Convergence of the Marker-and-Cell Scheme for the Incompressible Navier–Stokes Equations on Non-uniform Grids

T Gallouët,R Herbin,K Mallem,J.-C Latché

doi:10.1007/s10208-016-9338-4

Abstract

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.

Highlights

We prove in this paper the convergence of the Marker And Cell (MAC) scheme for the discretization of the steady-state and time-dependent incompressible Navier-Stokes equations in primitive variables, on non-uniform Cartesian grids, without any regularity assumption on the solution
The aim of this paper is to show, under minimal regularity assumptions on the solution, that sequences of approximate solutions obtained by the discretization of problem (1)(resp. (3)) by the MAC scheme converge to a solution of (2)(resp. (4)) as the mesh size tends to 0
The mathematical analysis of the scheme was performed for the steadystate Stokes equations in [26] for uniform rectangular meshes with H2-regularity assumption on the pressure

Summary

Space discretization

These latters coincide with the primal mesh: for any sub-volume Dǫ of such a partition, there is K ∈ M such that Dǫ = K, and we may write equivalently (ðiui)Dǫ or (ðiui)K We choose this latter notation in the definition of the discrete divergence below for the sake of consistency, since, if we adopt a variational point of view for the description of the scheme, the discrete velocity divergence has to belong (and does belong) to the space of discrete pressures (see Sections 3 and 4 below for a varitional form of the scheme, in the steady and time-dependent case, respectively). Discrete convection operator – Let us consider the momentum equation (1b) for the ith component of the velocity, and integrate it on a dual cell Dσ, σ ∈ E(i).

The scheme

Stability and existence of a solution

Convergence analysis

Time discretization

Case of the unsteady Stokes equations