A generalized scalar auxiliary variable approach for the Navier–Stokes-ω/Navier–Stokes-ω equations based on the grad-div stabilization

In this article, based on the grad-div stabilization, we propose a generalized scalar auxiliary variable approach for solving a fluid–fluid interaction problem governed by the Navier–Stokes-$\omega$/Navier–Stokes-$\omega$ equations. We adopt the backward Euler scheme and mixed finite element approximation for temporal-spatial discretization, and explicit treatment for the interface terms and nonlinear terms. The proposed scheme is almost unconditionally stable and requires solving only the linear equation with constant coefficient at each time step. It can also penalize for lack of mass conservation and improve the accuracy. Finally, a series of numerical experiments are carried out to illustrate the stability and effectiveness of the proposed scheme.