High order semi-lagrangian methods for the kinetic description of plasmas

The Convected Scheme (CS) is a family of `forward-trajectory´ semi-Lagrangian schemes for the numerical solution of transport equations (most often the Boltzmann equation), which uses a method of characteristics in an integral form to project an initial `moving cell´ (MC) forward to a group of final cells.The main drawback of the CS to date has been its high numerical diffusion in physical space, because of the 2nd order remapping that takes place at the end of each time step. A `long-lived moving cell´ version of the CS was able to suppress such a numerical diffusion, but at the expense of reduced numerical efficiency and increased memory requirement. Recently a high order cell-centered version of the CS was proposed, suitable for the kinetic simulation of neutral gas flows, which is 4th order accurate in space, while retaining the desirable properties of the CS. This was achieved by compensating the remapping error via a `small correction´ to the final position of the MC prior to remapping. Here a similar procedure is used to obtain a high-order face centered version of the CS, which is suitable for the solution of the Boltzmann-Poisson system for a low temperature plasma. Such a scheme is 3rd-order accurate in space, conservative, positivity preserving, very simple to implement and fast to run. The convergence properties of the new high-order scheme, as well as its numerical stability, are analyzed in classical electrostatic test-cases. The impact of the reduced numerical diffusion in space is discussed in detail. When applied to kinetic simulations of plasmas, the CS is also affected by numerical diffusion in velocity. Recent advances are presented and preliminary results are shown, for a scheme which is high-order accurate in both space and velocity.