In this work, we aim at constructing numerical schemes, that are as efficient as possible in terms of cost and conservation of invariants, for the Vlasov--Fokker--Planck system coupled with Poisson or Amp\`ere equation. Splitting methods are used where the linear terms in space are treated by spectral or semi-Lagrangian methods and the nonlinear diffusion in velocity in the collision operator is treated using a stabilized Runge--Kutta--Chebyshev (RKC) integrator, a powerful alternative of implicit schemes. The new schemes are shown to exactly preserve mass and momentum. The conservation of total energy is obtained using a suitable approximation of the electric field. An H-theorem is proved in the semi-discrete case, while the entropy decay is illustrated numerically for the fully discretized problem. Numerical experiments that include investigation of Landau damping phenomenon and bump-on-tail instability are performed to illustrate the efficiency of the new schemes.

翻译：本文旨在构建用于Vlasov-Fokker-Planck系统（耦合泊松方程或安培方程）的数值格式，这些格式在计算成本与不变量守恒方面尽可能高效。采用分裂方法处理问题，其中空间线性项通过谱方法或半拉格朗日方法求解，而碰撞算子中速度方向的非线性扩散则使用稳定化的Runge-Kutta-Chebyshev（RKC）积分器处理——该积分器是隐式格式的有力替代方案。新格式被证明能精确保持质量和动量。通过电场适当近似即可实现总能量守恒。在半离散情形下证明了H定理，同时在完全离散问题中通过数值方法展示了熵衰减特性。通过包括朗道阻尼现象和束尾不稳定性在内的数值实验，验证了新格式的高效性。