Macroscale, slowly varying, models emerge from the microscale dynamics in long thin domains

Abstract

Many practical approximations in science 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 approximation methodologies are typically either self-consistency or limits 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. 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 of finite size. Our resolution of the coupling between neighbouring locales leads to novel quantitative estimates of the error induced by long slow space variations. Two examples help us 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.