Physical feature preserving and unconditionally stable SAV fully discrete finite element schemes for incompressible flows with variable density

In this paper, we construct new positive preserving, unconditionally stable and fully discrete finite element schemes for incompressible flows with variable density. The proposed schemes employ the positive function transform rho = F(phi) for the density equation and scalar auxiliary variable (SAV) q = exp(-t/T) for the momentum equation in its reformulation system. The SAV schemes are unconditionally energy stable and second-order accurate in time and lead to two decoupled generalized Stokes equations for the momentum equation to be solved at each time step. Thus, it is easy to implement and extremely efficient for these schemes when combined with an adaptive time stepping method. Numerical experiments demonstrate the stability of computations and verify the second-order accuracy of the proposed schemes.