Stability and error analysis of a semi-implicit scheme for incompressible flows with variable density and viscosity

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 in L 2 norm with order one in time. Numerical illustrations confirm that the algorithm is stable and convergent under classic CFL condition even for sharp density profiles.