Asymptotic analysis of two reduction methods for systems of chemical reactions

Abstract

This paper concerns two methods for reducing large systems of chemical kinetics equations, namely, the method of intrinsic low-dimensional manifolds (ILDMs) due to Maas and Pope [Combust. Flame 88 (1992) 239] and an iterative method due to Fraser [J. Chem. Phys. 88 (1988) 4732] and further developed by Roussel and Fraser [J. Chem. Phys. 93 (1990) 1072]. Both methods exploit the separation of fast and slow reaction time scales to find low-dimensional manifolds in the space of species concentrations where the long-term dynamics are played out. The asymptotic expansions of these manifolds ( ε↓0, where ε measures the ratio of the reaction time scales) are compared with the asymptotic expansion of M ε , the slow manifold given by geometric singular perturbation theory. It is shown that the expansions of the ILDM and M ε agree up to and including terms of O(ε) ; the former has an error at O(ε 2) that is proportional to the local curvature of M 0 . The error vanishes if and only if the curvature is zero everywhere. The iterative method generates, term by term, the asymptotic expansion of M ε . Starting from M 0 , the ith application of the algorithm yields the correct expansion coefficient at O(ε i) , while leaving the lower-order coefficients invariant. Thus, after ℓ applications, the expansion is accurate up to and including the terms of O(ε ℓ) . The analytical results are illustrated on a planar system from enzyme kinetics (Michaelis–Menten–Henri) and a model planar system due to Davis and Skodje.