  Many practical approximations in physics and engineering invoke a relatively
long physical domain with a relatively thin cross-section. In this scenario we
typically expect the system to have structures that vary slowly in the long
dimension. Extant mathematical approximation methodologies are typically
self-consistency or limit arguments as the aspect ratio becomes unphysically
infinite. The proposed new approach is to analyse the dynamics based at each
cross-section in a rigorous Taylor polynomial. Then centre manifold theory
supports the local modelling of the system's dynamics with coupling to
neighbouring locales treated as a non-autonomous forcing. The union over all
cross-sections then provides powerful new support for the existence and
emergence of a centre manifold model global in the long domain, albeit finite
sized. Our resolution of the coupling between neighbouring locales leads to
novel quantitative estimates of the error induced by long slow space
variations. Four examples help develop and illustrate the approach and results.
The approach developed here may be used to quantify the accuracy of known
approximations, to extend such approximations to mixed order modelling, and to
open previously intractable modelling issues to new tools and insights.
