A new energy stable fractional time stepping scheme for the Navier–Stokes/Allen–Cahn diffuse interface model

In this work, we present an original time-discrete formulation of the coupled equations relating the velocity and pressure of an unsteady flow of two immiscible fluids and the Allen–Cahn equation describing the interface between both of them. We first prove this time-discrete formulation (based on the concept of coupled projection scheme (Deteix et al., 2014)) to be well posed and energy stable. We then propose a new family of iterative schemes for the actual approximation of solutions. We complete this work with numerical tests illustrating the order of accuracy and applying the new scheme on the well known rising bubble benchmark. •A new scheme for diffuse interface problems modelled by Allen–Cahn.•An extended projection method for variable density and viscosity.•Existence, maximum principle and stability of the discrete solution are demonstrated.•Temporal accuracy is illustrated using finite element on a manufactured solution.•Physical relevance is illustrated using the single bubble benchmark.