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

TL;DR

This paper introduces a rigorous method for deriving macroscale models from microscale dynamics in long, thin domains using Taylor polynomial approximations and centre manifold theory, providing new error estimates and broad applicability.

Contribution

It develops a novel approach combining Taylor polynomial analysis and centre manifold theory to rigorously derive macroscale models in long, thin domains, with quantifiable error bounds.

Findings

Provides a new framework for macroscale modeling in long domains.

Offers quantitative error estimates for slow variation approximations.

Demonstrates the approach with four illustrative examples.

Abstract

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.