A time-implicit numerical method and benchmarks for the relativistic Vlasov–Ampere equations

We present a time-implicit numerical method to solve the relativistic Vlasov–Ampere system of equations on a two dimensional phase space grid. The time-splitting algorithm we use allows the generalization of the work presented here to higher dimensions keeping the linear aspect of the resulting discrete set of equations. The implicit method is benchmarked against linear theory results for the relativistic Landau damping for which analytical expressions using the Maxwell-Juttner distribution function are derived. We note that, independently from the shape of the distribution function, the relativistic treatment features collective behaviours that do not exist in the nonrelativistic case. The numerical study of the relativistic two-stream instability completes the set of benchmarking tests