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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.

We associate to a finite digraph D a lattice polytope PD whose vertices are the rows of the Laplacian matrix of D. This generalizes a construction introduced by Braun and the third author. As a consequence of the Matrix-Tree Theorem, we show that the normalized volume of PD equals the complexity of D, and PD contains the origin in its relative interior if and only if D is strongly connected. Interesting connections with other families of simplices are established and then used to describe reflexivity, the h∗-polynomial, and the integer decomposition property of PD in these cases. We extend Braun and Meyer’s study of cycles by considering cycle digraphs. In this setting, we characterize reflexivity and show there are only four non-trivial reflexive Laplacian simplices having the integer decomposition property.