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Study of conservation and recurrence of Runge–Kutta discontinuous Galerkin schemes for Vlasov–Poisson systems

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In this paper we consider Runge–Kutta discontinuous Galerkin (RKDG) schemes for Vlasov–Poisson systems that model collisionless plasmas. One-dimensional systems are emphasized. The RKDG method, originally devised to solve conservation laws, is seen to have excellent conservation properties, be readily designed for arbitrary order of accuracy, and capable of being used with a positivity-preserving limiter that guarantees positivity of the distribution functions. The RKDG solver for the Vlasov equation is the main focus, while the electric field is obtained through the classical representation by Green’s function for the Poisson equation. A rigorous study of recurrence of the DG methods is presented by Fourier analysis, and the impact of different polynomial spaces and the positivity-preserving limiters on the quality of the solutions is ascertained. Several benchmark test problems, such as Landau damping, the two-stream instability, and the Kinetic Electro static Electron Nonlinear wave, are given.

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Acknowledgments

YC was supported by grant NSF DMS-1016001, IMG was supported by grants NSF DMS-0807712 and DMS-0757450, and PJM was supported by U.S. Dept. of Energy Contract # DE-FG05-80ET-53088. Also, support from the Department of Mathematics at Michigan State University and the Institute of Computational Engineering and Sciences at the University of Texas Austin are gratefully acknowledged.

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Authors and Affiliations

  1. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Yingda Cheng

  2. Department of Mathematics and ICES, University of Texas at Austin, Austin, TX, 78712, USA

    Irene M. Gamba

  3. Department of Physics and Institute for Fusion Studies, University of Texas at Austin, Austin, TX, 78712, USA

    Philip J. Morrison

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  1. Yingda Cheng
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Correspondence to Yingda Cheng.

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Cheng, Y., Gamba, I.M. & Morrison, P.J. Study of conservation and recurrence of Runge–Kutta discontinuous Galerkin schemes for Vlasov–Poisson systems. J Sci Comput 56, 319–349 (2013). https://doi.org/10.1007/s10915-012-9680-x

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