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An h-Adaptive RKDG Method for the Two-Dimensional Incompressible Euler Equations and the Guiding Center Vlasov Model

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Abstract

In this paper, we generalize an h-adaptive Runge–Kutta discontinuous Galerkin scheme developed earlier in Zhu et al. (J Sci Comput 69:1346–1365, 2016) for the 1D Vlasov–Poisson system to the guiding center Vlasov model and the 2D time dependent incompressible Euler equations in the vorticity-stream function formulation. The main difficulty of this generalization lies in solving the 2D Poisson equation due to the irregular adaptive mesh with hanging nodes. We adopt a local discontinuous Galerkin method to solve the Poisson equation. The full adaptive algorithm and the related numerical implementation details are included. Extensive numerical tests have been performed to showcase the effectiveness of the adaptive scheme and its advantage over the fixed-mesh scheme in saving computational cost and improving solution quality.

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Authors and Affiliations

  1. School of Natural Sciences, Nanjing University of Posts and Telecommunications, Nanjing, 210023, Jiangsu, People’s Republic of China

    Hongqiang Zhu

  2. School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing, Xiamen University, Xiamen, 361005, Fujian, People’s Republic of China

    Jianxian Qiu

  3. Department of Mathematics, University of Houston, Houston, TX, 77204-3008, USA

    Jing-Mei Qiu

Authors
  1. Hongqiang Zhu
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  2. Jianxian Qiu
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  3. Jing-Mei Qiu
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Correspondence to Jing-Mei Qiu.

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Dedicated to Prof. Chi-Wang Shu on the occasion of his 60th birthday.

The research is partially supported by NSFC Grants 11201242 and 11571290, NSAF Grant U1630247, NSF Grant NSF-DMS-1522777, and Air Force Office of Scientific Computing FA9550-16-1-0179.

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Zhu, H., Qiu, J. & Qiu, JM. An h-Adaptive RKDG Method for the Two-Dimensional Incompressible Euler Equations and the Guiding Center Vlasov Model. J Sci Comput 73, 1316–1337 (2017). https://doi.org/10.1007/s10915-017-0440-9

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  • DOI: https://doi.org/10.1007/s10915-017-0440-9

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