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, égale à la densité multipliée par la vitesse, 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.

Cet article explore les excursions dans le temps et leurs implications.

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