Fully conservative scheme for relativistic Landau–Fokker–Planck equation

In our previous work, a charge-momentum-energy-conserving algorithm was developed for the relativistic Vlasov–Maxwell system by the finite-difference method [T. Shiroto et al., arXiv 1802.07238 (2018)]. The scheme is based on the central difference scheme, so numerical dissipations are not included in the algorithm. This is why we extend the conservative algorithm to the relativistic Vlasov–Fokker–Planck–Maxwell system in order to introduce the effect of dissipation by the collision terms. In the non-relativistic regime, a conservative scheme has already been developed [W.T. Taitano et al., JCP 297, 357 (2015)], and our approach is similar to theirs but based on the potential equations of Braams and Karney [B.J. Braams and C.F.F. Karney, PRL 59, 1817 (1987)]. At the moment, a mass-momentum-energy-conserving scheme for the relativistic Landau–Fokker–Planck equation is composed with explicit time integration, and one-dimensional and three-velocity-components numerical experiments are performed. However, our conservative Vlasov–Maxwell scheme is based on an implicit time integration. Therefore, we will improve the conservative scheme with the combination of the implicit scheme and the Jacobian-Free Newton–Krylov method.