Mathematics of Computation

A stochastic particle numerical method for 3D Boltzmann equations without cutoff

Author(s): Nicolas Fournier; Sylvie Méléard.
Journal: Math. Comp. 71 (2002), 583-604.
MSC (2000): Primary 60J75, 60H10, 60K35, 82C40
Posted: October 25, 2001
Retrieve article in: PDF DVI PostScript
This article is available free of charge

Abstract: Using the main ideas of Tanaka, the measure-solution $\{P_t\}_t$ of a

-dimensional spatially homogeneous Boltzmann equation of Maxwellian molecules without cutoff is related to a Poisson-driven stochastic differential equation. Using this tool, the convergence to $\{P_t\}_t$ of solutions $\{P^l_t\}_t$ of approximating Boltzmann equations with cutoff is proved. Then, a result of Graham-Méléard is used and allows us to approximate $\{P^l_t\}_t$ with the empirical measure $\{\mu^{l,n}_t\}_t$ of an easily simulable interacting particle system. Precise rates of convergence are given. A numerical study lies at the end of the paper.

1.: Babovsky, H.; Illner, R.: A convergence proof for Nanbu's simulation method for the full Boltzmann equation, SIAM J. Num. Anal. 26 1, 46-65 (1989). MR 90e:76104
2.: Cercignani, C.; Illner, R.; Pulvirenti, M.: The mathematical theory of dilute gases, Applied Math. Sciences, Springer-Verlag, Berlin (1994). MR 96g:82046
3.: Desvillettes, L.: About the regularizing properties of the non-cut-off Kac equation. Comm. Math. Physics 168, 417-440, (1995). MR 96d:82052
4.: Desvillettes, L.; Graham, C.; Méléard, S.: Probabilistic interpretation and numerical approximation of a Kac equation without cutoff, Stoch. Proc. and Appl., 84, 1, 115-135 (1999). MR 2000k:60127
5.: Fournier, N.: Calcul des variations stochastiques sur l'espace de Poisson, applications à des E.D.P.S. paraboliques avec sauts et à certaines équations de Boltzmann. Thèse de l'Université Paris 6, Chapitre 4, (1999).
6.: Graham, C.; Méléard, S.: Stochastic particle approximations for generalized Boltzmann models and convergence estimates, Ann. Prob.25, 115-132 (1997). MR 98k:60181
7.: Graham, C.; Méléard, S.: Existence and regularity of a solution of a Kac equation without cutoff using the stochastic calculus of variations, Commun. Math. Phys. 205, 551-569 (1999). MR 2001c:60091
8.: Horowitz, J.; Karandikar, R.L.: Martingale problems associated with the Boltzmann equation, Seminar on Stochastic Processes, 1989 (E. Sinlar, K.L. Chung, R.K. GeFoor, eds.), Birkhäuser, Boston (1990). MR 92e:60194
9.: Jacod, J.; Shiryaev, A.N.: Limit theorems for stochastic processes, Springer-Verlag (1987). MR 89k:60044
10.: Méléard, S.: Asymptotic behaviour of some interacting particle systems, McKean-Vlasov and Boltzmann models, cours du CIME 95, Probabilistic models for nonlinear pde's, L.N. in Math. 1627, Springer (1996). MR 98f:60211
11.: Méléard, S.: Convergence of the fluctuations for interacting diffusions with jumps associated with Boltzmann equations, Stochastics and Stoch. Rep., Vol 63, pp 195-225, (1998). MR 99g:60103
12.: Nanbu, K.: Interrelations between various direct simulation methods for solving the Boltzmann equation, J. Phys. Soc. Japan 52, 3382-3388 (1983).
13.: Shiga, T.; Tanaka, H.: Central limit theorem for a system of Markovian particles with mean-field interactions, Z. Wahrsch. Verw. Geb. 69, 439-459 (1985). MR 88a:60056
14.: Sznitman, A.S.: Équations de type de Boltzmann, spatialement homogènes, Z. Wahrsch. verw. Geb. 66, 559-592 (1984). MR 86g:60081
15.: Tanaka, H.: On the uniqueness of Markov process associated with the Boltzmann equation of Maxwellian molecules, Proc. Intern. Symp. SDE, Kyoto, 409-425 (1976). MR 81d:60063
16.: Tanaka, H.: Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrsch. Verw. Geb. 46, 67-105 (1978). MR 80b:60083
17.: Toscani, G.; Villani, C.: Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas, J. Stat. Phys., 619-637 (1999). MR 2000f:82089
18.: Wagner, W.: A convergence proof for Bird's direct simulation method for the Boltzmann equation, J. Stat. Phys. 66, 1011-1044 (1992). MR 93f:82053

Retrieve articles in Mathematics of Computation with MSC (2000): 60J75, 60H10, 60K35, 82C40

Nicolas Fournier
Affiliation: Institut Elie Cartan, Faculté de Sciences, B.P. 239, 54506 Vandoeune-les-Nancy Cedex, France
Email: Nicolas.Fournier@antares.iecn.u-nancy.fr

Sylvie Méléard
Affiliation: Laboratoire de Probabilités, Paris 6, 4 place Jussieu, 75252 Paris cedex 05, France; and MODALX, UFR SEGMI, Université Paris 10, 92000 Nanterre, France
Email: sylm@ccr.jussieu.fr

DOI: 10.1090/S0025-5718-01-01339-4
PII: S 0025-5718(01)01339-4
Keywords: Boltzmann equations without cutoff, stochastic differential equations, jump measures, interacting particle systems
Received by editor(s): March 14, 2000
Posted: October 25, 2001
Copyright of article: Copyright 2001, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS