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In this paper, we present an approach to numerical solution of a model that simulates the process of pollutant dispersion in the planetary boundary layer. The foundation of model itself comes from the current research of a wind and diffusion modeling. From a technical point of view it is described by the advection-diffusion equation in two dimensions - the distance from a source and the height above a terrain. Both the wind and the diffusion coefficients depend on a given height, where such condition also brings relatively high degree of variability. Our numerical approach is based on the well known method of lines which is an elegant and much simpler method than an analytical one. However, the most obvious problem can be evidently recognized behind the coverage of boundary conditions in the proposed model and that is why we had to come up with modified version of them. Extensive set of experiments have been performed with our model. Subsequently, the obtained results were thoroughly compared with the analytical solution and with corresponding results of Copenhagen experiments. As it is clearly demonstrated in our paper all results agree very well.