Entropy stable high order discontinuous Galerkin methods for ideal compressible MHD on structured meshes

Yong Liu Chi-Wang Shu Brown University Mengping Zhang

Numerical Analysis and Scientific Computing mathscidoc:1804.25024

Journal of Computational Physics, 354, 163-178, 2018
We present a discontinuous Galerkin (DG) scheme with suitable quadrature rules \cite{Tchen2017entropy} for ideal compressible magnetohydrodynamic (MHD) equations on structural meshes. The semi-discrete scheme is analyzed to be entropy stable by using the symmetrizable version of the equations as introduced by Godunov \cite{godunov1972symmetric}, the entropy stable DG framework with suitable quadrature rules \cite{Tchen2017entropy}, the entropy conservative flux in \cite{chandrashekar2016entropy} inside each cell and the entropy dissipative approximate Godunov type numerical flux at cell interfaces to make the scheme entropy stable. {\color{blue}{ The main difficulty in the generalization of the results in \cite{Tchen2017entropy} is the appearance of the non-conservative ``source terms'' added in the modified MHD model introduced by Godunov \cite{godunov1972symmetric}, which do not exist in the general hyperbolic system studied in \cite{Tchen2017entropy}. Special care must be taken to discretize these ``source terms'' adequately so that the resulting DG scheme satisfies entropy stability. }} Total variation diminishing / bounded (TVD/TVB) limiters and bound-preserving limiters are applied to control spurious oscillations. We demonstrate the accuracy and robustness of this new scheme on standard MHD examples.
compressible MHD; symmetrization; entropy stability; discontinuous Galerkin method; summation-by-parts
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@inproceedings{yong2018entropy,
  title={Entropy stable high order discontinuous Galerkin methods for ideal compressible MHD on structured meshes},
  author={Yong Liu, Chi-Wang Shu, and Mengping Zhang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180416105043581061059},
  booktitle={Journal of Computational Physics},
  volume={354},
  pages={163-178},
  year={2018},
}
Yong Liu, Chi-Wang Shu, and Mengping Zhang. Entropy stable high order discontinuous Galerkin methods for ideal compressible MHD on structured meshes. 2018. Vol. 354. In Journal of Computational Physics. pp.163-178. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180416105043581061059.
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