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) 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.

Analysis of the Computational Singular Perturbation Reduction Method for Chemical Kinetics

“…In such cases, the system of kinetic equations can be reduced to a much smaller system for the evolution of only the slow variables, and the fast variables can be determined simply by table look-ups or by direct computation. Over the years, a large number of reduction methods have been proposed and implemented in computer codes; references can be found in our earlier article [12], and additional references are [1], [7], [22].…”

“…The focus of [12] was on the Intrinsic Low-Dimensional Manifold (ILDM) method due to Maas and Pope [19] and an iterative method proposed by Fraser [6] and further developed by Roussel and Fraser [31]. In this article, the focus is on the Computational Singular Perturbation (CSP) method developed by Lam and Goussis [8], [9], [13], [15], [16], [17], [18], [20], [21], [32].…”

“…The goal of the one-step CSP method is to reduce the matrix to lower block-triangular form-that is, to push the matrix 12 to increasingly higher order in ε. The method is identical to the full CSP method except for the updating of the matrices A and B.…”

“…In [12], we showed that the ILDM method yields an approximate slow manifold that is asymptotically accurate up to and including terms of O(ε), with an error of O(ε 2 ) proportional to the curvature of M 0 . The CSP method, on the other hand, can generate an approximate slow manifold that is asymptotically accurate up to any order.…”

Two perspectives on reduction of ordinary differential equations

“…Many techniques have been proposed for this purpose, especially in the chemical kinetics literature; see the references in [8,22,23]. Generally, they involve the construction of a coordinate system where the slow manifold has a simple functional representation-the ideal system being one where a number of components (the "fast" components) of the state vector are identically zero, as in the Fenichel normal form [7].…”