Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

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


Code(s) de Classification MSC:

Résumé: Using the main ideas of Tanaka \cite{Tanaka:78}, 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-M\'el\'eard, \cite{Graham:96} is used, and allows 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.

Mots Clés: Boltzmann equations without cutoff ; Stochastic differential equations ; Jump measures ; Interacting particle systems

Date: 2000-02-03

Prépublication numéro: PMA-563

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