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Consider the macroscale modelling of microscale spatiotemporal dynamics. Herewe develop a new approach to ensure coarse scale discrete models preserveimportant self-adjoint properties of the fine scale dynamics. The first partexplores the discretisation of microscale continuum dynamics. The secondaddresses how dynamics on a fine lattice are mapped to lattice a factor of twocoarser (as in multigrids). Such mapping of discrete lattice dynamics may beiterated to empower us in future research to explore scale dependent emergentphenomena. The support of dynamical systems, centre manifold, theory ensuresthat the coarse scale modelling applies with a finite spectral gap, in a finitedomain, and for all time. The accuracy of the models is limited by theasymptotic resolution of subgrid coarse scale processes, and is controlled bythe level of truncation. As given examples demonstrate, the novel feature ofthe approach developed here is that it ensures the preservation of importantconservation properties of the microscale dynamics.

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