Implicit reduced Vlasov–Fokker–Planck–Maxwell model based on high-order mixed elements

Abstract

Detailed description of the transport processes in plasma is crucial for many disciplines. When the mean-free-path of the electrons is comparable or exceeds a characteristic length scale of the plasma profile, non-local behavior can be observed. Predictions of the diffusion theory are not valid and non-local electric and magnetic fields are generated. Kinetic modeling of these phenomena on time scales several orders of magnitude longer than the electron–electron collision time has proven to be cumbersome due to prohibitive requirements on the time step and violation of the conservation laws in the classical explicit Vlasov–Fokker–Planck methods. Therefore, a multi-dimensional conservative implicit Vlasov–Fokker–Planck–Maxwell method is proposed, where the distribution function is approximated by a truncated Cartesian tensor expansion. The electric and magnetic fields are modeled self-consistently, describing the generation process and emergence of non-locality in detail. Mixed finite elements are employed in space and the velocity dimension is discretized by staggered finite differences. Conservation properties are proved theoretically and the overall features are benchmarked on a series of physically representative problems. The second order convergence in velocity and the spatial order proportional to the polynomial order of the finite elements is shown. Further possible extensions of the method are discussed.