  In this paper, an algorithm is presented to calculate the transition rates
between adjacent mesoscopic subvolumes in the presence of flow and diffusion.
These rates can be integrated in stochastic simulations of reaction-diffusion
systems that follow a mesoscopic approach, i.e., that partition the environment
into homogeneous subvolumes and apply the spatial stochastic simulation
algorithm (spatial SSA). The rates are derived by integrating Fick's second law
over a single subvolume in one dimension (1D), and are also shown to apply in
three dimensions (3D). The proposed algorithm corrects the derived rates to
ensure that they are physically meaningful and it is implemented in the AcCoRD
simulator (Actor-based Communication via Reaction-Diffusion). Simulations using
the proposed method are compared with a naive mesoscopic approach, microscopic
simulations that track every molecule, and analytical results that are exact in
1D and an approximation in 3D. By choosing subvolumes that are sufficiently
small, such that the Peclet number associated with a subvolume is sufficiently
less than 2, the accuracy of the proposed method is comparable with the
microscopic method, thus enabling the simulation of
advection-reaction-diffusion systems with the spatial SSA.
