Second order approximation for a quasi-incompressible Navier-Stokes Cahn-Hilliard system of two-phase flows with variable density

Phase-field model has been applied extensively and successfully for simulating two-phase flows with variable density. Several stable numerical methods have been proposed for solving such a highly nonlinear and coupled system. The key issue is to design a method such that it can preserve the (exact or slightly modified) conservative/dissipative law of energy of the fluid system at the discrete level. However, most of the existing energy stable numerical methods are restricted to only the first order accuracy in time for two-phase flow with different density. The design of a temporally second order accurate numerical method for the two-phase flows with variable density still remains challenging. In this paper, we develop several second order, robust and accurate numerical schemes for a quasi-incompressible Navier-Stokes Cahn-Hilliard model of two-phase flows with variable density which is thermodynamically consistent and was originally developed in [1] for simulating two-phase flows in complex geometries. Especially, numerical schemes proposed in this paper can preserve the mass conservation or energy dissipative law. Several numerical examples are presented to validate the robustness and accuracy of our numerical schemes.