Theory of multicolor lattice gas: A cellular automaton poisson solver

doi:10.1016/0021-9991(90)90188-7

Journal of Computational Physics

Volume 88, Issue 2, June 1990, Pages 433-466

https://doi.org/10.1016/0021-9991(90)90188-7 Get rights and content

Abstract

A class of cellular automata models is considered, consisting of a quiescent hydrodynamic lattice gas with multiple-valued passive labels or “colors.” Controlled sources of particle color are introduced on the lattice, as are collisions that change individual particle colors while preserving net color. This lattice gas model is shown to be equivalent, in steady state, to a solution to a Poisson equation, with source function proportional to the rate of color introduction and inversely proportional to the intrinsic color diffusivity. The rigorous proofs of the essential features of the multicolor lattice gas are facilitated by use of an equivalent “subparticle” representation in which the color is represented by underlying two-state “spins.” Theorems deduced in this way are valid for arbitrary numbers of allowed color values. For example, it is shown that the color diffusivity depends only on the density, for all models of this type. Some preliminary investigations of the efficiency and accuracy of the method are also discussed. Rates of relaxation to the steady state are estimated and schemes for introducing Dirichlet and Neumann boundary conditions are described. Two simple numerical test cases are presented that verify the theory. These results, most of which easily generalize to three dimensions, suggest that a lattice gas of this type may be a useful tool for solution of the Poisson equation.

References (15)

D. Montgomery et al.
Phys. Lett. A
(1987)
U. Frisch et al.
Phys. Rev. Lett.
(1986)
U. Frisch et al.
Complex Systems
(1987)
S. Wolfram
J. Slat. Phys.
(1986)
D. Montgomery et al.
Complex Systems
(1986)
D. Boghosian et al.
Complex Systems
(1987)
H. Chen et al.
Phys. Fluids
(1987)

There are more references available in the full text version of this article.

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Theory of multicolor lattice gas: A cellular automaton poisson solver

Abstract

Phys. Lett. A

Phys. Rev. Lett.

Complex Systems

J. Slat. Phys.

Complex Systems

Complex Systems

Phys. Fluids