We propose and analyze a fully decoupled finite difference scheme for the Abels–Garcke–Grün model for a binary mixture of two viscous incompressible fluids with unmatched densities and viscosities. The coupled system consists of the Navier–Stokes equations for the volume-averaged fluid velocity and a convective Cahn–Hilliard equation with Flory–Huggins potential for the phase-field variable. We first establish the unique solvability and unconditional energy stability of the numerical scheme. Then we verify the positivity-preserving property, which means that the discrete solution of the phase-field always stays in the physical interval ( − 1 , 1 ) (-1,1) at a point-wise level. This crucial fact ensures the feasibility of our scheme for the Cahn–Hilliard equation with a singular (logarithmic) potential. Afterwards, we perform a detailed optimal rate convergence analysis and derive error estimates that are first-order accurate in time and second-order accurate in space. Numerical experiments are presented to validate the theoretical results.
Chen et al. (Wed,) studied this question.