Model dynamics across multiple length and time scales on a spatial multigrid

Abstract

Most methods for modeling dynamics posit just two time scales: a fast and a slow scale. But many applications, including many in continuum mechanics, possess a wide variety of space-time scales. Often they possess a continuum of space-time scales. I develop an approach to analytically model the spatially discretized dynamics of advection and diffusion with rigorous support for changing the resolved spatial grid scale by just a factor of two. The analytic mapping of dynamics from a finer grid to a coarser grid is then iterated to generate a hierarchy of models on a spatial multigrid across a wide range of space-time scales, all with rigorous support across the whole hierarchy. This approach will empower us with great flexibility in modeling complex dynamics over multiple scales and promises to provide sound analytical closures for numerical simulations.