Convergence of the MAC scheme for the incompressible Navier-Stokes equations
Provides theoretical convergence guarantees for a widely used numerical scheme, addressing a gap in rigorous analysis for non-uniform grids.
Proved convergence of the MAC scheme for incompressible Navier-Stokes equations on non-uniform grids without regularity assumptions, establishing that the limit is a weak solution.
We prove in this paper the convergence of the Marker and cell (MAC) scheme for the dis-cretization of the steady-state and unsteady-state incompressible Navier-Stokes equations in primitive variables on non-uniform Cartesian grids, without any regularity assumption on the solution. A priori estimates on solutions to the scheme are proven ; they yield the existence of discrete solutions and the compactness of sequences of solutions obtained with family of meshes the space step of which tends to zero. We then establish that the limit is a weak solution to the continuous problem.