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 ) f(t,x,v) and the electric field E ( t , x ) E(t,x) converge in the L 2 L^2 norm with a rate of \[ O ( Δ t 2 + h m + 1 + h m + 1 Δ t ) , O (Δ t^2 +hᵐ⁺¹+ {hᵐ⁺¹}{Δ t} ), \] where m m is the degree of the polynomial reconstruction, and Δ t Δ t and h h are respectively the time and the phase-space discretization parameters.