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

Abstract

We study a time implicit Finite Volume scheme for degenerate Cahn-Hilliard model 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 is shown to preserve the key properties of the continuous model, namely mass conservation, positivity of the concentrations, the decay of the energy and the control of the entropy dissipation rate. This allows to establish the existence of a solution to the nonlinear algebraic system corresponding to the scheme. Further, we show thanks to compactness arguments that the approximate solution converges towards a weak solution of the continuous problems as the discretization parameters tend to 0. Numerical results illustrate the behavior of the numerical model.

A convergent finite volume scheme for dissipation driven models with volume filling constraint

“…. , n obtained in Theorem 7 and the entropy dissipation inequality (34). Indeed applying the inequality D σ u k i ≤ 2D σ u k i for all σ ∈ E and summing over k (34) yield…”

“…Indeed, there exists a wide literature concerning the design and the analysis of entropy stable finite volume schemes. For instance, let us mention the following (non-exhaustive) list of contributions [7,13,15,16,17,18,22,24,30,34,35,36,51,66,54,70]. Now, in the particular framework of cross-diffusion systems which admit an entropy structure as described in [60,61], we would like to mention the papers [1,3] where the authors proposed some convergent two-point flux approximation with upwind mobilities schemes for a seawater intrusion model in an unconfined and confined aquifer.…”

“…Our first aim is to prove some compactness properties on (u m ) m∈N . In this purpose we will use the black-box result [6, Theorem 3.9] (see also [34,Lemma B.1]).…”