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The semi-Lagrangian discontinuous Galerkin method, coupled with a splittingapproach in time, has recently been introduced for the Vlasov--Poissonequation. Since these methods are conservative, local in space, and able tolimit numerical diffusion, they are considered a promising alternative to moretraditional semi-Lagrangian schemes. In this paper we study the conservation ofimportant physical invariants and the long time behavior of the semi-Lagrangiandiscontinuous Galerkin method. To that end we conduct a theoretical analysisand perform a number of numerical simulations. In particular, we find that theentropy is nondecreasing for the discontinuous Galerkin scheme, whileunphysical oscillations in the entropy are observed for the traditional methodbased on cubic spline interpolation.

A study on conserving invariants of the Vlasov equation in semi-Lagrangian computer simulations