Low Mach number limit of some staggered schemes for compressible barotropic flows

In this paper, we study the behaviour at low Mach number of numerical schemes based on staggered discretizations for the barotropic Navier-Stokes equations. Three time discretizations are considered: the implicit-in-time scheme and two non-iterative pressure correction schemes. The last two schemes differ by the discretization of the convection term: linearly implicit for the first one, so the resulting scheme is unconditionnally stable, and explicit for the second one, so the scheme is stable under a CFL condition involving the material velocity only. We rigorously prove that these three variants are asymptotic preserving in the following sense: for a given mesh and a given time step, a sequence of solutions obtained with a sequence of vanishing Mach numbers tend to a solution of a standard scheme for incompressible flows. This convergence result is obtained by mimicking the proof already known in the continuous case.