Bound-Preserving Discontinuous Galerkin Method for Compressible Miscible Displacement in Porous Media

Hui Guo China University of Petroleum Yang Yang Michigan Technological University

Numerical Analysis and Scientific Computing mathscidoc:1907.25003

SIAM Journal on Scientific Computing, 39, (5), A1969–A1990, 2017
In this paper, we develop bound-preserving discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacement problems. We consider the problem with two components and the (volumetric) concentration of the $i$th component of the fluid mixture, $c_i$, should be between $0$ and $1$. However, $c_i$ does not satisfy the 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 both $c_1$ and $c_2$, respectively and enforce $c_1+c_2=1$ 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. Moreover, $c_i's$ are not the conservative variables, as a result, the classical bound-preserving limiter in (X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), 3091-3120) cannot be applied. Therefore, another limiter will be introduced. Numerical experiments will be given to demonstrate the accuracy in $L^\infty$-norm and good performance of the numerical technique.
compressible miscible displacement problem, discontinuous Galerkin method, bound-preserving
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@inproceedings{hui2017bound-preserving,
  title={Bound-Preserving Discontinuous Galerkin Method for Compressible Miscible Displacement in Porous Media},
  author={Hui Guo, and Yang Yang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190701093817161752383},
  booktitle={SIAM Journal on Scientific Computing},
  volume={39},
  number={5},
  pages={A1969–A1990},
  year={2017},
}
Hui Guo, and Yang Yang. Bound-Preserving Discontinuous Galerkin Method for Compressible Miscible Displacement in Porous Media. 2017. Vol. 39. In SIAM Journal on Scientific Computing. pp.A1969–A1990. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190701093817161752383.
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