A structure and asymptotic preserving scheme for the Vlasov-Poisson-Fokker-Planck model

TL;DR

This paper introduces a spectral Hermite-based numerical scheme for the Vlasov-Poisson-Fokker-Planck model that preserves key solutions and guarantees exponential relaxation to equilibrium across regimes.

Contribution

It develops an asymptotic preserving, structure-preserving scheme with proven stability and relaxation properties, applicable to various collisional plasma regimes.

Findings

Scheme naturally preserves stationary solutions.

Quantitative estimates show exponential relaxation to equilibrium.

Numerical simulations demonstrate robustness across regimes.

Abstract

We propose a numerical method for the Vlasov-Poisson-Fokker-Planck model written as an hyperbolic system thanks to a spectral decomposition in the basis of Hermite functions with respect to the velocity variable and a structure preserving finite volume scheme for the space variable. On the one hand, we show that this scheme naturally preserves both stationary solutions and linearized free-energy estimate. On the other hand, we adapt previous arguments based on hypocoercivity methods to get quantitative estimates ensuring the exponential relaxation to equilibrium of the discrete solution for the linearized Vlasov-Poisson-Fokker-Planck system, uniformly with respect to both scaling and discretization parameters. Finally, we perform substantial numerical simulations for the nonlinear system to illustrate the efficiency of this approach for a large variety of collisional regimes (plasma echos for weakly collisional regimes and trend to equilibrium for collisional plasmas) and to highlight its robustness (unconditional stability, asymptotic preserving properties).