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

Clément Cancès (1) , Flore Nabet (2)

1 RAPSODI - Reliable numerical approximations of dissipative systems
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

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:R3844R3846, 1997] and studied mathematically by the authors in [C. Cancès, 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.

Mots clés: two-phase flow; degenerate Cahn-Hilliard system; finite volumes; convergence

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Clément Cancès, Flore Nabet. Finite Volume approximation of a two-phase two fluxes degenerate Cahn-Hilliard model. ESAIM: Mathematical Modelling and Numerical Analysis, 2021, 55 (3), pp.969--1003. ⟨10.1051/m2an/2021002⟩. ⟨hal-02561981v3⟩