zbMATH — the first resource for mathematics

A lattice Boltzmann equation for diffusion. (English) Zbl 1106.82363
Summary: The formulation of lattice gas automata (LGA) for given partial differential equations is not straightforward and still requires ’some sort of magic”. Lattice Boltzmann equation (LBE) models are much more flexible than LGA because of the freedom in choosing equilibrium distributions with free parameters which can be set after a multiscale expansion according to certain requirements. Here a LBE is presented for diffusion in an arbitrary number of dimensions. The model is probably the simplest LBE which can be formulated. It is shown that the resulting algorithm with relaxation parameter \(\omega =1\) is identical to an explicit finite-difference (EFD) formulation at its stability limit. Underrelaxation \((0<\omega<1)\) allows stable integration beyond the stability limit of EFD. The time step of the explicit LBE integration is limited by accuracy and not by stability requirements.

82C40 Kinetic theory of gases in time-dependent statistical mechanics
76M99 Basic methods in fluid mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
Full Text: DOI
[1] W. F. Ames,Numerical Methods for Partial Differential Equations (Academic Press, New York, 1977). · Zbl 0577.65077
[2] P. Bhatnagar, E. P. Gross, and M. K. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems,Phys. Rev. 94(3):511–525 (1954). · Zbl 0055.23609
[3] H. Chen, and W. H. Matthaeus, New cellular automaton model for magnetohydrodynamics,Phys. Rev. Lett. 58(18):1845–1848 (1987).
[4] S. Chen, D. O. Martinez, W. H. Matthaeus, and H. Chen, Magnetohydrodynamics computations with lattice gas automata.J. Stat. Phys. 68(3/4):533–556 (1992). · Zbl 0923.76227
[5] S. Chen, G. D. Doolen, and W. H. Mattheus, Lattice gas automata for simple and complex fluids,J. Stat. Phys. 64(5/6):1133–1162 (1991).
[6] D. Dab, and J.-P. Boon, Cellular automata approach to reaction-diffusion systems, InCellular Automata and Modeling of Complex Physical Systems P. Manneville, N. Boccara, G. Y. Vichniac, and R. Bidaux, eds. (Springer, Berlin, 1989), pp. 257–273.
[7] D. d’Humières, P. Lallemand, and U. Frisch, Lattice gas models for 3D hydrodynamics,Europhys. Lett. 2(4):291–297 (1986).
[8] U. Frisch, B. Hasslacher, and Y. Pomeau, Lattice-gas automata for Navier-Stokes equations,Phys. Rev. Lett. 56:1505–1508 (1986).
[9] U. Frisch, D. d’Humières, B. Hasslacher, P. Lallemand, Y. Pomeau, and J.-P. Rivet, Lattice gas hydrodynamics in two and three dimensions,Complex Systems 1:649–707 (1987). · Zbl 0662.76101
[10] M. Hénon, Isometric collision rules for four-dimensional FCHC lattice gas.Complex Systems 1(3):475–494 (1987). · Zbl 0652.76051
[11] F. Higuera, S. Succi, and R. Benzi, Lattice gas dynamics with enhanced collisions,Europhys. Lett. 9(4):345–349 (1989).
[12] T. Karapiperis, and B. Blankleider, Cellular automaton model of reaction-transport processes,Physica D 78:30–64 (1994). · Zbl 0816.58025
[13] C. F. Kougias, Numerical simulations of small-scale oceanic fronts of river discharge type with the lattice gas automata method,J. Geophys. Res. 98(C10):18243–18255 (1993).
[14] D. O. Martinez, W. H. Matthaeus, S. Chen, and D. C. Montgomery, Comparison of spectral method and lattice Boltzmann simulations of two-dimensional hydrodynamics,Phys. Fluids 6(3):1285–1298 (1994). · Zbl 0826.76069
[15] G. McNamara, and G. Zanetti, Use of the Boltzmann equation to simulate lattice-gas automata,Phys. Rev. Lett. 61:2332–2335 (1988).
[16] D. Montgomery, and G. D. Doolen, Two cellular automata for plasma computations,Complex Systems 1:830–838 (1987). · Zbl 0652.76073
[17] R. Nasilowski, A cellular-automaton fluid model with simple rules in arbitrary many dimensions,J. Stat. Phys. 65(1/2):97–138 (1991). · Zbl 0943.82581
[18] Y. H. Qian, D. d’Humières, and P. Lallemand, Lattice BGK models for Navier-Stokes equation,Europhys. Lett. 17(6):479–484 (1992). · Zbl 1116.76419
[19] P. C. Rem, and J. A. Somers, Cellular automata algorithms on a transputer network, InDiscrete Kinematic Theory, Lattice Gas Dynamics, and Foundations of Hydrodynamics, R. Monaco, ed. (World Scientific, Singapore, 1989), pp. 268–275.
[20] D. H. Rothman, Cellular-automaton fluids: A model for flow in porous media.Geophysics 53(4):509–518, (1988).
[21] D. H. Rothman, and J. M. Keller, Immiscible cellular-automaton fluids,J. Stat. Phys. 52:1119–1127 (1988). · Zbl 1084.82504
[22] T. Toffoli, and N. Margolus, Invertible cellular automata: A review,Physica D 45:229–253 (1990). · Zbl 0729.68066
[23] D. V. van Coevorden, M. H. Ernst, R. Brito, and J. A. Somers, Relaxation and transport in FCHC lattice gases.J. Stat. Phys. 74(5/6):1085–1115 (1994).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.