Exploiting the natural partitioning in the numerical solution of ODE systems arising in atmospheric chemistry

Abstract

Large air pollution models are commonly used to study transboundary transport of air pollutants. Such models are described mathematically by systems of partial differential equations (the number of equations being equal to the number of pollutants involved in the model). The use of appropriate splitting procedures leads to several sub-models. If the model is discretized on a (96×96×10) grid and if the number of pollutants is 35, then a system of ODE's containing 3225600 equations is to be treated in each sub-model. The ODE system of the chemical sub-model can be decoupled to (96×96×10) small systems, each of them containing 35 equations. The number of time-steps, needed in each submodel, is typically several thousand.

The chemical part of an air pollution model is one of the most difficult parts for the numerical algorithms. Therefore it is desirable to apply reliable and sufficiently accurate algorithms during the numerical treatment of the chemical sub-models. Moreover, it is also desirable to apply fast numerical algorithms that can be run efficiently on the 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 the disadvantages of several numerical methods (which are often used in the treatment of such ODE systems) will be discussed. All conclusions are made for the particular situation where large air pollution models are to be treated on big modern highspeed computers. Moreover, it is also assumed that a particular air pollution model, 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 ODEs are to be handled numerically.