The long-term dynamics of many dynamical systems evolve on an attracting, invariant "slow manifold" that can be parameterized by a few observable variables. Yet a simulation using the full model of the problem requires initial values for all variables. Given a set of values for the observables parameterizing the slow manifold, one needs a procedure for finding the additional values such that the state is close to the slow manifold to some desired accuracy. We consider problems whose solution has a singular perturbation expansion, although we do not know what it is nor have any way to compute it. We show in this paper that, under some conditions, computing the values of the remaining variables so that their (m + 1)st time derivatives are zero provides an estimate of the unknown variables that is an mth-order approximation to a point on the slow manifold in sense to be defined. We then show how this criterion can be applied approximately when the system is defined by a legacy code rather than directly through closed form equations.

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

Abstract. 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 (m = 0, 1, . . .) 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, ε, measuring the separation of time scales. We show that, for each m = 0, 1, . . ., the fixed point of the iterative algorithm approximates the slow manifold up to and including terms of O(ε m ). Moreover, for each m, we identify explicitly the conditions under which the mth iterative algorithm converges to this fixed point. Finally, we show that when the iteration is unstable (or converges slowly) it may be stabilized (or its convergence may be accelerated) by application of the Recursive Projection Method. Alternatively, the Newton-Krylov Generalized Minimal Residual Method may be used. In the subsequent article, we will consider the accuracy and convergence of the iterative algorithms for a broader class of systems -in which there need not be an explicit small parameter -to which the algorithms also apply.

The Computational Singular Perturbation (CSP) method of Lam and Goussis is an iterative method to reduce the dimensionality of systems of ordinary differential equations with multiple time scales. In [J. Nonlin. Sci., to appear], the authors 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.