High-order bound-preserving discontinuous Galerkin methods for compressible miscible displacements in porous media on triangular meshes

Nattaporn Chuenjarern Michigan Technological University Ziyao Xu Michigan Technological University Yang Yang Michigan Technological University

Numerical Analysis and Scientific Computing mathscidoc:1907.25001

Journal of Computational Physics, 378, (1), 110-128, 2019.2
In this paper, we develop high-order bound-preserving (BP) discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacements on triangular meshes. We consider the problem with multi-component fluid mixture and the (volumetric) concentration of the $j$th component, $c_j$, should be between 0 and 1. There are three main difficulties. Firstly, $c_j$ does not satisfy a maximum-principle. Therefore, the numerical techniques introduced in (X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), 3091-3120) cannot be applied directly. The main idea is to apply the positivity-preserving techniques to all $c_j's$ and enforce $\sum_jc_j=1$ simultaneously to obtain physically relevant approximations. By doing so, we have to treat the time derivative of the pressure $dp/dt$ as a source in the concentration equation and choose suitable fluxes in the pressure and concentration equations. Secondly, it is not easy to construct first-order numerical fluxes for interior penalty DG methods on triangular meshes. One of the key points in the high-order BP technique applied in this paper is the combination of high-order and lower-order numerical fluxes. We will construct second-order BP schemes and use the second-order numerical fluxes as the lower-order one. Finally, the classical slope limiter cannot be applied to $c_j$. To construct the BP technique, we will not approximate $c_j$ directly. Therefore, a new limiter will be introduced. Numerical experiments will be given to demonstrate the high-order accuracy and good performance of the numerical technique.
compressible miscible displacements, bound-preserving, high-order, discontinuous Galerkin method, triangular meshes, multi-component uid, ux limiter
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@inproceedings{nattaporn2019high-order,
  title={High-order bound-preserving discontinuous Galerkin methods for compressible miscible displacements in porous media on triangular meshes},
  author={Nattaporn Chuenjarern, Ziyao Xu, and Yang Yang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190701051005214174381},
  booktitle={Journal of Computational Physics},
  volume={378},
  number={1},
  pages={110-128},
  year={2019},
}
Nattaporn Chuenjarern, Ziyao Xu, and Yang Yang. High-order bound-preserving discontinuous Galerkin methods for compressible miscible displacements in porous media on triangular meshes. 2019. Vol. 378. In Journal of Computational Physics. pp.110-128. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190701051005214174381.
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