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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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


Authors: Nicolas Fournier and Sylvie Méléard
Journal: Math. Comp. 71 (2002), 583-604
MSC (2000): Primary 60J75, 60H10, 60K35, 82C40
Published electronically: October 25, 2001
MathSciNet review: 1885616
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Abstract: Using the main ideas of Tanaka, the measure-solution $\{P_t\}_t$ of a $3$-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.


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Additional Information

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: http://dx.doi.org/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
Published electronically: October 25, 2001
Article copyright: © Copyright 2001 American Mathematical Society