An efficient iterative algorithm for the approximation of the fast and slow dynamics of stiff systems

Abstract

The relation between the iterative algorithms based on the computational singular perturbation (CSP) and the invariance equation (IE) methods is examined. The success of the two methods is based on the appearance of fast and slow time scales in the dynamics of stiff systems. Both methods can identify the low-dimensional surface in the phase space (slow invariant manifold, SIM), where the state vector is attracted under the action of fast dynamics. It is shown that this equivalence of the two methods can be expressed by simple algebraic relations. CSP can also construct the simplified non-stiff system that models the slow dynamics of the state vector on the SIM. An extended version of IE is presented which can also perform this task. This new IE version is shown to be exactly similar to a modified version of CSP, which results in a very efficient algorithm, especially in cases where the SIM dimension is small, so that significant model simplifications are possible.