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Equation-free patch scheme for efficient computational homogenisation via self-adjoint coupling

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Abstract

Equation-free macroscale modelling is a systematic and rigorous computational methodology for efficiently predicting the dynamics of a microscale complex system at a desired macroscale system level. In this scheme, a given microscale model is computed in small patches spread across the space-time domain, with patch coupling conditions bridging the unsimulated space. For accurate predictions, care must be taken in designing the patch coupling conditions. Here we construct novel coupling conditions which preserve self-adjoint symmetry, thus guaranteeing that the macroscale model maintains some important conservation laws of the original microscale model. Consistency of the patch scheme’s macroscale dynamics with the original microscale model is proved for systems in 1D and 2D space, and these proofs immediately extend to higher dimensions. Expanding from a system with a single configuration to an ensemble of configurations establishes that the proven consistency also holds for cases where the microscale periodicity does not integrally fill the patches. This new self-adjoint patch scheme provides an efficient, flexible, and accurate computational homogenisation, as demonstrated here with canonical examples in 1D and 2D space based on heterogenous diffusion, and is applicable to a wide range of multiscale scenarios of interest to scientists and engineers.

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Notes

  1. Often \({\mathcal {L}}\) is a real matrix, and then this self-adjoint property is the usual matrix symmetry.

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Funding

This research was funded by the Australian Research Council under grants DP150102385 and DP200103097. The work of I.G.K. was also partially supported by the DARPA PAI program.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Adelaide, Adelaide, South Australia, 5005, Australia

    J. E. Bunder & A. J. Roberts

  2. Departments of Chemical and Biomolecular Engineering and Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, Maryland, USA

    I. G. Kevrekidis

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  1. J. E. Bunder
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Correspondence to J. E. Bunder.

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Appendix A provides computer algebra code; numerical code is available from the GitHub Equation-Free Matlab/Octave Toolbox [41]

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A Macroscale homogenised 1D diffusion

A Macroscale homogenised 1D diffusion

This is computer algebra script homo1Ddiff.txt for Sect. 3.1.

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Bunder, J.E., Kevrekidis, I.G. & Roberts, A.J. Equation-free patch scheme for efficient computational homogenisation via self-adjoint coupling. Numer. Math. 149, 229–272 (2021). https://doi.org/10.1007/s00211-021-01232-5

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