In the atmospheric dynamics research community, many colleagues of mine happen to use atmospheric models that have non-hydrostatic equation solvers, e.g. the UK MetOffice model, and WRF-ARW:
The equation set for ARW is fully compressible, Eulerian and nonhydrostatic...
There are another group of models that use a hydrostatic solver by default (e.g. ECMWF), and presumably make the following assumption about the Earth's atmosphere.
Hydrostatic equilibrium describes the atmospheric state in which the upward directed pressure gradient force (the decrease of pressure with height) is balanced by the downward-directed gravitational pull of the Earth. On average the Earth’s atmosphere is always close to hydrostatic equilibrium.
My question is what are the applications of each type of solver, i.e. when would you chose to use a non-hydrostatic model over a hydrostatic one? (which is presumably less computationally expensive?)