Choose interelement coupling to preserve self-adjoint dynamics in multiscale modelling and computation

TL;DR

This paper introduces a new multiscale modelling approach that preserves self-adjoint properties of microscale dynamics in coarse-scale models, enabling accurate and consistent simulation of emergent phenomena.

Contribution

The paper develops a novel discretisation and mapping method that maintains self-adjoint properties across scales, enhancing the fidelity of multiscale models.

Findings

Preserves conservation properties in coarse models

Ensures finite spectral gap for stability

Applicable to iterative multiscale lattice mappings

Abstract

Consider the macroscale modelling of microscale spatiotemporal dynamics. Here we develop a new approach to ensure coarse scale discrete models preserve important self-adjoint properties of the fine scale dynamics. The first part explores the discretisation of microscale continuum dynamics. The second addresses how dynamics on a fine lattice are mapped to lattice a factor of two coarser (as in multigrids). Such mapping of discrete lattice dynamics may be iterated to empower us in future research to explore scale dependent emergent phenomena. The support of dynamical systems, centre manifold, theory ensures that the coarse scale modelling applies with a finite spectral gap, in a finite domain, and for all time. The accuracy of the models is limited by the asymptotic resolution of subgrid coarse scale processes, and is controlled by the level of truncation. As given examples demonstrate, the novel feature of the approach developed here is that it ensures the preservation of important conservation properties of the microscale dynamics.