Positivity-preserving finite volume scheme for compressible two-phase flows in anisotropic porous media: The densities are depending on the physical pressures

•A finite volume scheme is proposed to approximate a compressible diphasic model in anisotropic media.•A discrete maximum principle as well as a priori estimates are established.•The convergence of the numerical scheme is proved under general assumptions on the data and mesh.•Numerical results illustrate the efficiency and the robustness of the methodology with respect to the anisotropy. We are concerned with the approximation of solutions to a compressible two-phase flow model in porous media thanks to an enhanced control volume finite element discretization. The originality of the methodology consists in treating the case where the densities are depending on their own pressures without any major restriction neither on the permeability tensor nor on the mesh. Contrary to the ideas of [23], the point of the current scheme relies on a phase-by-phase “sub”-unpwinding approach so that we can recover the coercivity-like property. It allows on a second place for the preservation of the physical bounds on the discrete saturation. The convergence of the numerical scheme is therefore performed using classical compactness arguments. Numerical experiments are presented to exhibit the efficiency and illustrate the qualitative behavior of the implemented method.