@TechReport{ it:2013-010,
author = {Mark B. Flegg and Stefan Hellander and Radek Erban},
title = {Convergence of Methods for Coupling of Microscopic and
Mesoscopic Reaction-Diffusion Simulations},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2013},
number = {2013-010},
month = apr,
abstract = {In this paper, three multiscale methods for coupling of
mesoscopic (compartment-based) and microscopic
(molecular-based) stochastic reaction-diffusion simulations
are investigated. Two of the three methods that will be
discussed in detail have been previously reported in the
literature; the two-regime method (TRM) and the
compartment-placement method (CPM). The third method that
is introduced and analysed in this paper is the ghost cell
method (GCM). Presented is a comparison of sources of
error. The convergent properties of this error are studied
as the time step $\Delta t$ (for updating the
molecular-based part of the model) approaches zero. It is
found that the error behaviour depends on another
fundamental computational parameter $h$, the compartment
size in the mesoscopic part of the model. Two important
limiting cases, which appear in applications, are
considered: \par $\,$(i) $\Delta t \to 0$ and $h$ is fixed;
\par (ii) $\Delta t \to 0$ and $h \rightarrow 0$ such that
$\sqrt{\Delta t}/h$ is fixed. \\ The error for previously
developed approaches (the TRM and CPM) converges to zero
only in the limiting case (ii), but not in case (i). It is
shown that the error of the GCM converges in the limiting
case (i). Thus the GCM is superior to previous coupling
techniques if the mesoscopic description is much coarser
than the microscopic part of the model.}
}