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 Δt tend to zero provided the initial data is Lipschitz and has a finite total curvature, or in other words, that it belongs to $${W^{1,\infty} \cap W^{2,1}}$$ . The rate of convergence is $${\mathcal{O}({\Delta}t^2 + \varepsilon/{\Delta}t)}$$ , which should be compared to the results of Besse who recently established in (SIAM J Numer Anal 42(1):350–382, 2004) similar rates for a uniform semi-Lagrangian scheme, but requiring that the initial data are in $${{\mathcal C}^2}$$ . Several numerical tests illustrate the effectiveness of our approach for generating the optimal adaptive discretizations.