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

We consider dynamical systems possessing an attracting, invariant "slow manifold" that can be parameterized by a few observable variables. We present a procedure that, given a process for integrating the system step by step and a set of values of the observables, finds the values of the remaining system variables such that the state is close to the slow manifold to some desired accuracy. It should be noted that this is not equivalent to "integrating down to the manifold" since the latter process may significantly change the values of the observables. 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 (because the system is not necessarily expressed in a singular perturbation form). 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 a 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. This procedure can be valuable when one wishes to start a simulation of the detailed model on the slow manifold with particular values of observable variables characterizing the slow manifold.