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

Abstract

International audienceAn 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 infinity metric. The numerical solutions are proved to converge in L infinity 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

Convergence of a semi-Lagrangian scheme for the reduced Vlasov–Maxwell system for laser–plasma interaction

“…More precisely, we focus our attention to the convergence analysis of a semi-Lagrangian scheme for approximating these models. Similar convergence studies have been achieved by several authors for the Vlasov-Poisson equations, we cite here [11,1,2,7]. The present work intends to adapt these results within the framework of the reduced Vlasov-Maxwell models.…”