Asymptotic Analysis of Two Reduction Methods for Systems of Chemical Reactions

TitleAsymptotic Analysis of Two Reduction Methods for Systems of Chemical Reactions
Publication TypeReport
Year of Publication2001
AuthorsKaper, HG, Kaper, TJ
Series TitlePhysica D
Date Published10/2001
Other NumbersANL/MCS-P912-1001

<p>This article concerns two methods for reducing large systems of chemical kinetics equations, namely, the method of intrinsiclow-dimensional manifolds (ILDMs) due to Maas and Pope [U. Maas and S. B. Pope, Combustion and Flame 88 (1992) 239-264] and an iterative method due to Fraser [S. J. Fraser, J. Chem. Phys. 88 (1988) 4732-4738] and further developed by Roussel and Fraser. 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 analysis is carried out in the context of systems of ordinary differential equations with multiple time scales and geometric singular perturbation theory (GSPT). A small parameter e measures the separation of time scales. The underlying assumption is that the system of equations has an asymptotically stable slow manifold M0 in the limit as e � 0. Then it follows from GSPT that there exists a slow manifold Me for all sufficiently small positive e, which is asymptotically close to M0. It is shown that the ILDM method yields a low-dimensional manifold whose asymptotic expansion agrees with the asymptotic expansion of Me up to and including terms of O(e). At O(e2), an error appears that is proportional to the local curvature of M0; it vanishes if and only if the curvature is zero everywhere. The iterative method generates, term by term, the asymptotic expansion of the slow manifold Me. Starting from M0, the ith application of the algorithm yields the correct expansion coefficient at O(e i-1) invariant. Thus, after l applications, the expansion is accurate up to and including the terms of O(el). The analytical results are illustrated in two examples: a planar system from enzyme kinetics (Michaelis-Menten-Henri) and a model planar system due to Davis and Skodje.</p>