  An important class of problems exhibits smooth behaviour on macroscopic space
and time scales, while only a microscopic evolution law is known. For such
time-dependent multi-scale problems, an "equation-free" framework has been
proposed, of which patch dynamics is an essential component. Patch dynamics is
designed to perform numerical simulations of an unavailable macroscopic
equation on macroscopic time and length scales; it uses appropriately
initialized simulations of the available microscopic model in a number of small
boxes (patches), which cover only a fraction of the space-time domain. To
reduce the effect of the artificially introduced box boundaries, we use buffer
regions to "shield" the boundary artefacts from the interior of the domain for
short time intervals. We analyze the accuracy of this scheme for a diffusion
homogenization problem with periodic heterogeneity, and propose a simple
heuristic to determine a sufficient buffer size. The algorithm performance is
illustrated through a set of numerical examples, which include a non-linear
reaction-diffusion equation and the Kuramoto--Sivashinsky equation.
