Analytical and Numerical Approaches in Coagulation of Particles

Authors: Bilal Barakeh

Abstract:

In this paper we discuss the effect of unbounded particle interaction operator on particle growth and we study how this can address the choice of appropriate time steps of the numerical simulation. We provide also rigorous mathematical proofs showing that large particles become dominating with increasing time while small particles contribute negligibly. Second, we discuss the efficiency of the algorithm by performing numerical simulations tests and by comparing the simulated solutions with some known analytic solutions to the Smoluchowski equation.

Keywords: Stochastic processes, coagulation of particles, numerical scheme.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1056266

References:

[1] Smoluchowski, M.V, Versuch einer mathematischen Theorie der Koagulationskinetik kolloider L&o&sungen. Z. Phys. Chem. 92, 1916, pp. 129-168,
[2] Babovsky, H.: On a Monte Carlo scheme for Smoluchowski-s coagulation equation. Monte Carlo Methods Appl. vol. 5, 1999, pp. 1- 18.
[3] Barakeh, B.: Convergence proof of a Monte Carlo scheme for the resolution of the Smoluchowski coagulation equation. Aplimat-Journal of Applied Mathematics, vol. 2, 2009, pp. 149-158.
[4] Sabelfeld, K.K, Rogasinsky, S.V, Kolodka, A.A and levykin A.I.: Stochastic algorithms for Flow solving Smoluchowski coagulation 0 250 500 750 1000 1250 1500 0 0.2 0.4 0.6 0.8 1 0 250 500 750 1000 1250 1500 0 0.2 0.4 0.6 0.8 1 0 250 500 750 1000 1250 1500 0 0.2 0.4 0.6 0.8 1 0 250 500 750 1000 1250 1500 0 0.01 0.02 0.03 0.04 0.05 0.06 0 250 500 750 1000 1250 1500 -0.0125 -0.01 -0.0075 -0.005 -0.0025 0 0.0025 0 250 500 750 1000 1250 1500 0 0.001 0.002 0.003 0.004 0.005 equation & applications to aerosol growth simulation. Monte Carlo Methods Appl. vol. 2, 1996, pp. 41-87.
[5] Eibeck, A. and Wagner, W.: Stochastic particle approximations for smoluchowski-s coagulation equation. The Annals of Applied Probability, vol.11, 2001, pp.1137-1165.
[6] Eibeck, A. and Wagner, W.: An efficient stochastic algorithm for studying coagulation dynamics and gelation phenomena. Siam J. Sci . Comput. vol. 22, 2000, pp. 802-821.
[7] Babovsky, H. and Illner, R.: A convergence proof for Nanbu-s simulation method for the full Boltzmann equation. SIAM J. Numer. Anal., vol. 26, 1989, pp. 45-65.
[8] Kolodko, A. and Wagner, W.: Convergence of a Nanbu type method for the Smoluchowski equation. Monte Carlo Methods and Applications. Vol. 3, 1999, pp. 255-273.
[9] Lécot, C. and Wagner, W.: A quasi Monte Carlo scheme for Smoluchowski-s coagulation equation. Mathematics of Computation. 73 2004, pp.1953-1966.
[10] Hairer, E., Norsett, S.P, and Wanner, G.: Solving Ordinary Differential Equations I, Springer-Verlag, Berlin 1987.
[11] Gronwall, T.H.: Note on the derivative with respect to a parameter of the solutions of a system of and differential equations. Ann. of Math. 20, 1919, pp. 292- 296.
[12] Aldous, D.J.: Deterministic and Stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists, Bernoulli. vol. 5, 1999, pp. 3-48.