A structure and asymptotic preserving scheme for the Vlasov-Poisson-Fokker-Planck model
This work provides an incremental improvement in numerical methods for plasma physics simulations, addressing specific computational challenges in collisional plasma modeling.
The authors developed a numerical scheme for the Vlasov-Poisson-Fokker-Planck model that preserves stationary solutions and free-energy estimates, and demonstrated exponential relaxation to equilibrium uniformly across scaling and discretization parameters, with simulations showing efficiency across collisional regimes and robustness including unconditional stability.
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).