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Bound-preserving discontinuous Galerkin method for compressible miscible displacement in porous media
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 th component of the fluid mixture, , should be between and
. However, 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 and
, respectively and enforce simultaneously to obtain physically
relevant approximations. By doing so, we have to treat the time derivative of
the pressure as a source in the concentration equation. Moreover,
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 -norm and good performance of the numerical technique