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

Abstract

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, J. Comput. Phys., 229 (2010), pp. 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, J. Comput. Phys., 229 (2010), pp. 3091--3120] cannot be applied. Therefore, another limiter will be introduced. Numerical experiments will be given to demonstrate the accuracy in the $L^\infty$-norm and good performance of the numerical technique.