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

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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 \({W^{1,\infty} \cap W^{2,1}}\) . 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.

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Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions, UMR CNRS 7598, Université Pierre et Marie Curie, Paris, France

    Martin Campos Pinto

  2. Institut de Recherche Mathématique Avancée, UMR CNRS 7501, Université Louis Pasteur, Strasbourg, France

    Martin Campos Pinto & Michel Mehrenberger

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  1. Martin Campos Pinto
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  2. Michel Mehrenberger
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Correspondence to Martin Campos Pinto.

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Campos Pinto, M., Mehrenberger, M. Convergence of an adaptive semi-Lagrangian scheme for the Vlasov-Poisson system. Numer. Math. 108, 407–444 (2008). https://doi.org/10.1007/s00211-007-0120-z

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  • DOI: https://doi.org/10.1007/s00211-007-0120-z

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