Artificial compressibility method for the incompressible Navier-Stokes equations with variable density

We introduce a novel artificial compressibility technique to approximate the incompressible Navier-Stokes equations with variable fluid properties such as density and dynamical viscosity. The proposed scheme used the couple pressure and momentum, equal to the density times the velocity, as primary unknowns. It also involves an adequate treatment of the diffusive operator such that treating the nonlinear convective term explicitly leads to a scheme with time independent stiffness matrices that is suitable for pseudo-spectral methods. The stability and temporal convergence of the semi-implicit version of the scheme is established under the hypothesis that the density is approximated with a method that conserves the minimum-maximum principle. Numerical illustrations confirm that both the semi-implicit and explicit scheme are stable and converge with order one under classic CFL condition. Moreover, the proposed scheme is shown to perform better than a momentum based pressure projection method, previously introduced by one of the authors, on setups involving gravitational waves and immiscible multi-fluids in a cylinder.