An efficient stabilized multiple auxiliary variables method for the Cahn–Hilliard–Darcy two-phase flow system

Abstract

Herein, we develop a totally decoupled, linear, and temporally second-order accurate numerical scheme for the Cahn–Hilliard–Darcy system which models the two-phase incompressible fluid flows in porous medium or in Hele–Shaw cells. Our proposed scheme is based on a simple and efficient stabilized multiple scalar auxiliary variables (S-MSAV) approach. Two time-dependent variables are defined to change the original governing equations to be the equivalent forms and then the Crank–Nicolson type approximation and the explicit Adams–Bashforth approximation are used to temporally discretize the equivalent equations. All nonlinear parts and auxiliary variables are explicitly treated, thus we can decouple the phase-field variable and auxiliary variables in time. The velocity and pressure are decoupled by using a second-order accurate pressure correction method. Therefore, our numerical scheme is very simple and efficient. We analytically prove the discrete energy dissipation law and unique solvability of our scheme with the absence of external force. The benchmark tests are performed to show that our method has desired accuracy and energy stability. Moreover, our method works well in simulating buoyancy-driven pinchoff and viscous fingering.