Finite difference patch dynamics for advection homogenization problems

Abstract

We consider problems in which there is a separation between the (microscopic) scale at which the available model is defined, and the (macroscopic) scale of interest. For time-dependent multi-scale problems of this type, 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 only uses appropriately initialized simulations of the available microscopic model in a number of small boxes (patches), which cover a fraction of the space-time domain. We review some recent convergence results and demonstrate that the method allows to simulate advection-dominated problems accurately. © 2006 Springer-Verlag Berlin Heidelberg.