Application of predictor-corrector schemes with several correctors in solving air pollution problems

Systems of ordinary differential equations obtained by using splitting-up techniques in some air pollution models and a pseudospectral (Fourier) discretization of the first-order space derivatives are considered. The application of a fairly general class of predictor-corrector (PC) schemes in the time-discretization process is discussed. Several corrections with different corrector formulae are carried out in thesePC schemes. The classicalDahlquist theory valid for the case when the stepsize is constant is preserved (under very mild restrictions on the stepsize) when suchPC schemes are used as variable stepsize variable formula methods (VSVFM's). This fact is exploited by allowing the stepsize to follow the variation of a certain norm of the wind velocity vector in aVSVFM based on specially constructedPC schemes with large intervals of absolute stability on the imaginary axis. A device that attempts to check both the accuracy and the stability in the course of the integration process has been developed. The code based on the application of thisVSVFM in the time-integration part of the treatment of both 2-dimensional and 3-dimensional models has been tested by using meteorological data prepared at stations located in practically all European countries. The numerical results indicate thatPC schemes with several correctors can successfully be used for the class of problems under consideration. The main reason for this success is the special nature of the computational cost per time-step (due to the splitting approach used). Some short remarks on the possibility of extending the results for large systems ofODE's arising in the treatment of other classes of problems are made.Dedicated to Germund Dahlquist, on the occasion of his 60th birthday.