Patch Dynamics for Multiscale Problems Bridging the Macroscale and Microscale

There are important systems—those that couple molecular dynamics with a material’s macroscale behavior or use a Boltzmann particle model to predict large-scale patterns in a fluid flow, for example—that must be modeled on relatively long time or space scales, but the dynamics can be advanced only on short time or space scales. First developed by Yannis Kevrekidis and his collaborators,1–5 patch dynamics is an efficient approach for bridging these scales. Essentially, patch dynamics uses locally averaged properties of short space–time scales to advance and predict long space–time scale dynamics. Inspired by Kevrekidis’ early success, E. Weinan and Bjorn Engquist recast the approach into an algorithm they call the heterogeneous multiscale method6 and demonstrated its applicability to additional problems. In this article, I describe how patch dynamics bridges multiple scales and provides new tools for scientists and engineers trying to predict the dynamics of long space–time scales. When successfully applied, patch dynamics can use microscopic descriptions of a problem to create a system-level framework that helps predict macroscopic properties from direct numerical simulations of relevant microscopic models.