Convergence of an adaptive semi-Lagrangian scheme for the Vlasov-Poisson system |
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Authors: | Martin Campos Pinto Michel Mehrenberger |
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Institution: | (1) Laboratoire Jacques-Louis Lions, UMR CNRS 7598, Université Pierre et Marie Curie, Paris, France;(2) Institut de Recherche Mathématique Avancée, UMR CNRS 7501, Université Louis Pasteur, Strasbourg, France |
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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 Δt tend to zero provided the initial data is Lipschitz and has a finite total curvature, or in other words, that it belongs
to . The rate of convergence is , 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 . Several numerical tests illustrate the effectiveness of our approach for generating the optimal adaptive discretizations. |
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Keywords: | 65M12 65M50 82D10 |
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