  Many physical systems are well described on domains which are relatively
large in some directions but relatively thin in other directions. In this
scenario we typically expect the system to have emergent structures that vary
slowly over the large dimensions. For practical mathematical modelling of such
systems we require efficient and accurate methodologies for reducing the
dimension of the original system and extracting the emergent dynamics. Common
mathematical approximations for determining the emergent dynamics often rely on
self-consistency arguments or limits as the aspect ratio of the 'large' and
'thin' dimensions becomes unphysically infinite. Here we build on a new
approach, previously establish for systems which are large in only one
dimension, which analyses the dynamics at each cross-section of the domain with
a rigorous multivariate Taylor series. Then centre manifold theory supports the
local modelling of the system's emergent dynamics with coupling to neighbouring
cross-sections treated as a non-autonomous forcing. The union over all
cross-sections then provides powerful support for the existence and emergence
of a centre manifold model global in the large finite domain. Quantitative
error estimates are determined from the interactions between the cross-section
coupling and both fast and slow dynamics. Two examples provide practical
details of our methodology. 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.
