Better buffers for patches in macroscale simulation of systems with microscale randomness

We consider one dimensional lattice diffusion model on a microscale grid with many discrete diffusivity values which repeat periodicially. Computer algebra explores how the dynamics of small coupled `patches' predict the slow emergent macroscale dynamics. We optimise the geometry and coupling of patches by comparing the macroscale predictions of the patch solutions with the macroscale solution on the infinite domain, which is derived for a general diffusivity period. The results indicate that patch dynamics is a viable method for numerical macroscale modelling of microscale systems with fine scale roughness. Moreover, the minimal error on the macroscale is generally obtained by coupling patches via `buffers' that are as large as half of each patch.