A fully-decoupled second-order-in-time and unconditionally energy stable scheme for the Cahn-Hilliard-Navier-Stokes equations with variable density

In this paper, we develop a second-order, fully decoupled, and energy-stable numerical scheme for the Cahn-Hilliard-Navier-Stokes model for two phase flow with variable density and viscosity. We propose a new decoupling Constant Scalar Auxiliary Variable (D-CSAV) method which is easy to generalize to schemes with high order accuracy in time. The method is designed using the “zero-energy-contribution” property while maintaining conservative time discretization for the “non-zero-energy-contribution” terms. A new set of scalar auxiliary variables is introduced to develop second-order-in-time, unconditionally energy stable, and decoupling-type numerical schemes. We also introduce a stabilization parameter α to improve the stability of the scheme by slowing down the dynamics of the scalar auxiliary variables. Our algorithm simplifies to solving three independent linear elliptic systems per time step, two of them with constant coefficients. The update of all scalar auxiliary variables is explicit and decoupled from solving the phase field variable and velocity field. We rigorously prove unconditional energy stability of the scheme and perform extensive benchmark simulations to demonstrate accuracy and efficiency of the method.