Efficient high order positivity preserving DG-FEM methods for multi-dimensional Vlasov simulations of plasma

Abstract

Summary form only given. Discontinuous Galerkin finite element methods (DG-FEM) 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 work, we extend our single dimension, 1D-1V semi-Lagrangian discontinuous Galerkin (SLDG) method1 to 2D-2V, two dimensions for configuration space, and two dimensions for velocity space. The DG representation allows us to capture complicated geometries in configuration space through the use of unstructured grids. Our method uses operator splitting techniques that enable us to apply different time stepping options in each direction. For velocity space, we use the semi-Lagrangian DG method on a structured grid that removes CFL limitations on the electric field. For configuration space, we apply explicit Runge-Kutta time stepping on unstructured grids. In order to mitigate restrictive CFL conditions, each sub-problem is sub-cycled according to a local velocity. Due to the fact that the proposed scheme is mesh-based and high-order accurate, we can compute solutions with much less statistical noise than what is found in traditional particle-in-cell (PIC) solutions of comparable resolution. We present simulation results for the formation of a plasma sheath in a collisionless plasma. We start with a 1D problem, and then demonstrate how the multi-D extension performs on a radially symmetric cylinder. We argue that our high-order mesh based method allows us to simultaneously produce accurate results for the plasma sheath near the wall as well as the quasi-neutral region.