard method rather than the Recursive Projection Method (RPM) for the numerical bifurcation analysis. Second, we compare the obtained bifurcation diagram with the bifurcation diagrams of the corresponding macroscopic PDE and of the lattice Boltzmann model. Most importantly, we perform an extensive study of the influence of the lifting or reconstruction step on the minimal successful time step of the coarse-grained time stepper and the accuracy of the results. It is shown experimentally that this time step must often be much larger than the time it takes for the higher-order moments to become slaved by the lowest-order moment, which somewhat contradicts earlier claims.

MSC:
 82C80 Numerical methods of time-dependent statistical mechanics (MSC2010) 37L60 Lattice dynamics and infinite-dimensional dissipative dynamical systems
Software:
AUTO2000; HomCont; MATCONT; ODEPACK
Full Text:
References:
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