Equation-free computation: an overview of patch dynamics

Abstract

<jats:title>Abstract</jats:title><jats:p>This chapter overviews recent progress in the development of patch dynamics, an essential ingredient of the equation-free framework. In many applications we have a given detailed microscopic numerical simulator that we wish to use over macroscopic scales. Patch dynamics uses only simulations within a number of small regions (surrounding macroscopic grid points) in the space-time domain to approximate a discretization scheme for an unavailable macroscopic equation. The approach was first presented and analyzed for a standard diffusion problem in one space dimension; here, we will discuss subsequent efforts to generalize the approach and extend its analysis. We show how one can modify the definition of the initial and boundary conditions to allow patch dynamics to mimic any finite difference scheme, and we investigate to what extent (and at what computational cost) one can avoid the need for specifically designed patch boundary conditions. One can surround the patches with buffer regions, where one can impose (to some extent) arbitrary boundary conditions. The convergence analysis shows that the required buffer for consistency depends on the coefficients in the macroscopic equation; in general, for advection dominated problems, smaller buffer regions–as compared to those for diffusion-dominated problems–suffice.</jats:p>