Abstract
In this paper a synchronous multi-particle cellular automaton model of diffusion with self-annihilation is developed based on the multi-particle cellular automata suggested previously by other authors. The models of pure diffusion and diffusion with self-annihilation are described and investigated. The correctness of the models is tested separately against the exact solutions of the diffusion equation for different 3D domains. The accuracy of the cellular automata simulation results is investigated depending on the number of cells per a single physical unit. The calculation time of cellular automaton simulation of diffusion with self-annihilation is compared with the calculation time of the Monte Carlo random walk on parallelepipeds method for different domain sizes. The parallel implementation of the cellular automaton model is developed and efficiency of the parallel code is analyzed.
Supported by the Russian Science Foundation under Grant 19-11-00019.
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References
Smith, G.D.: Numerical Solution of Partial Differential Equations (Finite Difference Methods). Oxford University Press, Oxford (1990)
Courant, R., Friedrichsund, K., Lewy, H.: \(\ddot{\rm U}\)ber die partiellen Differentialgleichungen der mathematischen Physik. Math. Annalen 100, 32–74 (1928)
Sabelfeld, K.K.: Monte Carlo Methods in Boundary Value Problems. Springer, Heidelberg (1991)
Sabelfeld, K.K.: Random walk on spheres method for solving drift-diffusion problems. Monte Carlo Methods Appl. 22(4), 265–275 (2016)
Sabelfeld, K.K.: Random walk on spheres algorithm for solving transient drift-diffusion-reaction problems. Monte Carlo Methods Appl. 23(3), 189–212 (2017)
Sabelfeld, K.: Stochastic simulation methods for solving systems of isotropic and anisotropic drift-diffusion-reaction equations and applications in cathodoluminescence imaging. Submitted to Probabilistic Engineering Mechanics (2018)
Toffoli, T., Margolus, N.: Cellular Automata Machines: A New Environment for Modeling. MIT Press, USA (1987)
Weimar, J.R.: Cellular automata for reaction-diffusion systems. Parallel Comput. 23, 1699–1715 (1997)
Weimar, J.R.: Three-dimensional cellular automata for reaction-diffusion systems. Fundamenta Informaticae 52(1–3), 277–284 (2002)
Weimar, J.R., Tyson, J.J., Watson, L.T.: Diffusion and wave propagation in cellular automaton models of excitable media. Physica D 55(3–4), 309–327 (1992)
Chopard, B.: Cellular automata modeling of physical systems. In: Meyers, R. (ed.) Computational Complexity, pp. 407–433. Springer, New York (2012). https://doi.org/10.1007/978-1-4614-1800-9_27
Frenkel, D., Ernst, M.H.: Simulation of diffusion in a two-dimensional lattice-gas cellular automaton: a test of mode-coupling theory. Phys. Rev. Lett. 63(20), 2165–2168 (1989)
Chopard, B., Droz, M.: Cellular automata model for the diffusion equation. J. Stat. Phys. 64(3–4), 859–892 (1991)
Dab, D., Boon, J.-P.: Cellular automata approach to reaction-diffusion systems. In: Manneville, P., Boccara, N., Vichniac, G.Y., Bidaux, R. (eds.) Cellular Automata and Modeling of Complex Physical Systems, pp. 257–273. Springer, Heidelberg (1989). https://doi.org/10.1007/978-3-642-75259-9_23
Karapiperis, T., Blankleider, B.: Cellular automaton model of reaction-transport processes. Physica D 78, 30–64 (1994)
Bandman, O.L.: Comparative study of cellular-automata diffusion models. In: Malyshkin, V. (ed.) PaCT 1999. LNCS, vol. 1662, pp. 395–409. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48387-X_41
Bandman, O.: Cellular automata diffusion models for multicomputer implementation. Bull. Nov. Comp. Center Comp. Sci. 36, 21–31 (2014)
Medvedev, Y.: Multi-particle Cellular-automata models for diffusion simulation. In: Hsu, C.-H., Malyshkin, V. (eds.) MTPP 2010. LNCS, vol. 6083, pp. 204–211. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14822-4_23
Chopard, B., Frachebourg, L., Droz, M.: Multiparticle lattice gas automata for reaction diffusion systems. Int. J. Mod. Phys. C 05(01), 47–63 (1994)
Medvedev, Yu.: Automata noise in diffusion cellular-automata models. Bull. Nov. Comp. Center Comp. Sci. 30, 43–52 (2010)
Bandman, O.: The concept of invariants in reaction-diffusion cellular-automata. Bull. Nov. Comp. Center Comp. Sci. 33, 23–34 (2012)
Kortl\(\ddot{\rm u}\)ke, O.: A general cellular automaton model for surface reactions. J. Phys. A Math. Gen. 31(46), 9185–9197 (1998)
Mai, J., von Niessen, W.: Diffusion and reaction in multicomponent systems via cellular-automaton modeling: \(A+B_2\). J. Chem. Phys. 98(3), 2032–2037 (1993)
Rice, J.A.: Mathematical Statistics and Data Analysis, 3rd edn. Thomson Brooks/Cole, USA (2006)
Sabelfeld, K.K., Kireeva, A.E.: A meshless random walk on parallelepipeds algorithm for solving transient anisotropic diffusion-recombination equations and applications to cathodoluminescence imaging. Submitted to Numerische Mathematik (2018)
MVS-10P cluster, JSCC RAS. http://www.jscc.ru. Accessed 22 May 2019
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Kireeva, A., Sabelfeld, K.K., Kireev, S. (2019). Synchronous Multi-particle Cellular Automaton Model of Diffusion with Self-annihilation. In: Malyshkin, V. (eds) Parallel Computing Technologies. PaCT 2019. Lecture Notes in Computer Science(), vol 11657. Springer, Cham. https://doi.org/10.1007/978-3-030-25636-4_27
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