Model dynamics on a multigrid across multiple length and time scales

TL;DR

This paper introduces a multigrid modeling approach for complex dynamics across many space-time scales, enabling flexible and rigorous analysis of systems with a continuum of scales beyond traditional two-scale models.

Contribution

It presents a novel hierarchical method for discretized advection and diffusion dynamics that supports scale changes by a factor of two with rigorous validation across multiple levels.

Findings

Hierarchical multigrid models effectively capture a wide range of space-time scales.

The approach provides rigorous support for scale transitions in discretized dynamics.

Flexibility in modeling complex systems across multiple scales is demonstrated.

Abstract

Most methods for modelling 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 discuss an approach to modelling the discretised dynamics of advection and diffusion with rigorous support for changing the resolved spatial grid scale by just a factor of two. The mapping of dynamics from a finer grid to a coarser grid is then iterated to generate a hierarchy of models across a wide range of space-time scales, all with rigorous support across the whole hierarchy. This approach empowers us with great flexibility in modelling complex dynamics over multiple scales.