Runge-Kutta discontinuous Galerkin method with a simple and compact Hermite WENO limiter on unstructured meshes

Jun Zhu Nanjing University of Aeronautics and Astronautics Xinghui Zhong University of Utah Chi-Wang Shu Brown University Jianxian Qiu Xiamen University

Numerical Analysis and Scientific Computing mathscidoc:1610.25003

Communications in Computational Physics, 21, 623-649, 2017
In this paper we generalize a new type of compact Hermite weighted essentially non-oscillatory (HWENO) limiter for the Runge-Kutta discontinuous Galerkin (RKDG) method, which was recently developed for structured meshes, to two dimensional unstructured meshes. The main idea of this HWENO limiter is to reconstruct the new polynomial by the usage of the entire polynomials of the DG solution from the target cell and its neighboring cells in a least squares fashion while maintaining the conservative property, then use the classical WENO methodology to form a convex combination of these reconstructed polynomials based on the smoothness indicators and associated nonlinear weights. The main advantage of this new HWENO limiter is the robustness for very strong shocks and simplicity in implementation especially for the unstructured meshes considered in this paper, since only information from the target cell and its immediate neighbors is needed. Numerical results for both scalar and system equations are provided to test and verify the good performance of this new limiter.
Runge-Kutta discontinuous Galerkin method, HWENO limiter, unstructured mesh
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@inproceedings{jun2017runge-kutta,
  title={Runge-Kutta discontinuous Galerkin method with a simple and compact Hermite WENO limiter on unstructured meshes},
  author={Jun Zhu, Xinghui Zhong, Chi-Wang Shu, and Jianxian Qiu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161011084120817467117},
  booktitle={Communications in Computational Physics},
  volume={21},
  pages={623-649},
  year={2017},
}
Jun Zhu, Xinghui Zhong, Chi-Wang Shu, and Jianxian Qiu. Runge-Kutta discontinuous Galerkin method with a simple and compact Hermite WENO limiter on unstructured meshes. 2017. Vol. 21. In Communications in Computational Physics. pp.623-649. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161011084120817467117.
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