Arbitrarily high order Convected Scheme solution of the Vlasov–Poisson system

The Convected Scheme (CS) is a ‘forward-trajectory’ semi-Lagrangian method for solution of transport equations, which has been most often applied to the kinetic description of plasmas and rarefied neutral gases. In its simplest form, the CS propagates the solution forward in time by advecting the so-called ‘moving cells’ along their characteristic trajectories, and by remapping them on the mesh at the end of the time step. The CS is conservative, positivity preserving, simple to implement, and it is not subject to time step restriction to maintain stability. Recently (Güçlü and Hitchon, 2012 [1]) a new methodology was introduced for reducing numerical diffusion, based on a modified equation analysis: the remapping error was compensated by applying small corrections to the final position of the moving cells prior to remapping. While the spatial accuracy was increased from 2nd to 4th order, the new scheme retained the important properties of the original method, and was shown to be extremely simple and efficient for constant advection problems.