Because of the enormous range of spatial and temporal scales in geophysical flows, explicit integration of generic compressible equations with existing computational resources are prohibitively expensive for the majority of applications. In order to account for a broad range of scales, one has to invoke analytic or numerical approximations that allow for a relatively large time-step integrations of the governing equations. In effect, meteorological models encompass a variety of approximate systems of fluid equations tailored to the spatial and temporal scales of the investigated phenomena.

This talk summarizes overall author's  research presented as the habilitation achievement, i.e. the development of computational methods for solving Navier-Stokes equations in the anelastic approximation. Methods these are based on accurate non-oscillatory forward-in-time (NFT) algorithms formulated on structured refined grid or unstructured meshes within the framework of EULAG model and applied to many research problems in a broad range - from micro O(10^-2 m) to planetary (10⁴ m) - of physical scales.