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

Fournier, Nicolas; Méléard, Sylvie

doi:10.1090/S0025-5718-01-01339-4

A stochastic particle numerical method for 3D Boltzmann equations without cutoff
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by Nicolas Fournier and Sylvie Méléard PDF

Math. Comp. 71 (2002), 583-604 Request permission

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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