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

¹ Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, 67000 Strasbourg, France & Centre INRIA université de Lorraine Antenne de Strasbourg, MACARON Project, Strasbourg, France
² CNRS/Univ Pau & Pays Adour/E2S UPPA, Laboratoire de Mathématiques et de leurs Applications de Pau - Fédération IPRA, UMR5142 64000, Pau, France & Cagire team, Inria Bordeaux Sud-Ouest, France
³ Sorbonne Université, CNRS UMR 7190, Institut Jean Le Rond d’Alembert, 75005 Paris, France

Abstract

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.