This work originated from a collaboration with the swiss weather prediction service MeteoSwiss which computes its predictions using the COSMO model and uses a terrain-following grid on a non-convex domain. The goal of this thesis is to present a numerical method adapted to the grid to improve the quality of the predictions. The chosen method is the DDFV method, which originated in the late 90s. We first analyse the way turbulent diffusion is treated in the COSMO model, as well in space as in time. We then introduce the DDFV method and prove its convergence on non-convex domains. In order to compensate for the loss of regularity of the solution near the nonsmooth part of the boundary, we introduce a refinement of the grid at the reentrant corner. Finally, we analyse the stability of several time-schemes and compute stability criterions when the DDFV method is used as a space discretisation.