Convergence of classes of high-order semi-Lagrangian schemes for the Vlasov-Poisson system

In this paper we present some classes of high-order semi-Lagran- gian schemes for solving the periodic one-dimensional Vlasov-Poisson system in phase-space on uniform grids. We prove that the distribution function $f(t,x,v)$ and the electric field $E(t,x)$ converge in the $L^2$ norm with a rate of

$\mathcal{O}\left(\Delta t^2 +h^{m+1}+ \frac{h^{m+1}}{\Delta t}\right),$

where $m$ is the degree of the polynomial reconstruction, and $\Delta t$ and $h$ are respectively the time and the phase-space discretization parameters.