Hyperbolic reduced model for Vlasov-Poisson equation with Fokker-Planck collision

Emmanuel Franck (1, 2, 3) , Ibtissem Lannabi (4) , Youssouf Nasseri (5, 1) , Laurent Navoret (1, 5) , Giuseppe Parasiliti Rantone (6, 7) , Guillaume Steimer (1, 5)

1 IRMA - Institut de Recherche Mathématique Avancée
2 Inria Nancy - Grand Est
3 MACARON - Apprentissage automatique pour des méthodes numériques optimisées
4 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]
5 TONUS - TOkamaks and NUmerical Simulations
6 ANGE - Numerical Analysis, Geophysics and Ecology
7 SU - Sorbonne Université

This paper proposes a reduced model to simulate the one-dimensional Vlasov-Poisson equation with the non-linear Fokker-Planck operator. The model provides the space-time dynamics of a few macroscopic quantities constructed following the Reduced Order Method (ROM) in the velocity variable: the compression is thus applied to the semi-discretization of the Vlasov equation. To gain efficiency, a Discrete Empirical Interpolation Method (DEIM) is applied to the compressed non-linear Fokker-Planck operator. The size of the resulting reduced model is chosen empirically according to the Knudsen number. Furthermore, we propose a correction to the reduced collision operator that ensures the reduced moments to satisfy an Euler-type system. Numerical simulations of the reduced model show that the model can capture the plasma dynamics in different collisional regimes and initial conditions at a low cost.

Domains: Mathematics [math]

Emmanuel Franck, Ibtissem Lannabi, Youssouf Nasseri, Laurent Navoret, Giuseppe Parasiliti Rantone, et al.. Hyperbolic reduced model for Vlasov-Poisson equation with Fokker-Planck collision. 2024. ⟨hal-04099697v2⟩