Articles you may be interested inThe modified moment method and theory of nonlinear transport processes in gases: Third order cumulant approximation

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) 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 applications, the expansion is accurate up to and including the terms of O(ε ). 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.

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Summary. This article is concerned with the asymptotic accuracy of the Computational Singular Perturbation (CSP) method developed by Lam and Goussis [The CSP method for simplifying kinetics, Int. J. Chem. Kin. 26 (1994) to reduce the dimensionality of a system of chemical kinetics equations. The method, which is generally applicable to multiple-time scale problems arising in a broad array of scientific disciplines, exploits the presence of disparate time scales to model the dynamics by an evolution equation on a lower-dimensional slow manifold. In this article it is shown that the successive applications of the CSP algorithm generate, order by order, the asymptotic expansion of a slow manifold. The results are illustrated on the Michaelis-Menten-Henri equations of enzyme kinetics. PAC numbers. 82.33.Vx, 87.15.Rn, 82.33.Tb, 02.60.Lj, 02.30.Yy