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Cellular automata methods of the numerical solution of mathematical physics equations for the hexagonal grid  

Authors
 Matyushkin I.V.
 Zapletina M.A.
Date of publication
 2016

Abstract
 Different CA types, which solve elliptic (Laplace), parabolic (diffusion) and hyperbolic (wave) partial differential equations is formulated for the hexagonal grid. The continuous CA is only used. As it turned out, the iteration patterns of Laplace and diffusion equations solutions have close connections. A local function of transition for boundary cells has been empirically proved, but not obtained. The connection between micro- (for local function of transition) and macro- (for diffusion equation) coefficients have been obtained for the quasi one-dimensional problem of the matter diffusion in the half-space. CA behavior was discussed in the context of Courant-Lewy conditions. In comparison with seven-dot scheme a block automaton, which we suggest as generalization of Margolus automaton to hexagonal grid, has been demonstrated a better behavior. The method of Neumann conditions writing with using CA has been proposed. This method is based on continuity equation. The method of coming out to complex plane is proposed for two-dimensional wave equation. This method allows avoiding the computation of second derivatives, which demand using CA with memory for time parameter.
Keywords
 cellular automata, partial differential equations, hexagonal grid, diffusion, wave equation
Library reference
 Matyushkin I.V., Zapletina M.A. Cellular automata methods of the numerical solution of mathematical physics equations for the hexagonal grid // Problems of Perspective Micro- and Nanoelectronic Systems Development - 2016. Proceedings / edited by A. Stempkovsky, Moscow, IPPM RAS, 2016. Part 4. P. 82-87.
URL of paper
 http://www.mes-conference.ru/data/year2016/pdf/D030.pdf

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