Patch dynamics with buffers for homogenization problems

Abstract

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.

Finite Difference Patch Dynamics for Advection Homogenization Problems

“…In this section, we briefly review some theoretical convergence results that were obtained for the parabolic homogenization problem (8), see [32] for details. In this case, we know that the order of the macroscopic equation d = 2.…”

“…For more details, including a discussion of the additional issues that need to be addressed for truly microscopic models, we refer to [32]. We emphasize that an initialization according to equation (23) has the important advantage that one can choose a suitable finite difference approximation for each derivative independently, as opposed to the method described in [21,33], which automatically leads to central finite differences.…”

“…Here, we consider situations where the macroscopic model (2) is assumed to be a partial differential equation in one space dimension, so X = x. For this type of problems, the patch dynamics scheme was proposed [21,32,33], which only performs appropriately initialized microscopic simulations in a small fraction of the space-time domain to reduce the computational cost. The general idea is the following.…”

“…We showed that the patch dynamics scheme approximates a finite difference scheme for the effective (homogenized) equation, using only the microscopic (homogenization) equation in a set of small boxes [31,32,33]. A major issue is the imposition of appropriate box boundary conditions.…”

“…A major issue is the imposition of appropriate box boundary conditions. For example, when the macroscopic behaviour is governed by diffusion, we can impose the average gradient as a boundary condition [33], or we can take arbitrary boundary conditions, provided we surround the computational boxes by buffer boxes to reduce the artefacts [32]. This latter technique is especially suited when a (e.g.…”