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

Abstract

Consider the macroscale modelling of microscale spatio-temporal dynamics. Here we develop an approach to ensure coarse scale discrete models preserve important self-adjoint properties of the microscale dynamics. The first part explores the discrete modelling of microscale continuum dynamics in multiple spatial dimensions. The second part addresses how dynamics on a fine lattice are mapped to lattice a factor of two coarser (as in multigrids); for simplicity we address only one-dimensional lattices. Such mapping of discrete lattice dynamics may be iterated to empower future research to explore scale dependent emergent phenomena. The support of the dynamical systems theory of centre manifolds ensures that the coarse scale modelling applies with a finite spectral gap, in a finite domain, and for all time. The accuracy of the modelling is limited by the asymptotic resolution of subgrid scale processes. As given examples demonstrate, the approach developed here ensures the preservation of important symmetries of the microscale dynamics.