  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.
