| The Vlasov-Poisson equations describe the evolution of a collisionless plasma, represented through a probability density function (PDF) that self-interacts via an electrostatic force. One of the main difficulties in numerically solving this system is the severe time-step restriction that arises from parts of the PDF associated with moderate-to-large velocities. The dominant approach in the plasma physics community for removing these time-step restrictions is the so-called particle-in-cell (PIC) method, which discretizes the distribution function into a set of macro-particles, while the electric field is represented on a mesh. Several alternatives to this approach exist, including fully Lagrangian, fully Eulerian, and so-called semi-Lagrangian methods. The focus of this work is the semi-Lagrangian approach, which begins with a grid-based Eulerian representation of both the PDF and the electric field, then evolves the PDF via Lagrangian dynamics, and finally projects this evolved field back onto the original Eulerian mesh.;We present a semi-Lagrangian and a hybrid semi-Lagrangian method for solving the Vlasov Poisson equations, based on high-order discontinuous Galerkin (DG) spatial representations of the solution. The Poisson equation is solved via a high-order local discontinuous Galerkin (LDG) scheme. The resulting methods are high-order accurate, which is demonstrably important for this problem in order to retain the rich phase-space structure of the solution; mass conservative; and provably positivity-preserving. We argue that our approach is a promising method that can produce very accurate results at relatively low computational expense. We demonstrate this through several examples for the (1+1)D case, using both the hybrid as well as the full semi-Lagrangian method. In particular, the methods are validated on several numerical test cases, including the two-stream instability problem, Landau damping, and the formation of a plasma sheath. In addition, we propose a (2+2)D method that promises to be a productive avenue of future research. The (2+2)D method incorporates local time-stepping methods on unstructured grids in physical space and semi-Lagrangian time stepping on Cartesian grids in velocity space. This method is again high-order, mass conservative, and provably positivity-preserving. |
