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Numerische Mathematik

, Volume 108, Issue 3, pp 407–444 | Cite as

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

  • Martin Campos Pinto
  • Michel Mehrenberger
Article

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.

Mathematical Subject Classification (2000)

65M12 65M50 82D10 

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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis Lions, UMR CNRS 7598Université Pierre et Marie CurieParisFrance
  2. 2.Institut de Recherche Mathématique Avancée, UMR CNRS 7501Université Louis PasteurStrasbourgFrance

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