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.

翻译：本文件的重点是证明离散隐含标记和标记(MAC)办法对时间依赖型纳维埃-斯托克方程式具有可变密度和可变粘度的离散隐含标记和标记(MAC)办法的趋同性证据,问题以单一的迪里赫莱边界条件完成,根据非统一的笛卡尔电网分解,先验估计未知数,同时从地形学角度论证,每个步骤都存在离散式办法的解决办法。我们最后证明,由于网目大小和时间步骤趋向于零,离散解决办法的顺序限制是解决问题的薄弱办法。