First we summarize recent efforts to accurately assess the regime of validity of sound-proof model equations for atmospheric flows. We find that for length scales comparable to the pressure scale height, and for background potential temperature stratifications weaker than the Mach number to the power 2/3, atmospheric flows are well described by either the anelastic or the pseudo-incompressible models. There are regimes, in which the pseudo-incompressible model even holds under strong stratification as found, e.g., in the stratosphere.
On large scales comparable to the Earth radius, compressibility effects become non-negligible even for internal waves, and accurate computations appear to require using the compressible flow equations as the starting point.
We then proceed to demonstrate that a compressible flow solver can be constructed using multi-grid techniques, which guarantees control of an anelastic or pseudo-incompressible divergence on small scales, while it uses, e.g., a symplectic integrator to represent the long-wave compressible dynamics.