Higher Order Corrections in the Approximation of Low Dimensional Manifolds and the Construction of Simplified Problems with the CSP Method

In systems of stiff Ordinary Differential Equations (ODEs) both fast and slow time scales are encountered. The fast time scales are responsible for the development of low-dimensional manifolds on which the solution moves according to the slow time scales. In this paper, methodologies for constructing highly accurate (i) expressions describing the manifold, and (ii) simplified non-stiff equations governing the slow evolution of the solution on the manifold are developed, according to an iterative procedure proposed in the Computational Singular Perturbation (CSP) method. It is shown that the increasing accuracy achieved with each iteration is directly related to the time rates of change of the CSP vectors spanning the manifold along the solution trajectory. Here, an algorithm is presented which implements these calculations and is validated on the basis of two simple examples.