Implementation of a Variable Stepsize Variable Formula Method in the Time-Integration Part of a Code for Treatment of Long-Range Transport of Air Pollutants
A mathematical model consisting of two partial differential equations is used to study the long-range transport of sulphur di-oxide and sulphate over Europe. The discretization of the first-order space derivatives ( the advection terms) is carried out by a pseudospectral (Fourier) algorithm. A special technique is applied in the discretization of the second-order space derivatives ( the diffusion terms). Two large systems of ordinary differential equations are solved at each time-step. It is shown that these systems can efficiently be treated by a variable stepsize variable formula method (VSVFM) based on the use of predictor-corrector schemes. The stepsize selection strategy and the formula selection strategy are discussed in detail. An attempt to carry out both an accuracy control and a stability control is made at each time-step. The great efficiency of the VSVFM implemented in our software as well as the reliability of the results are illustrated by numerical experiments, in which real meteorological data (for 1979) at the grid-points of a space domain covering the whole of Europe were used. The main ideas, implemented in the time-integration part, might be applied in many other situations, where the systems of ordinary differential equations arising after the space discretization are only moderately stiff (so that the stability requirements are dominant over the accuracy requirements on a large part of the time-interval but the use of implicit time-integration algorithms that require solving systems of algebraic equations at each time-step is not justified). As an illustration only it should be mentioned that such an application has been carried out in connection with models describing long-range transport of nitrogen pollutants over Europe.