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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
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[1] Gear, C.W.; Kevrekidis, I.G.; Theodoropoulos, C., ‘coarse’ integration/bifurcation analysis via microscopic simulators: micro-Galerkin methods, Comput. chem. eng., 26, 7-8, 941-963, (2002)
[2] Kevrekidis, I.G.; Gear, C.W.; Hyman, J.M.; Kevrekidis, P.G.; Runborg, O.; Theodoropoulos, C., Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis, Commun. math. sci., 1, 4, 715-762, (2003) · Zbl 1086.65066
[3] Theodoropoulos, C.; Qian, Y.H.; Kevrekidis, I.G., “coarse” stability and bifurcation analysis using time-steppers: a reaction-diffusion example, Proc. natl. acad. sci., 97, 18, 9840-9843, (2000) · Zbl 1064.65121
[4] Samaey, G.; Roose, D.; Kevrekidis, I.G., The gap-tooth scheme for homogenization problems, Multiscale model. simul., 4, 1, 278-306, (2005) · Zbl 1092.35009
[5] E, W.; Engquist, B., The heterogeneous multiscale methods, Commun. math. sci., 1, 1, 87-133, (2003) · Zbl 1093.35012
[6] E.J. Doedel, R.C. Paffenroth, A.R. Champneys, T.F. Fairgrieve, Y.A. Kuznetsov, B.E. Oldeman, B. Sandstede, X. Wang, AUTO2000: continuation and bifurcation software for ordinary differential equations (with HomCont), Tech. rep., Concordia University, 2002.
[7] Dhooge, A.; Govaerts, W.; Kuznetsov, Y.A., MATCONT: A MATLAB package for numerical bifurcation analysis of odes, ACM trans. math. software, 29, 2, 141-164, (2003) · Zbl 1070.65574
[8] Shroff, G.M.; Keller, H.B., Stabilization of unstable procedures: the recursive projection method, SIAM J. numerical anal., 30, 4, 1099-1120, (1993) · Zbl 0789.65037
[9] Lust, K.; Roose, D., Computation and bifurcation analysis of periodic solutions of large-scale systems, (), 265-302 · Zbl 0958.65141
[10] Lust, K.; Roose, D.; Spence, A.; Champneys, A., An adaptive newton – picard algorithm with subspace iteration for computing periodic solutions, SIAM J. sci. comput., 19, 4, 1188-1209, (1998) · Zbl 0915.65088
[11] Qian, Y.H.; Orszag, S.A., Scalings in diffusion-driven reaction \(A + B \rightarrow C\): numerical simulations by lattice BGK models, J. stat. phys., 81, 1-2, 237-253, (1995) · Zbl 1106.82357
[12] Ginzbourg, I.; Adler, P.M., Boundary flow condition analysis for the three-dimensional lattice Boltzmann model, J. phys. II (France), 4, 191-214, (1994)
[13] He, X.; Zou, Q.; Luo, L.-S.; Dembo, M., Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model, J. stat. phys., 87, 1/2, 115-136, (1997) · Zbl 0937.82043
[14] Chen, H.; Chen, S.; Matthaeus, W.H., Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method, Phys. rev. A, 45, 8, R5339-R5342, (1992)
[15] Qian, Y.H.; D’Humières, D.; Lallemand, P., Lattice BGK models for navier – stokes equation, Europhys. lett., 17, 6, 479-484, (1992) · Zbl 1116.76419
[16] Chopard, B.; Dupuis, A.; Masselot, A.; Luthi, P., Cellular automata and lattice Boltzmann techniques: an approach to model and simulate complex systems, Adv. complex syst., 5, 2/3, 103-246, (2002) · Zbl 1090.82030
[17] Succi, S., The lattice Boltzmann equation for fluid dynamics and beyond, numerical mathematics and scientific computation, (2001), Oxford University Press
[18] Dawson, S.P.; Chen, S.; Doolen, G.D., Lattice Boltzmann computations for reaction-diffusion equations, J. chem. phys., 98, 2, 1514-1523, (1993)
[19] Dab, D.; Boon, J.-P.; Li, Y.-X., Lattice-gas automata for coupled reaction-diffusion equations, Phys. rev. lett., 66, 19, 2535-2538, (1991)
[20] Hindmarsch, A.C., ODEPACK, A systematized collection of ODE solvers, (), 55-64
[21] Makeev, A.G.; Maroudas, D.; Kevrekidis, I.G., “coarse” stability and bifurcation analysis using stochastic simulators: kinetic Monte Carlo examples, J. chem. phys., 116, 23, 10083-10091, (2002)
[22] Makeev, A.G.; Maroudas, D.; Panagiotopoulos, A.Z.; Kevrekidis, I.G., Coarse bifurcation analysis of kinetic Monte Carlo simulations: A lattice-gas model with lateral interactions, J. chem. phys., 117, 18, 8229-8240, (2002)
[23] Keller, H.B., Numerical solution of bifurcation and nonlinear eigenvalue problems, (), 359-384 · Zbl 0581.65043
[24] Saad, Y., Numerical methods for large eigenvalue problems, algorithms and architectures for advanced scientific computing, (1992), Manchester University Press Manchester
[25] Keller, H., Numerical methods for two-point boundary value problems, (1968), Blaisdell New-York · Zbl 0172.19503
[26] Ascher, U.; Mattheij, R.; Russell, R., Numerical solution of boundary value problems for ordinary differential equations, (1988), Prentice-Hall Englewood Cliffs, NJ
[27] K. Lust, PDEcont, URL: http://www.math.rug.nl/ kurt/r_PDEcont.html.
[28] van der Sman, R.G.M.; Ernst, M.H., Convection-diffusion lattice Boltzmann scheme for irregular lattices, J. comput. phys., 160, 2, 766-782, (2000) · Zbl 1040.76514
[29] C.W. Gear, I.G. Kevrekidis, Constraint-defined manifolds: a legacy code approach to low-dimensional computation, Tech. Rep. physics/0312094, arXiv e-Print archive, 2003. · Zbl 1203.37005
[30] C.W. Gear, T.J. Kaper, I.G. Kevrekidis, A. Zagaris, Projecting to a slow manifold: singularly perturbed systems and legacy codes, SIAM J. Appl. Dyn. Syst., 2005, in press. · Zbl 1170.34343
[31] Van Leemput, P.; Lust, K., Numerical bifurcation analysis of lattice Boltzmann models: a reaction-diffusion example, (), 572-579 · Zbl 1107.65115
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