TY - JOUR
T1 - Coarse-grained numerical bifurcation analysis of lattice Boltzmann models
AU - Van Leemput, P.
AU - Lust, K. W.A.
AU - Kevrekidis, I. G.
N1 - Funding Information:
This work was done while KL was a postdoctoral fellow of the Fund for Scientific Research—Flanders which also provided further funding through project G.0130.03 (PVL, KL). This paper presents research results of the Belgian Programme on Interuniversity Attraction Poles, initiated by the Belgian Federal Science Policy Office (PVL, KL). The work of IGK was partially supported by AFOSR and by an NSF/ITR grant. The scientific responsibility rests with its authors.
PY - 2005/10/1
Y1 - 2005/10/1
N2 - In this paper we study the coarse-grained bifurcation analysis approach proposed by I.G. Kevrekidis and collaborators in PNAS [C. Theodoropoulos, Y.H. Qian, I.G. Kevrekidis, "Coarse" stability and bifurcation analysis using time-steppers: a reaction-diffusion example, Proc. Natl. Acad. Sci. 97 (18) (2000) 9840-9843]. 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.
AB - In this paper we study the coarse-grained bifurcation analysis approach proposed by I.G. Kevrekidis and collaborators in PNAS [C. Theodoropoulos, Y.H. Qian, I.G. Kevrekidis, "Coarse" stability and bifurcation analysis using time-steppers: a reaction-diffusion example, Proc. Natl. Acad. Sci. 97 (18) (2000) 9840-9843]. 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.
KW - Coarse-grained modeling
KW - Lattice Boltzmann method
KW - Newton-Picard method
KW - Numerical bifurcation analysis
KW - Reaction-diffusion systems
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U2 - 10.1016/j.physd.2005.06.033
DO - 10.1016/j.physd.2005.06.033
M3 - Article
AN - SCOPUS:24644483244
SN - 0167-2789
VL - 210
SP - 58
EP - 76
JO - Physica D: Nonlinear Phenomena
JF - Physica D: Nonlinear Phenomena
IS - 1-2
ER -