A new efficient fully-decoupled and second-order time-accurate scheme for Cahn–Hilliard phase-field model of three-phase incompressible flow

Abstract

For the highly coupled and nonlinear Cahn–Hilliard phase-field model of three-phase incompressible flow, how to establish a fully-decoupled numerical scheme with second-order time accuracy has always been a very difficult and unsolved problem. In this paper, we propose a novel decoupling method, which only needs to solve several decoupling linear elliptic equations with constant coefficients at each time step to obtain a numerical solution with second-order time accuracy. The key idea is to introduce two nonlocal auxiliary variables into the system, one of which is used to linearize the nonlinear potential, and the other is used to introduce an ordinary differential equation to deal with the nonlinear coupling terms with “zero-energy-contribution” characteristics. We strictly prove the solvability and unconditional energy stability of the scheme, and conduct numerical simulations in 2D and 3D to show the accuracy and stability of the scheme numerically. To the best of the author’s knowledge, the method developed in this paper is the first second-order fully-decoupled scheme for the hydrodynamics coupled phase-field model.