Energy conserving discontinuous Galerkin spectral element method for the Vlasov-Poisson system

We propose a new, energy conserving, spectral element, discontinuous Galerkin method for the approximation of the Vlasov-Poisson system in arbitrary dimension, using Cartesian grids. The method is derived from the one proposed in [4], with two modifications: energy conservation is obtained by a suitable projection operator acting on the solution of the Poisson problem, rather than by solving multiple Poisson problems, and all the integrals appearing in the finite element formulation are approximated with Gauss-Lobatto quadrature, thereby yielding a spectral element formulation. The resulting method has the following properties: exact energy conservation (up to errors introduced by the time discretization), stability (thanks to the use of upwind numerical fluxes), high order accuracy and high locality. For the time discretization, we consider both Runge-Kutta methods and exponential integrators, and show results for 1D and 2D cases (2D and 4D in phase space, respectively).