Title: Couple microscale periodic patches to simulate macroscale emergent dynamics,
Abstract: This article proposes a new way to construct computationally efficient
`wrappers' around fine scale, microscopic, detailed descriptions of dynamical
systems, such as molecular dynamics, to make predictions at the macroscale
`continuum' level. It is often significantly easier to code a microscale
simulator with periodicity: so the challenge addressed here is to develop a
scheme that uses only a given periodic microscale simulator; specifically, one
for atomistic dynamics. Numerical simulations show that applying a suitable
proportional controller within `action regions' of a patch of atomistic
simulation effectively predicts the macroscale transport of heat. Theoretical
analysis establishes that such an approach will generally be effective and
efficient, and also determines good values for the strength of the proportional
controller. This work has the potential to empower systematic analysis and
understanding at a macroscopic system level when only a given microscale
simulator is available.