  We present in this paper a pressure correction scheme for barotropic
compressible Navier-Stokes equations, which enjoys an unconditional stability
property, in the sense that the energy and maximum-principle-based a priori
estimates of the continuous problem also hold for the discrete solution. The
stability proof is based on two independent results for general finite volume
discretizations, both interesting for their own sake: the $L^2$-stability of
the discrete advection operator provided it is consistent, in some sense, with
the mass balance and the estimate of the pressure work by means of the time
derivative of the elastic potential. The proposed scheme is built in order to
match these theoretical results, and combines a fractional-step time
discretization of pressure-correction type to a space discretization
associating low order non-conforming mixed finite elements and finite volumes.
Numerical tests with an exact smooth solution show the convergence of the
scheme.
