Summary
We propose a stochastic particle method for diffusive dynamics which allows coupling with kinetic reactions. This is realized by constructing and simulating the infinitesimal transitions of a Markov process which models the elementary processes taking place in the system. For this, we divide the domain into a finite number of cells. The simulation of the diffusive motion of the particles is based on assigning to the particles a velocity vector. Instead of considering jumps in all directions, we compute only the flux between neighbouring cells. This approach is a strong and effective improvement on the use of random walks. It allows also the approximation of coagulation equations with diffusion in a bounded domain: in every cell we simulate a coagulation process according to a usual method (direct simulation or mass flow algorithm) and we couple these dynamics with the spatial motion of particles.
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References
Bénilan, P., Wrzosek, D.: On an infinite system of reaction-diffusion equations. Adv. Math. Sci. Appl., 7(1), 351–366 (1997)
Deaconu, M., Fournier, N.: Probabilistic approach of some discrete and continuous coagulation equations with diffusion. Stochastic Processes Appl. 101(1), 83–111 (2002)
Ethier, S., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley, New York, 1986.
Eibeck, A., Wagner, W.: Stochastic particle approximations for Smoluchowski’s coagulation equation. Ann. Appl. Prob., 11(4), 1137–1165 (2001)
Guiaş, F: Convergence properties of a stochastic model for coagulationfragmentation processes with diffusion. Stochastic Anal. Appl., 19(2), 254–278 (2001)
Kurtz, T.G.: Limit theorems for sequences of jump Markov processes approximating ordinary differential processes, J. Appl. Prob., 8(1971) 344–356
Lions, P.L., Mas-Gallic, S.: Une méthode particulaire déterministe pour des équations diffusives non linéaires. C.R. Acad. Sci. Paris, 332(1), 369–376 (2001)
Laurençot, P., Mischler, S.: Global existence for the discrete diffusive coagulation-fragmentation equations in L1. Proc.R.Soc.Edinb., Sect. A, Math. 132(5), 1219–1248 (2002)
Russo, G.: Deterministic diffusion of particles. Comm. Pure Appl. Math., XLIII, 697–733 (1990)
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© 2006 Springer-Verlag Berlin Heidelberg
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Guiaş, F. (2006). A Stochastic Numerical Method for Diffusion Equations and Applications to Spatially Inhomogeneous Coagulation Processes. In: Niederreiter, H., Talay, D. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2004. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-31186-6_10
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DOI: https://doi.org/10.1007/3-540-31186-6_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-25541-3
Online ISBN: 978-3-540-31186-7
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