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ABSTRACT 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 method uses 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 method with time‐independent stiffness matrices that is suitable for pseudo‐spectral methods. The stability and temporal convergence of a semi‐implicit version of the method are 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 methods are stable and converge with order one under the classic CFL condition. Moreover, the proposed method 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. 

Abstract We study the stability and convergence properties of a semi-implicit time stepping scheme for the incompressible Navier–Stokes equations with variable density and viscosity. The density is assumed to be approximated in a way that conserves the minimum-maximum principle. The scheme uses a fractional time-stepping method and the momentum, which is equal to the product of the density and velocity, as a primary unknown. The semi-implicit algorithm for the coupled momentum-pressure is shown to be conditionally stable and the velocity is shown to converge inL²norm with order one in time. Numerical illustrations confirm that the algorithm is stable and convergent under classic CFL condition even for sharp density profiles. 

Abstract Convergence of a finite element method with mass-lumping and flux upwinding is formulated for solving the immiscible two-phase flow problem in porous media. The method approximates directly the wetting phase pressure and saturation, which are the primary unknowns. Well-posedness is obtained in [J. Numer. Math., 29(2), 2021]. Theoretical convergence is proved via a compactness argument. The numerical phase saturation converges strongly to a weak solution in L 2 in space and in time whereas the numerical phase pressures converge strongly to weak solutions in L 2 in space almost everywhere in time. The proof is not straightforward because of the degeneracy of the phase mobilities and the unboundedness of the derivative of the capillary pressure. 

Abstract A finite element method with mass-lumping and flux upwinding is formulated for solving the immiscible two-phase flow problem in porous media. The method approximates directly thewetting phase pressure and saturation, which are the primary unknowns. The discrete saturation satisfies a maximum principle. Stability of the scheme and existence of a solution are established.