  We study the conservation properties of the Hermite-discontinuous Galerkin
(Hermite-DG) approximation of the Vlasov-Maxwell equations. In this
semi-discrete formulation, the total mass is preserved independently for every
plasma species. Further, an energy invariant exists if central numerical fluxes
are used in the DG approximation of Maxwell's equations, while a dissipative
term is present when upwind fluxes are employed. In general, traditional
temporal integrators might fail to preserve invariants associated with
conservation laws during the time evolution. Hence, we analyze the capability
of explicit and implicit Runge-Kutta (RK) temporal integrators to preserve such
invariants. Since explicit RK methods can only ensure preservation of linear
invariants but do not provide any control on the system energy, we consider
modified explicit RK methods in the family of relaxation Runge-Kutta methods
(RRK). These methods can be tuned to preserve the energy invariant at the
continuous or semi-discrete level, a distinction that is important when upwind
fluxes are used in the discretization of Maxwell's equations since upwind
provides a numerical source of energy dissipation that is not present when
central fluxes are used. We prove that the proposed methods are able to
preserve the energy invariant and to maintain the semi-discrete energy
dissipation (if present) according to the discretization of Maxwell's
equations. An extensive set of numerical experiments corroborates the
theoretical findings. It also suggests that maintaining the semi-discrete
energy dissipation when upwind fluxes are used leads to an overall better
accuracy of the method relative to using upwind fluxes while forcing exact
energy conservation.
