Convergence of the MAC scheme for the compressible stationary Navier-Stokes equations

TL;DR

This paper proves the convergence of the MAC scheme for discretizing steady compressible Navier-Stokes equations, establishing existence, estimates, and that the limit solutions satisfy key physical equations on Cartesian grids.

Contribution

It provides a rigorous proof of convergence for the MAC scheme applied to steady compressible Navier-Stokes equations, including existence and limit analysis.

Findings

Existence of solutions to the MAC scheme is established.

Approximate solutions converge up to a subsequence.

Limit solutions satisfy mass, momentum, and equation of state.

Abstract

We prove in this paper the convergence of the Marker and Cell (MAC) scheme for the discretization of the steady state compressible and isentropic Navier-Stokes equations on two or three-dimensional Cartesian grids. Existence of a solution to the scheme is proven, followed by estimates on approximate solutions, which yield the convergence of the approximate solutions, up to a subsequence, and in an appropriate sense. We then prove that the limit of the approximate solutions satisfies the mass and momentum balance equations, as well as the equation of state, which is the main difficulty of this study.