We study a non-local version of the Cahn-Hilliard dynamics for phase separation in a two-component incompressible and immiscible mixture with linear mobilities. Differently to the celebrated local model with nonlinear mobility, it is only assumed that the divergences of the two fluxes-but not necessarily the fluxes themselves-annihilate each other.
We propose a time implicit Finite Volume scheme for this problem. 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. We prove existence of a solution to the nonlinear scheme and convergence of the approximate solution towards a weak solution of the continuous problems. The existence of a weak solution has been established by showing the convergence of a minimizing movement scheme à la Jordan et al.
Numerical results illustrate the behavior of the numerical model and we also compare the non-local model to the classical Cahn-Hilliard model.