  If the dynamics of an evolutionary differential equation system possess a
low-dimensional, attracting, slow manifold, there are many advantages to using
this manifold to perform computations for long term dynamics, locating features
such as stationary points, limit cycles, or bifurcations. Approximating the
slow manifold, however, may be computationally as challenging as the original
problem. If the system is defined by a legacy simulation code or a microscopic
simulator, it may be impossible to perform the manipulations needed to directly
approximate the slow manifold. In this paper we demonstrate that with the
knowledge only of a set of ``slow'' variables that can be used to {\it
parameterize} the slow manifold, we can conveniently compute, using a legacy
simulator, on a nearby manifold. Forward and reverse integration, as well as
the location of fixed points are illustrated for a discretization of the
Chafee-Infante PDE for parameter values for which an Inertial Manifold is known
to exist, and can be used to validate the computational results.
