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 ) induced by a space and velocity transport operator. First, we investigate the properties (collisional invariants and equilibrium) of the asymptotic model obtained when . 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 that enables to capture numerically the solution of the asymptotic model. Numerous numerical results are displayed, illustrating the behavior of the method.
Keywords: Vlasov equation, Fokker–Planck collision operator, highly oscillatory systems, multiscale numerical schemes, Particle method.
Anaïs Crestetto 1; Nicolas Crouseilles 2; Damien Prel 1
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@article{SMAI-JCM_2024__10__141_0, author = {Ana{\"\i}s Crestetto and Nicolas Crouseilles and Damien Prel}, title = {Deterministic particle method for {Fokker{\textendash}Planck} equation with strong oscillations}, journal = {The SMAI Journal of computational mathematics}, pages = {141--173}, publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles}, volume = {10}, year = {2024}, doi = {10.5802/smai-jcm.109}, language = {en}, url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.109/} }
TY - JOUR AU - Anaïs Crestetto AU - Nicolas Crouseilles AU - Damien Prel TI - Deterministic particle method for Fokker–Planck equation with strong oscillations JO - The SMAI Journal of computational mathematics PY - 2024 SP - 141 EP - 173 VL - 10 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.109/ DO - 10.5802/smai-jcm.109 LA - en ID - SMAI-JCM_2024__10__141_0 ER -
%0 Journal Article %A Anaïs Crestetto %A Nicolas Crouseilles %A Damien Prel %T Deterministic particle method for Fokker–Planck equation with strong oscillations %J The SMAI Journal of computational mathematics %D 2024 %P 141-173 %V 10 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.109/ %R 10.5802/smai-jcm.109 %G en %F SMAI-JCM_2024__10__141_0
Anaïs Crestetto; Nicolas Crouseilles; Damien Prel. Deterministic particle method for Fokker–Planck equation with strong oscillations. The SMAI Journal of computational mathematics, Volume 10 (2024), pp. 141-173. doi : 10.5802/smai-jcm.109. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.109/
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