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Convergence of an adaptive semi-Lagrangian scheme for the Vlasov-Poisson system (2007)
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BibTeX
@MISC{Pinto07convergenceof,
author = {Martin Campos Pinto and Michel Mehrenberger},
title = {Convergence of an adaptive semi-Lagrangian scheme for the Vlasov-Poisson system},
year = {2007}
}
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Abstract
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 2,1. The rate of convergence is O(∆t 2 + 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
Keyphrases
adaptive semi-lagrangian scheme vlasov-poisson system time step initial data exact one 1-dimensional vlasov-poisson system cauchy problem numerical flow rigorous analysis prescribed tolerance parameter local interpolation error uniform semi-lagrangian scheme next one numerical solution finite total curvature accurate evolution adaptive mesh local regularity key feature two-dimensional phase space similar rate
