  Many physical systems are formulated on domains which are relatively large in
some directions but relatively thin in other directions. We expect such systems
to have emergent structures that vary slowly over the large dimensions. 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 nonphysically infinite. Here we extend to nonlinear
dynamics a new approach [IMA J. Appl. Maths, DOI: 10.1093/imamat/hxx021] which
analyses the dynamics at each cross-section of the domain via a rigorous
multivariate Taylor series. Then centre manifold theory supports the global
modelling of the system's emergent dynamics in the large but finite domain.
Interactions between the cross-section coupling and both fast and slow dynamics
determines quantitative error bounds for the nonlinear modelling. We illustrate
the methodology by deriving the large-scale dynamics of a thin liquid film,
where the film is subject to a Coriolis force induced by a rotating substrate.
The approach developed here quantifies the accuracy of known approximations,
extends such approximations to mixed order modelling, and may open previously
intractable modelling issues to new tools and insights.
