  The present paper is focused on the proof of the convergence of the discrete
implicit Marker-and-Cell (MAC) scheme for time-dependent Navier--Stokes
equations with variable density and variable viscosity. The problem is
completed with homogeneous Dirichlet boundary conditions and is discretized
according to a non-uniform Cartesian grid. A priori-estimates on the unknowns
are obtained, and along with a topological degree argument they lead to the
existence of a solution of the discrete scheme at each time step. We conclude
with the proof of the convergence of the scheme toward the continuous problem
as mesh size and time step tend toward zero with the limit of the sequence of
discrete solutions being a solution to the weak formulation of the problem.
