Analysis of the approximate slow invariant manifold method for reactive flow equations

The Approximate Slow Invariant Manifold method of Singh, Powers and Paolucci is a useful method for addressing model reduction in systems of reactive flow equations. It exploits separations of time scales between slow and fast species, and it generalizes the Intrinsic Low-Dimensional Manifold method, which was developed for model reduction in the context of reaction kinetics, to systems with diffusive and active transport. In this article, we present a mathematical analysis of the Approximate Slow Invariant Manifold method in the context of systems of reaction–diffusion equations with slow and fast reaction kinetics. Beginning with systems of two species (one slow and one fast), and then treating general systems with multiple slow and fast species, we explicitly determine the accuracy of the Approximate Slow Invariant Manifold method. We find that it is correct up to and including the terms of first order in the small parameter that measures the separation of the kinetics time scales, and that it captures many of the terms at second order, as well. Our analysis includes precise statements of the errors at second order, and we find that these are proportional to the slow components of the reaction–diffusion equation, as well as to the curvature of the critical manifold. We illustrate the results analytically on two prototypical examples, the Michaelis–Menten–Henri model with diffusion of the slow species and the Davis–Skodje model in which both the slow and fast species diffuse.