  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 by advecting the `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 not subject to time step restriction to maintain stability.
Recently [Y. G\"u\c{c}l\"u and W.N.G. Hitchon, 2012] 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 simple and
efficient for constant advection problems. Here the CS is applied to the
solution of the Vlasov-Poisson system: the Vlasov equation is split into two
constant advection equations, one in configuration space and one in velocity
space, and high order time accuracy is achieved by proper composition of the
operators. The splitting procedure enables us to use the constant advection
solver, which we extend to arbitrarily high order of accuracy: a new improved
procedure is given, which makes the calculation of the corrections
straightforward. Focusing on periodic domains, we describe a spectrally
accurate scheme based on the fast Fourier transform; the proposed
implementation is strictly conservative and positivity preserving. The ability
to correctly reproduce the system dynamics, as well as resolving small-scale
features in the solution, is shown in classical 1D-1V test cases, both in the
linear and the non-linear regimes.
