Mixed stabilized finite element methods based on backward difference/Adams–Bashforth scheme for the time-dependent variable density incompressible flows

In this paper, we consider a second-order mixed stabilized finite element method based on pressure projection method for variable density incompressible flows. The originality of the proposed approach is to use a stabilized method based on the difference between a consistent and an under-integrated mass matrix of the pressure for variable density incompressible flows approximated by the lowest equal-order finite element pairs. A second-order backward difference (BDF) for the temporal term and a second-order Adams–Bashforth (AB) for the explicit treatment of the nonlinear term lead to the presented second-order BDF/AB scheme. The stability of the method was proved and the performance of the method is numerically illustrated. Finally, comparison with some established methods, a series of numerical experiments are given to show that this method has better stability and accuracy.