A finite difference model for air pollution simulation N. Sanı ´n, G. Montero * University Institute for Intelligent Systems and Numerical Applications in Engineering (IUSIANI), University of Las Palmas de Gran Canaria, Spain Available online 23 October 2006 Abstract A 3-D model for atmospheric pollutant transport is proposed considering a set of coupled convection–diffusion–reaction equations. The convective phenomenon is mainly produced by a wind field obtained from a 3-D mass consistent model. In particular, the modelling of oxidation and hydrolysis of sulphur and nitrogen oxides released to the surface layer is carried out by using a linear module of chem- ical reactions. The dry deposition process, represented by the so-called deposition velocity, is introduced as a boundary condition. More- over, the wet deposition is included in the source term of the governing equations using the washout coefficient. Before obtaining a numerical solution, the problem is transformed using a terrain conformal coordinate system. This allows to work with a simpler domain in order to build a mesh that provides finite difference schemes with high spatial accuracy. The convection–diffusion–reaction equations are solved with a high order accurate time-stepping discretization scheme which is constructed following the technique of Lax and Wendroff. Finally, the model is tested with a numerical experiment in La Palma Island (Canary Islands). Ó 2006 Elsevier Ltd. and Civil-Comp Ltd. All rights reserved. Keywords: Wind modelling; Mass consistent models; Air pollution modelling; Eulerian model; Finite differences; Accurate time-stepping 1. Introduction The difficulty for developing a complete understanding of environmental impacts taking into account few measures of atmospheric phenomena arises from the com- plexity of atmospheric physical and chemical processes, the spatial dimension of the atmosphere and the time scaling of the episodes. Numerical simulations of the atmosphere play an important role in reaching a more complete under- standing of environmental impacts. In this paper, a numer- ical model for wind and air pollution simulation is proposed. On the one hand, the wind computation is car- ried out using a mass consistent model which is based on the mass conservation law, several measures of wind velo- cities at different points of the domain and the physics of the atmosphere [1]. More details about the general proce- dure and mathematical supports for this kind of models, with particular reference to the approximations that char- acterize them, may be seen in [2]. On the other hand, an Eulerian model is proposed for the resolution of pollutant concentrations. This model is based on the convection–dif- fusion–reaction equation of pollutant species and includes emission sources, dry and wet depositions and chemical reactions; see, e.g. [3]. In this paper, we have used the finite difference method in spite of other discretization techniques, like finite ele- ment or finite volume methods, are also suitable. The finite difference method is based on local approximations of the partial derivatives in a PDE, which are derived by low order Taylor series expansions. In general, this method is quite simple to define and rather easy to implement. Also, it is particularly appealing for simple regions, such as the cube obtained in our model after a terrain conformal coor- dinate transformation. In addition, it is profitable if meshes are uniform, e.g., the structured meshes used here with reg- ular horizontal and variable vertical discretization. The matrices that result from this technique are often well structured and they typically consist of a few nonzero diag- onals. Another advantage is that there are a number of fast solvers for constant coefficient problems, which can deliver the solution in logarithmic time per grid point, i.e., the 0965-9978/$ - see front matter Ó 2006 Elsevier Ltd. and Civil-Comp Ltd. All rights reserved. doi:10.1016/j.advengsoft.2006.09.013 * Corresponding author. Tel.: +34 28 458831; fax: +34 28 458811. E-mail address: gustavo@dma.ulpgc.es (G. Montero). www.elsevier.com/locate/advengsoft Advances in Engineering Software 38 (2007) 358–365