Some mathematical and numerical aspects of reduction in chemical kinetics

Bernard Larrouturou and Bruno Sportisse

Conference Computational Science for the 21st century, Tours (F), May 97, Published by John Wiley Sons, p422-432.

Several applications, such as combustion processes, hypersonic flow or air pollution, require to take into account chemical phenomena. This has many consequences for the numerical simulation. First, the wide range covered by time scales implies the well known {\em stiffness} of the resulting equations. Moreover a precise description of the chemistry often requires to consider a tremendous number of species and chemical reactions, which dramatically increases the computational cost. One classical way to handle stiffness is the use of specific (implicit) numerical solvers. Another way is to build so-called ``reduced models'' by getting rid of the stiffness and lowering the dimension of the system. Conventional methods have been employed for a long time by chemists for this purpose (Bodenstein \& al 1924, Semenov 1939) ; they are mainly based on considerations about the lifetimes of species or reactions. Some computational methods (Lam \& al 1991, Maas \& al 1992) have been proposed more recently : most of them study the local behaviour of the underlying dynamical system and the reduction process involves the search for a dominant invariant subspace of the Jacobian matrix. By forgetting the ``chemical structure'' such methods are double-edged: they may be applied to any stiff system, but the counterpart is however an increasing cost of algebraic manipulations and the loss of physical understanding. These observations motivate the need for well defined conventional methods. The mathematical background of such methods is known to be provided by the singular perturbation and the center manifold theories. Many authors have already used these tools in order to justify the conventional reducing mechanisms, especially the Quasi-Steady State Assumption (QSSA). Our purpose is therefore to describe and investigate the framework of conventional methods. We present in the first section the chemical and mathematical backgrounds. We prove some results in the general monomolecular case in the second section, the most interesting point being the exhaustiveness of conventional methods for building the reduced model. In the last section we raise some questions, with the emphasis put on numerical considerations.