High-order discontinuous galerkin methods for the Vlasov-Poisson system

Discontinuous Galerkin (DG) finite element methods are a powerful solution technique for nonlinear hyperbolic conservation laws, such as those that arise in the modeling of plasma. DG-FEM can be applied to achieve high-order spatial accuracy; however, one drawback of classical DG-FEM with explicit time-stepping is their poor CFL restriction compared to high-order finite difference or finite volume counterparts. In kinetic models of collisionless plasma, i.e. kinetic Vlasov models, this small time step problem is further exacerbated due to the possibility that some particles in the system may travel at moderate to large velocities. In this talk I will describe two high-order DG methods for the Vlasov-Poisson system for circumventing the small time-step restriction of explicit DG schemes. The first is an operator split semi-Lagrangian DG method, and the second is an implicit spacetime DG method. Both of these methods are unconditionally stable. In both methods we make use of a Taylor series in time approximation of the electric field in order to locally linearize the evolution operators. In the semi- Lagrangian case this allows us to develop a high-order time discretization based on series of operator split steps that are each one-dimensional constant-coefficient advection steps. In the space-time DG case the Taylor series in time approximation of the electric field allows to solve in each time slab a linear algebra problem rather than a nonlinear one. For these methods we also consider positivity-preserving limiters on the discrete distribution function that guarantee that the algorithms produce physically meaningful approximations. The proposed methods are applied to several test cases.