Geodesic
Hyperbolic reduced model for Vlasov-Poisson equation with Fokker-Planck collision
Emmanuel Franck1 ; Ibtissem Labanni2 ; Youssouf Nasseri1 ; Laurent Navoret1 ; Giuseppe Parasiliti Rantone3 ; Guillaume Steimer1
1 Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, 67000 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 &
3 Sorbonne Université, CNRS UMR 7190, Institut Jean Le Rond d’Alembert, 75005 Paris, France
ESAIM. Proceedings, Tome 77 (2024), pp. 213-228.

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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.
DOI : 10.1051/proc/202477213

Emmanuel Franck 1 ; Ibtissem Labanni 2 ; Youssouf Nasseri 1 ; Laurent Navoret 1 ; Giuseppe Parasiliti Rantone 3 ; Guillaume Steimer 1

1 Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, 67000 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 &
3 Sorbonne Université, CNRS UMR 7190, Institut Jean Le Rond d’Alembert, 75005 Paris, France 
@article{EP_2024_77_a10,
     author = {Emmanuel Franck and Ibtissem Labanni and Youssouf Nasseri and Laurent Navoret and Giuseppe Parasiliti Rantone and Guillaume Steimer},
     title = {Hyperbolic reduced model for {Vlasov-Poisson} equation with {Fokker-Planck} collision},
     journal = {ESAIM. Proceedings},
     pages = {213--228},
     publisher = {mathdoc},
     volume = {77},
     year = {2024},
     doi = {10.1051/proc/202477213},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/articles/10.1051/proc/202477213/}
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Emmanuel Franck; Ibtissem Labanni; Youssouf Nasseri; Laurent Navoret; Giuseppe Parasiliti Rantone; Guillaume Steimer. Hyperbolic reduced model for Vlasov-Poisson equation with Fokker-Planck collision. ESAIM. Proceedings, Tome 77 (2024), pp. 213-228. doi : 10.1051/proc/202477213. https://geodesic-test.mathdoc.fr/articles/10.1051/proc/202477213/

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