Nonlinear emergent macroscale PDEs, with error bound, for nonlinear microscale systems

Abstract

Many multiscale physical scenarios have a spatial domain which is large in some dimensions but relatively thin in other dimensions. These scenarios includes homogenization problems where microscale heterogeneity is effectively a ‘thin dimension’. In such scenarios, slowly varying, pattern forming, emergent structures typically dominate the large dimensions. Common modelling approximations of the emergent dynamics usually rely on self-consistency arguments or on a nonphysical mathematical limit of an infinite aspect ratio of the large and thin dimensions. Instead, here we extend to nonlinear dynamics a new modelling approach which analyses the dynamics at each cross-section of the domain via a multivariate Taylor series (Roberts and Bunder in IMA J Appl Math 82(5):971–1012, 2017. 10.1093/imamat/hxx021). Centre manifold theory extends the analysis at individual cross-sections to a rigorous global model of the system’s emergent dynamics in the large but finite domain. A new remainder term quantifies the error of the nonlinear modelling and is expressed in terms of the interaction between cross-sections and the fast and slow dynamics. We illustrate the rigorous approach by deriving the large-scale nonlinear dynamics of a thin liquid film on a rotating substrate. The approach developed here empowers new mathematical and physical insight and new computational simulations of previously intractable nonlinear multiscale problems.

Linking Machine Learning with Multiscale Numerics: Data-Driven Discovery of Homogenized Equations

“…One may loosely think of this as a PDE for the locally averaged (over the fine-scale variations) material response. In applied mathematics, this problem is the purview of homogenization theory [3] (see also [5]): using asymptotic techniques, and under specific assumptions, one can derive a closed-form analytical expression for the homogenized PDE. The homogenized PDE solution is the part of the original solution that does not exhibit micro-scale variation.…”