Convergence of methods for coupling of microscopic and mesoscopic
reaction-diffusion simulations

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: (i) \Delta t approaches 0
and h is fixed; and (ii) \Delta t approaches 0 and h approaches 0 such that
\Delta t/h^2 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.