  In this article we present a novel staggered semi-implicit hybrid
finite-volume/finite-element (FV/FE) method for the resolution of weakly
compressible flows in two and three space dimensions. The pressure-based
methodology introduced in Berm\'udez et al. 2014 and Busto et al. 2018 for
viscous incompressible flows is extended here to solve the compressible
Navier-Stokes equations. Instead of considering the classical system including
the energy conservation equation, we replace it by the pressure evolution
equation written in non-conservative form. To ease the discretization of
complex spatial domains, face-type unstructured staggered meshes are
considered. A projection method allows the decoupling of the computation of the
density and linear momentum variables from the pressure. Then, an explicit
finite volume scheme is used for the resolution of the transport diffusion
equations on the dual mesh, whereas the pressure system is solved implicitly by
using continuous finite elements defined on the primal simplex mesh.
Consequently, the CFL stability condition depends only on the flow velocity,
avoiding the severe time restrictions that might be imposed by the sound
velocity in the weakly compressible regime. High order of accuracy in space and
time of the transport diffusion stage is attained using a local ADER (LADER)
methodology. Moreover, also the CVC Kolgan-type second order in space and first
order in time scheme is considered. To prevent spurious oscillations in the
presence of shocks, an ENO-based reconstruction, the minmod limiter or the
Barth-Jespersen limiter are employed. To show the validity and robustness of
our novel staggered semi-implicit hybrid FV/FE scheme, several benchmarks are
analysed, showing a good agreement with available exact solutions and numerical
reference data from low Mach numbers, up to Mach numbers of the order of unity.
