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).

翻译：我们针对Vlasov-Poisson-Fokker-Planck模型提出一种数值方法，该方法通过速度变量上Hermite函数基的谱分解将模型转化为双曲型系统，并结合空间变量上的结构保持有限体积格式。一方面，我们证明该格式自然地保持稳态解和线性化自由能估计。另一方面，我们采用基于弱耗散正则性方法的已有论证思路，得到定量估计以确保线性化Vlasov-Poisson-Fokker-Planck系统离散解的指数弛豫过程在尺度与离散参数上保持一致。最后，我们对非线性系统开展大量数值模拟，以展示该方法在多种碰撞机制（弱碰撞机制下的等离子体回声与碰撞等离子体趋向平衡的过程）下的有效性，并突出其鲁棒性（无条件稳定性与渐进保持特性）。