Convergence of the implicit MAC-discretized Navier--Stokes equations with variable density and viscosity on non-uniform grids

TL;DR

This paper proves the convergence of an implicit MAC scheme for variable density and viscosity Navier--Stokes equations on non-uniform grids, ensuring the discrete solutions approximate the continuous problem as discretization parameters tend to zero.

Contribution

It provides a rigorous convergence proof for the implicit MAC discretization of variable density and viscosity Navier--Stokes equations on non-uniform grids.

Findings

Existence of solutions at each time step established.

Convergence of the scheme to the weak solution of the continuous problem proved.

A priori estimates and topological degree argument used in the proof.

Abstract

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.