Convergence of an adaptive semi-Lagrangian scheme for the Vlasov-Poisson system

An adaptive semi-Lagrangian scheme for solving the Cauchy problem associated to the periodic 1+1-dimensional Vlasov-Poisson system in the two-dimensional phase space is proposed and analyzed. A key feature of our method is the accurate evolution of the adaptive mesh from one time step to the next one, based on a rigorous analysis of the local regularity and how it gets transported by the numerical flow. The accuracy of the scheme is monitored by a prescribed tolerance parameter ε which represents the local interpolation error at each time step, in the L∞ metric. The numerical solutions are proved to converge in L∞ towards the exact ones as ε and ∆t tend to zero provided the initial data is Lipschitz and has a finite total curvature, or in other words, that it belongs to W 1,∞∩W . The rate of convergence is O(∆t + ε/∆t), which should be compared to the results of Besse, who recently established [6] similar rates for a uniform semi-Lagrangian scheme, but requiring that the initial data are in C. Several numerical tests illustrate the effectiveness of our approach for generating the optimal adaptive discretizations.