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We developed a global semi-implicit two-time-level semi-Lagrangian model for the atmospheric adiabatic primitive equations discretized through finite-differences, on locally refined grids. The principal aim of the technique is to have high resolution over a region of interest through successive grid refinements of a relatively coarse global uniform mesh, and in this way, to obtain accurate forecasts over this region at much lower computational costs. We tested the method through experiments with standard test cases for global atmospheric dynamical cores and also with real meteorological data. For a few days of integration, we obtained forecasts over the region of interest with similar accuracy to predictions computed on very fine globally uniform meshes, in a very efficient method. For longer periods, unresolved features outside the refined areas eventually reach the region of interest and affect the quality of the results. Our conclusion is that the use of locally refined meshes, combined with a stable semi-Lagrangian semi-implicit method, provides an effective way to compute accurate short range regional predictions. [ABSTRACT FROM AUTHOR]