A novel fully-decoupled, linear, and unconditionally energy-stable scheme of the conserved Allen–Cahn phase-field model of a two-phase incompressible flow system with variable density and viscosity

Abstract

In this article, we first use the nonlocal conserved Allen–Cahn dynamics to reformulate the phase-field model with variable density and viscosity for the two-phase incompressible flow system, and then construct a novel fully-decoupled, linear, and unconditionally energy stable scheme to solve the model. The development of the scheme is based on a combination of a novel decoupling technique, the penalty method of the Navier–Stokes equation, the quadratization method of linearizing the double-well potential, and the operator Strang-splitting method. The scheme is highly efficient, and it only needs to solve a series of fully-decoupled linear elliptic equations at each time step, in which the Allen–Cahn equation and the pressure Poisson equation only have constant coefficients. We rigorously prove the unconditional energy stability of the scheme and conduct numerical simulations in 2D and 3D to demonstrate the accuracy, stability, and effectiveness of the scheme numerically.