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:在这项工作中,我们的目标是为Vlasov-Fokker-Planck-Planck系统以及Poisson或Amp ⁇ ere等式制定尽可能高效的数值计划,在用光谱或半Lagrangian方法处理空间线性术语以及用稳定的Runge-Kutta-Chebyshev(RKC)集成器处理碰撞操作员速度的非线性扩散方面,这是隐含计划的一种强有力的替代方案。新的计划显示完全保持质量和势头。总能源的保护是利用电场的适当近似获得的。在半分解案中证明了一种H-理论,而对于完全分散的问题则用数字来说明酶衰变。进行了包括调查Landau阻力现象和连尾不稳现象在内的数字实验,以说明新计划的效率。