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A High Order Semi-Lagrangian Discontinuous Galerkin Method for the Two-Dimensional Incompressible Euler Equations and the Guiding Center Vlasov Model Without Operator Splitting

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Abstract

In this paper, we generalize a high order semi-Lagrangian (SL) discontinuous Galerkin (DG) method for multi-dimensional linear transport equations without operator splitting developed in Cai et al. (J Sci Comput 73(2–3):514–542, 2017) to the 2D time dependent incompressible Euler equations in the vorticity-stream function formulation and the guiding center Vlasov model. We adopt a local DG method for Poisson’s equation of these models. For tracing the characteristics, we adopt a high order characteristics tracing mechanism based on a prediction-correction technique. The SLDG with large time-stepping size might be subject to extreme distortion of upstream cells. To avoid this problem, we propose a novel adaptive time-stepping strategy by controlling the relative deviation of areas of upstream cells.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

    Xiaofeng Cai & Jing-Mei Qiu

  2. Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, 70409, USA

    Wei Guo

Authors
  1. Xiaofeng Cai
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  2. Wei Guo
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  3. Jing-Mei Qiu
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Correspondence to Jing-Mei Qiu.

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W. Guo: Research is supported by NSF Grant NSF-DMS-1620047. J.-M. Qiu: Research of first and last author is supported by NSF Grant NSF-DMS-1522777 and 1818924, Air Force Office of Scientific Computing FA9550-18-1-0257 and University of Delaware.

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Cai, X., Guo, W. & Qiu, JM. A High Order Semi-Lagrangian Discontinuous Galerkin Method for the Two-Dimensional Incompressible Euler Equations and the Guiding Center Vlasov Model Without Operator Splitting. J Sci Comput 79, 1111–1134 (2019). https://doi.org/10.1007/s10915-018-0889-1

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  • DOI: https://doi.org/10.1007/s10915-018-0889-1

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