Development of efficient partitioned ODE algorithms
with an application to air pollution models

Z. Zlatev

National Environmental Research Institute
Frederiksborgvej 399, P. O. Box 358
DK-4000 Roskilde, Denmark
e-mail: luzz@sun2.dmu.dk

Many scientific and engineering problems lead, very often
after the application of an appropriate discretization
procedure, to the solution of a large number of not very
big but very stiff ODE systems. The underlying problems
are as a rule very time-consuming. Therefore, it is
important to develop and use efficient numerical
methods for the arising ODE systems. The numerical treatment
of stiff ODE systems is normally carried out by using
implicit integration methods. A long sequence of non-linear
algebraic systems has normally to be treat numerically when
implicit integration methods are used. The well known Newton
iterative method is often used in the solution of these
non-linear algebraic systems. This leads to the calculation
of the elements of the Jacobian matrices and to the inversion
(normally by using some kind of factorization) of these matrices.
The computational work can in many cases, but not always, be
reduced considerably when some kind of partitioning is used.
Theoretical conditions, under which the partitioning procedures
can successfully been used, will be studied in this paper. An
example, taken from a large air pollution model, will
be given to illustrate the usefulness of the theoretical
results.