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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. Finite volume methods; MAC scheme; incompressible Navier-Stokes equations; variable density and viscosity; transport equations.

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.

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The purpose in this article is to design finite‐volume schemes on structured grids for the transport of piecewise‐constant functions (typically, indicator functions) with as low diffusion as possible. We first propose an extension of the so‐called Lagrange‐projection algorithm, or downwind scheme with an Ultrabee limiter, for the transport equation in one space dimension with a non‐constant velocity; as its constant velocity counterpart, this scheme is designed to capture the discontinuities separating two plateaus in only one cell, and is referred to as “anti‐diffusive.” It is shown to preserve the bounds of the solution. Then, for two and three dimensional problems, we introduce a conservative alternate‐directions algorithm, an show that this latter enjoys a discrete maximum principle, provided that the underlying one‐dimensional schemes satisfy a property which may be seen as a flux limitation, possibly incorporated a posteriori in any explicit scheme. Numerical tests of this alternate‐directions algorithm are performed, with a variety of one‐dimensional embedded schemes including the anti‐diffusive scheme developed here and the so called THINC scheme. The observed numerical diffusion is indeed very low. With the anti‐diffusive scheme, the above‐mentioned a posteriori limitation is necessary to preserve the solution bounds, but, in the performed tests, does not introduce any visible additional diffusion.