Convergence of a positivity-preserving DDFV scheme for compressible two-phase flow in porous media

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Abstract In this paper we study the convergence of the positivity-preserving discrete duality finite volume (PP-DDFV) scheme introduced in Crozon et al. (2025, Positivity-preserving DDFV scheme for compressible two-phase flow in porous media. Comput. Math. Appl., 194, 110–134). It approximates solutions to the immiscible compressible two-phase Darcy flow in anisotropic porous media with a degeneracy of the mobilities. The primary variables are the physical pressures, and the phase density depends on its own pressure. Additionally, the considered two-dimensional mesh is quite general including nonconforming, as well as distorted partitions. We propose a new penalization term because it is required in the DDFV framework to force the primal and dual parts of the global pressure and the capillary terms to converge towards the same limit. As a consequence, all the results are adapted to show that the approximated solutions converge to a weak solution of the diphasic model, up to a subsequence. This is made due to the compactness arguments of type Lions–Aubin–Simon theorem. In the end, we present some typical numerical tests to exhibit the good convergence of the numerical scheme and the impact of the penalization term on the solution behaviour.

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