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Many physical systems are well described on domains which are relativelylarge in some directions but relatively thin in other directions. In thisscenario we typically expect the system to have emergent structures that varyslowly over the large dimensions. For practical mathematical modelling of suchsystems we require efficient and accurate methodologies for reducing thedimension of the original system and extracting the emergent dynamics. Commonmathematical approximations for determining the emergent dynamics often rely onself-consistency arguments or limits as the aspect ratio of the 'large' and'thin' dimensions becomes unphysically infinite. Here we build on a newapproach, previously establish for systems which are large in only onedimension, which analyses the dynamics at each cross-section of the domain witha rigorous multivariate Taylor series. Then centre manifold theory supports thelocal modelling of the system's emergent dynamics with coupling to neighbouringcross-sections treated as a non-autonomous forcing. The union over allcross-sections then provides powerful support for the existence and emergenceof a centre manifold model global in the large finite domain. Quantitativeerror estimates are determined from the interactions between the cross-sectioncoupling and both fast and slow dynamics. Two examples provide practicaldetails of our methodology. The approach developed here may be used to quantifythe accuracy of known approximations, to extend such approximations to mixedorder modelling, and to open previously intractable modelling issues to newtools and insights.

Slowly varying, macroscale models emerge from microscale dynamics over multiscale domains