  It has been established that there is an inherent limit to the accuracy of
the reaction-diffusion master equation. Specifically, there exists a
fundamental lower bound on the mesh size, below which the accuracy deteriorates
as the mesh is refined further. In this paper we extend the standard
reaction-diffusion master equation to allow molecules occupying neighboring
voxels to react, in contrast to the traditional approach in which molecules
react only when occupying the same voxel. We derive reaction rates, in two
dimensions as well as three dimensions, to obtain an optimal match to the more
fine-grained Smoluchowski model, and show in two numerical examples that the
extended algorithm is accurate for a wide range of mesh sizes, allowing us to
simulate systems intractable with the standard reaction-diffusion master
equation. In addition, we show that for mesh sizes above the fundamental lower
limit of the standard algorithm, the generalized algorithm reduces to the
standard algorithm. We derive a lower limit for the generalized algorithm,
which, in both two dimensions and three dimensions, is on the order of the
reaction radius of a reacting pair of molecules.
