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We study a time implicit Finite Volume scheme for degenerate Cahn-Hilliardmodel proposed in [W. E and P. Palffy-Muhoray. Phys. Rev. E, 55:R3844-R3846,1997] and studied mathematically by the authors in [C. Canc\`es, D. Matthes,and F. Nabet. Arch. Ration. Mech. Anal., 233(2):837-866, 2019]. The scheme isshown to preserve the key properties of the continuous model, namely massconservation, positivity of the concentrations, the decay of the energy and thecontrol of the entropy dissipation rate. This allows to establish the existenceof a solution to the nonlinear algebraic system corresponding to the scheme.Further, we show thanks to compactness arguments that the approximate solutionconverges towards a weak solution of the continuous problems as thediscretization parameters tend to $0$. Numerical results illustrate thebehavior of the numerical model.

Finite Volume approximation of a two-phase two fluxes degenerate Cahn-Hilliard model