Efficient IMEX and consistently energy-stable methods of diffuse-interface models for incompressible three-component flows

In this study, we consider the numerical approximation of incompressible three-component fluids, in which the fluid interfaces are captured by ternary Cahn–Hilliard equations and the fluid flows are governed by Navier–Stokes equations. This system includes not only nonlinear effects but also coupling among phase-field variables, velocities, and pressure. The ternary Cahn–Hilliard–Navier–Stokes system also satisfies the energy dissipation law, which is a basic physical property. For the appropriate treatment of the nonlinear and coupling terms and preservation of the energy dissipation law in a discrete version, we develop second-order time-accurate, linearly implicit-explicit (IMEX) methods using a variation of the scalar auxiliary variable (SAV) method. To improve the consistency between the original and modified energies, a simple and effective energy relaxation technique is considered. We analytically proved the unique solvability and the relaxed energy dissipation law. The proposed schemes are highly efficient for implementation because only linear elliptic equations need to be solved separately. Extensive computational experiments are performed to validate the accuracy, energy stability, and performance of the proposed method. To facilitate further study, we provide the C codes for the typical numerical simulations at http://github.com/yang521.