Convergence of the MAC scheme for the incompressible Navier-Stokes equations with variable density and viscosity

The present paper addresses the convergence of the implicit Marker-and-Cell scheme for time-dependent Navier–Stokes equations with variable density and density-dependent viscosity and forcing term. A priori estimates on the unknowns are obtained, and thanks to a topological degree argument, they lead to the existence of an approximate solution at each time step. Then, by compactness arguments relying on these same estimates, we obtain the convergence (up to the extraction of a subsequence), when the space and time steps tend to zero, of the numerical solutions to a limit; this latter is shown to be a weak solution to the continuous problem by passing to the limit in the scheme.