A linear second-order in time unconditionally energy stable finite element scheme for a Cahn–Hilliard phase-field model for two-phase incompressible flow of variable densities

We propose a novel second-order BDF time stepping method of variable time step sizes combined with a classical residual-based stabilized finite element spatial discretization using the Streameline-Upwind Petrov–Galerkin (SUPG)/pressure stabilization Petrov–Galerkin (PSPG)/grad-div stabilization for solving the phase–field model for two-phase incompressible flow of different densities and viscosities in the advection dominated regime. In the case of uniform time step size and without extra stabilization, the scheme is shown to satisfy a discrete energy law. Benchmark test of the Rayleigh–Taylor instability under high Reynolds number and Péclect number demonstrates that the scheme captures details of the instability comparable to results in the literature by schemes based on sharp-interface models.