Antonios Zagaris

Analysis of the Computational Singular Perturbation Reduction Method for Chemical Kinetics

Abstract 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) 461–486] 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.

Fast and Slow Dynamics for the Computational Singular Perturbation Method

The computational singular perturbation (CSP) method ofLam and Goussis is an iterative method to reduce the dimensionality of systems of ordinary differential equations with multiple time scales. In [J. Nonlinear Sci., 14 (2004), pp. 59--91], the authors of this paper showed that each iteration of the CSP algorithm improves the approximation of the slow manifold by one order. In this paper, it is shown that the CSP method simultaneously approximates the tangent spaces to the fast fibers along which solutions relax to the slow manifold. Again, each iteration adds one order of accuracy. In some studies, the output of the CSP algorithm is postprocessed by linearly projecting initial data onto the slow manifold along these approximate tangent spaces. These projections, in turn, also become successively more accurate.

Analysis of the accuracy and convergence of equation-free projection to a slow manifold

In [C.W. Gear, T.J. Kaper, I.G. Kevrekidis and A. Zagaris, SIAM J. Appl. Dyn. Syst. 4 (2005) 711–732], we developed a class of iterative algorithms within the context of equation-free methods to approximate low-dimensional, attracting, slow manifolds in systems of differential equations with multiple time scales. For user-specified values of a finite number of the observables, the mth member of the class of algorithms (

Simplified yet highly accurate enzyme kinetics for cases of low substrate concentrations.

m = 0, 1, \ldots

Blooming in a nonlocal, coupled phytoplankton-nutrient model

) finds iteratively an approximation of the appropriate zero of the (m+1)st time derivative of the remaining variables and uses this root to approximate the location of the point on the slow manifold corresponding to these values of the observables. This article is the first of two articles in which the accuracy and convergence of the iterative algorithms are analyzed. Here, we work directly with fast-slow systems, in which there is an explicit small parameter,

The effect of slow spatial processes on emerging spatiotemporal patterns

\varepsilon

Emergence of steady and oscillatory localized structures in a phytoplankton–nutrient model

, measuring the separation of time scales. We show that, for each

Clusters of reaction rates and concentrations in protein networks such as the phosphotransferase system

m = 0, 1, \ldots

Projecting to a Slow Manifold: Singularly Perturbed Systems and Legacy Codes

, the fixed point of the iterative algorithm approximates the slow manifold up to and including terms of

Two perspectives on reduction of ordinary differential equations

{\mathcal O}(\varepsilon^m)