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

Highlights: • Construct special techniques to preserve two bounds without using the maximum-principle-preserving technique. • Treat the time derivative of the pressure as a source of the concentration equation. • Apply the algorithm on unstructured meshes. • In the flux limiter, use the second-order flux as the lower-order one. • Use L2-projection of the porosity and construct special limiters that suitable for multi-component fluid mixtures. -- Abstract: 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 jth 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 Zhang and Shu (2010) [44] cannot be applied directly. The main idea is to apply the positivity-preserving techniques to all $c_j^{\prime }s$ and enforce ${\sum }_j{c}_j=1$ 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.