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The purpose of the present paper is to compare two semi-Lagrangian methods inthe context of the four-dimensional Vlasov--Poisson equation. Morespecifically, our goal is to compare the performance of the more recentlydeveloped semi-Lagrangian discontinuous Galerkin scheme with the de factostandard in Eulerian Vlasov simulation (i.e. using cubic spline interpolation).To that end, we perform simulations for nonlinear Landau damping and atwo-stream instability and provide benchmarks for the SeLaLib and sldg codes(both on a workstation and using MPI on a cluster).  We find that the semi-Lagrangian discontinuous Galerkin scheme shows amoderate improvement in run time for nonlinear Landau damping and a substantialimprovement for the two-stream instability. It should be emphasized that theseresults are markedly different from results obtained in the asymptotic regime(which favor spline interpolation). Thus, we conclude that the traditionalapproach of evaluating numerical methods is misleading, even for short timesimulations. In addition, the absence of any All-to-All communication in thesemi-Lagrangian discontinuous Galerkin method gives it a decisive advantage forscaling to more than 256 cores.

A comparison of semi-Lagrangian discontinuous Galerkin and spline based Vlasov solvers in four dimensions