Convergence of an Adaptive Scheme for the one dimensional Vlasov-Poisson system
Abstract
An adaptive semi-Lagrangian scheme for solving the Cauchy problem associated to the periodic one-dimensional Vlasov-Poisson system 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 the 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 which represents the local interpolation error at each time step.The numerical solutions are proved to converge in sup-norm towards the exact ones as the toloerance parameter and the time step tend to zero provided the initial data is Lipschitz and has a finite total curvature. t)$, Several numerical tests illustrate the effectiveness of our approach for generating the optimal adaptive discretizations.