Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system

Blanca Ayuso; José A. Carrillo; Chi-Wang Shu

doi:10.3934/krm.2011.4.955

Article Contents

2011, Volume 4, Issue 4: 955-989. Doi: 10.3934/krm.2011.4.955

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Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system

1.
Centre de Recerca Matemática, Campus de Bellaterra, Ediﬁci C, E-08193 Bellaterra, Barcelona
2.
ICREA-Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra
3.
Division of Applied Mathematics, Brown University, Providence, RI 02912

Received: December 2010

Revised: July 2011

Published: December 2011

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Abstract

We construct a new family of semi-discrete numerical schemes for the approximation of the one-dimensional periodic Vlasov-Poisson system. The methods are based on the coupling of discontinuous Galerkin approximation to the Vlasov equation and several finite element (conforming, non-conforming and mixed) approximations for the Poisson problem. We show optimal error estimates for all the proposed methods in the case of smooth compactly supported initial data. The issue of energy conservation is also analyzed for some of the methods.

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Mathematics Subject Classification: Primary: 65N30, 65M12, 65M15; Secondary: 82D10.

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