Convergence of a finite volume scheme for coagulation-fragmentation equations

Bourgade, Jean-Pierre; Filbet, Francis

doi:10.1090/S0025-5718-07-02054-6

Convergence of a finite volume scheme for coagulation-fragmentation equations
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by Jean-Pierre Bourgade and Francis Filbet PDF

Math. Comp. 77 (2008), 851-882 Request permission

Abstract:

This paper is devoted to the analysis of a numerical scheme for the coagulation and fragmentation equation. A time explicit finite volume scheme is developed, based on a conservative formulation of the equation. It is shown to converge under a stability condition on the time step, while a first order rate of convergence is established and an explicit error estimate is given. Finally, several numerical simulations are performed to investigate the gelation phenomenon and the long time behavior of the solution.

References

Michael Aizenman and Thor A. Bak, Convergence to equilibrium in a system of reacting polymers, Comm. Math. Phys. 65 (1979), no. 3, 203–230. MR 530150
Hans Babovsky, On a Monte Carlo scheme for Smoluchowski’s coagulation equation, Monte Carlo Methods Appl. 5 (1999), no. 1, 1–18. MR 1684990, DOI 10.1515/mcma.1999.5.1.1
Thor A. Bak and Ole Heilmann, A finite version of Smoluchowski’s coagulation equation, J. Phys. A 24 (1991), no. 20, 4889–4893. MR 1131259
F. P. da Costa, A finite-dimensional dynamical model for gelation in coagulation processes, J. Nonlinear Sci. 8 (1998), no. 6, 619–653. MR 1650684, DOI 10.1007/s003329900061
R. L. Drake, A general mathematical survey of the coagulation equation, in Topics in Current Aerosol Research (Part 2), International Reviews in Aerosol Physics and Chemistry, Pergamon Press, Oxford, UK, (1972), pp. 203–376.
Andreas Eibeck and Wolfgang Wagner, An efficient stochastic algorithm for studying coagulation dynamics and gelation phenomena, SIAM J. Sci. Comput. 22 (2000), no. 3, 802–821. MR 1784826, DOI 10.1137/S1064827599353488
L. D. Erasmus, D. Eyre, and R. C. Everson, Numerical treatment of the population balance equation using a Spline-Galerkin method, Computers Chem. Engrg., 8 (1994), pp. 775–783.
M. Escobedo, S. Mischler, and B. Perthame, Gelation in coagulation and fragmentation models, Comm. Math. Phys. 231 (2002), no. 1, 157–188. MR 1947695, DOI 10.1007/s00220-002-0680-9
M. Escobedo, Ph. Laurençot, S. Mischler, and B. Perthame, Gelation and mass conservation in coagulation-fragmentation models, J. Differential Equations 195 (2003), no. 1, 143–174. MR 2019246, DOI 10.1016/S0022-0396(03)00134-7
Robert Eymard, Thierry Gallouët, and Raphaèle Herbin, Finite volume methods, Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, North-Holland, Amsterdam, 2000, pp. 713–1020. MR 1804748, DOI 10.1086/phos.67.4.188705
Francis Filbet and Philippe Laurençot, Mass-conserving solutions and non-conservative approximation to the Smoluchowski coagulation equation, Arch. Math. (Basel) 83 (2004), no. 6, 558–567. MR 2105334, DOI 10.1007/s00013-004-1060-9
Francis Filbet and Philippe Laurençot, Numerical simulation of the Smoluchowski coagulation equation, SIAM J. Sci. Comput. 25 (2004), no. 6, 2004–2028. MR 2086828, DOI 10.1137/S1064827503429132
Flavius Guiaş, A Monte Carlo approach to the Smoluchowski equations, Monte Carlo Methods Appl. 3 (1997), no. 4, 313–326. MR 1604836, DOI 10.1515/mcma.1997.3.4.313
Intae Jeon, Existence of gelling solutions for coagulation-fragmentation equations, Comm. Math. Phys. 194 (1998), no. 3, 541–567. MR 1631473, DOI 10.1007/s002200050368
Philippe Laurençot, On a class of continuous coagulation-fragmentation equations, J. Differential Equations 167 (2000), no. 2, 245–274. MR 1793195, DOI 10.1006/jdeq.2000.3809
Philippe Laurençot and Stéphane Mischler, From the discrete to the continuous coagulation-fragmentation equations, Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), no. 5, 1219–1248. MR 1938720, DOI 10.1017/S0308210500002080
Christian Lécot and Wolfgang Wagner, A quasi-Monte Carlo scheme for Smoluchowski’s coagulation equation, Math. Comp. 73 (2004), no. 248, 1953–1966. MR 2059745, DOI 10.1090/S0025-5718-04-01627-8
M. H. Lee, On the validity of the coagulation equation and the nature of runaway growth, Icarus, 143 (2000), pp. 74–86.
Randall J. LeVeque, Numerical methods for conservation laws, 2nd ed., Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992. MR 1153252, DOI 10.1007/978-3-0348-8629-1
F. Leyvraz, Existence and properties of post-gel solutions for the kinetic equations of coagulation, J. Phys. A 16 (1983), no. 12, 2861–2873. MR 715741
F. Leyvraz and H. R. Tschudi, Singularities in the kinetics of coagulation processes, J. Phys. A 14 (1981), no. 12, 3389–3405. MR 639565, DOI 10.1088/0305-4470/14/12/030
J. Makino, T. Fukushige, Y. Funato, and E. Kokubo, On the mass distribution of planetesimals in the early runaway stage, New Astronomy, 3 (1998), pp. 411–417.
Govind Menon and Robert L. Pego, Approach to self-similarity in Smoluchowski’s coagulation equations, Comm. Pure Appl. Math. 57 (2004), no. 9, 1197–1232. MR 2059679, DOI 10.1002/cpa.3048
H. Tanaka, S. Inaba, and K. Nakaza, Steady-state size distribution for the self-similar collision cascade, Icarus, 123 (1996), pp. 450–455.
Robert M. Ziff and E. D. McGrady, The kinetics of cluster fragmentation and depolymerisation, J. Phys. A 18 (1985), no. 15, 3027–3037. MR 814641