Analysis of practical applications of lattice gas models

S. P. Bobkov ; A. S. Chernyavskaya ; V. V. Shergin

Modelirovanie i analiz informacionnyh sistem, Tome 26 (2019) no. 2, pp. 256-266

Résumé

In recent years, discrete approaches have been widely used in mathematical modeling of physicochemical processes. Cellular automata-based methods greatly simplify modeling procedures in many cases. In particular, this is important when using models in the form of partial differential equations systems to analyze the transfer of a substance in inhomogeneous media. In some cases, it is quite difficult to set the boundary conditions correctly if the object of study has boundaries of complex shape. It is also difficult to use mathematical physics classical equations if one cannot neglect the influence of stochastic effects on the process flow. The lattice gas models considered in the article are one of the types of cellular automata. Until now they have not been widely adopted, despite the fact that the first works on their use appeared about forty years ago. It is known, however, that lattice gases successfully describe a number of hydrodynamic phenomena, and the results obtained do not contradict the generally accepted views on the physical nature of continuous media motion processes. When using models of lattice gases, there are often questions about the correctness of the use of discrete models in various flow regimes. The second problem is a large-scale transition from model discrete parameters to generally accepted macroscopic characteristics of flows, such as flow velocity, viscosity and density of the medium, etc. It is also necessary to take into account that the indicated parameters in the lattice model are dimensionless, and the corresponding real macroscopic parameters have dimension. In this paper, an attempt is made to propose a method of large-scale transition, as well as to indicate the areas of practical use of some models of lattice gases.

Détail

Keywords: discrete approach, lattice gas, HPP, FHP and LBM models.

Bibliographie

[1] Wolfram S., A new kind of science, Wolfram media, Champaign, IL, 2002 | MR | Zbl

[2] Toffoli T., Margolus N., Cellular Automata Machines:a new environment for modeling, The MIT Press, Cambridge, Massachusetts, 1987

[3] Bandman O. L., “Kletochno-avtomatnye modeli prostranstvennoj dinamiki”, Sistemnaya informatika, 10 (2006), 59–113 (in Russ.)

[4] Wolfram S., “Cellular automaton fluids 1: Basic theory”, Stat. Phys., 45:3-4 (1986), 471–526 | DOI | MR | Zbl

[5] Clavin P. et al, “Simulation of free boundaries in flow system by lattice-gas models”, Journal of Fluid Mechanics, 188 (1988), 437–464 | DOI | Zbl

[6] Frish U., Hasslacher B., Pomeau Y., “Lattice-gas automata for the Navier-Stokes equation”, Physical Review Letters, 56:14 (1986), 1505–1508 | DOI

[7] Hardy J., de Pazzis O., Pomeau Y., “Molecular dynamics of a classical lattice gas: transport properties and time correlation functions”, Physical Review, 13:5 (1976), 1949–1961 | DOI

[8] Bandman O. L., “Diskretnoe modelirovanie fiziko-himicheskih processov”, Prikladnaya diskretnaya matematika, 3 (2009), 33–49 (in Russ.)

[9] Wolfram S., “Statistical mechanics of cellular automata”, Reviews of Modern Physics, 5:3 (1983), 601–644 | DOI | MR

[10] Wolf-Gladrow D., Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction, Springer, 2005 | MR

[11] Chopard B. et al., “Cellular automata and lattice Boltzmann techniques: an approach to model and simulate complex systems”, Advances in Complex Systems, 5:2 (2002), 103–246 | DOI | MR | Zbl

[12] Shan X., Chen H., “Simulation of nonideal gases and liquid-gas phase transitions by the lattice Boltzmann equation”, Phys. Rev. E, 49:4 (1994), 2941–2948 | DOI

[13] He X., Luo L., “Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation”, Phys. Rev. E, 56:6 (1997), 6811–6817 | DOI | MR

[14] Nourgaliev R. R et al., “The lattice Boltzmann equation method: Theoretical interpretation, numerics and implications”, Int. J. Multiphase Flow, 29:1 (2003), 117–169 | DOI | Zbl

[15] Frish U., Crutchfild J. P., Hasslacher B., “Lattice Gas hydrodynamics in two and three dimensions”, Complex Systems, 1 (1987), 649–707 | MR

[16] Abramovich G. N., Prikladnaya gasovaya dinamika, v. 2, Nauka, M., 1991 (in Russ.)