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Studying the trans boundary transport of air pollutants is an important environmental problem. Systems of partial differential equations are normally arising when mathematical models are used in the solution of this problem. The interrelations between the different air pollutants are rather complicated. ; Therefore many air pollutants are to be included in the models. This leads to very large and very complicated numerical tasks. The chemical part of an air pollution model is one of the most difficult parts for the numerical algorithms. It is necessary to apply reliable and sufficiently accurate algorithms during the numerical treatment of the chemical sub-models. ; Moreover, it is also necessary to apply fast numerical algorithms that can be run efficiently on modern high-speed computers. These two important requirements work, as often happens in practice, in opposite directions. Therefore a good compromise is needed. Some results achieved in the efforts to find a good compromise will be described. ; The advantages and disadvantages of several numerical methods will be discussed. All conclusions are made for the particular situation where large air pollution models are to be treated on big modern high-speed computers. ; Moreover, it is also assumed that a particular air pollution model, namely the Danish Eulerian Model, is used. However, the ODE systems that arise in the chemical sub-models have at least three rather common properties, which appear again and again when large scientific and engineering problems are studied. ; These systems are large, stiff and badly scaled. Therefore some of the conclusions are also valid in a much more general context, i.e. in all cases where large, stiff and badly scaled systems of ODE's are to be handled numerically.