Deterministic particle method for Fokker–Planck equation with strong oscillations
Crestetto, Anaïs1 ; Crouseilles, Nicolas2 ; Prel, Damien1
1Nantes Université, CNRS, Laboratoire de Mathématiques Jean Leray, LMJL, UMR 6629, F-44000 Nantes, France & Inria (MINGuS), France
2Univ Rennes, Inria (MINGuS) & IRMAR UMR 6625 & ENS Rennes, France
The SMAI Journal of computational mathematics, Tome 10 (2024), pp. 141-173

The aim of this paper is to investigate a deterministic particle method for a model containing a Fokker–Planck collision operator in velocity and strong oscillations (characterized by a small parameter ε) induced by a space and velocity transport operator. First, we investigate the properties (collisional invariants and equilibrium) of the asymptotic model obtained when ε→0. Second a numerical method is developed to approximate the solution of the multiscale Fokker–Planck model. To do so, a deterministic particle method (recently introduced for the Landau equation in Carrillo et al. 2020) is proposed for Fokker–Planck type operators. This particle method consists in reformulating the collision operator in an advective form and in regularizing the advection field in such a way that it conserves the geometric bracket structure. In the Fokker–Planck homogeneous case, the properties of the resulting method are analysed. In the non homogeneous case, the particle method is coupled with a uniformly accurate time discretization in ε that enables to capture numerically the solution of the asymptotic model. Numerous numerical results are displayed, illustrating the behavior of the method.

Publié le :
DOI : 10.5802/smai-jcm.109
Classification : 65M25, 65M75, 35Q83
Keywords: Vlasov equation, Fokker–Planck collision operator, highly oscillatory systems, multiscale numerical schemes, Particle method.

Crestetto, Anaïs  1   ; Crouseilles, Nicolas  2   ; Prel, Damien  1

1 Nantes Université, CNRS, Laboratoire de Mathématiques Jean Leray, LMJL, UMR 6629, F-44000 Nantes, France & Inria (MINGuS), France
2 Univ Rennes, Inria (MINGuS) & IRMAR UMR 6625 & ENS Rennes, France 
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     title = {Deterministic particle method for {Fokker{\textendash}Planck} equation with strong oscillations},
     journal = {The SMAI Journal of computational mathematics},
     pages = {141--173},
     year = {2024},
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Crestetto, Anaïs; Crouseilles, Nicolas; Prel, Damien. Deterministic particle method for Fokker–Planck equation with strong oscillations. The SMAI Journal of computational mathematics, Tome 10 (2024), pp. 141-173. doi: 10.5802/smai-jcm.109

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