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

Autor: J. E. Bunder, A. J. Roberts
Jazyk: angličtina
Rok vydání: 2021
Předmět:
Zdroj: SN Applied Sciences, Vol 3, Iss 7, Pp 1-28 (2021)
Druh dokumentu: article
ISSN: 2523-3963
2523-3971
DOI: 10.1007/s42452-021-04229-9
Popis: 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. https://doi.org/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.
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