  The efficiency of exact simulation methods for the reaction-diffusion master
equation (RDME) is severely limited by the large number of diffusion events if
the mesh is fine or if diffusion constants are large. Furthermore, inherent
properties of exact kinetic-Monte Carlo simulation methods limit the efficiency
of parallel implementations. Several approximate and hybrid methods have
appeared that enable more efficient simulation of the RDME. A common feature to
most of them is that they rely on splitting the system into its reaction and
diffusion parts and updating them sequentially over a discrete timestep. This
use of operator splitting enables more efficient simulation but it comes at the
price of a temporal discretization error that depends on the size of the
timestep. So far, existing methods have not attempted to estimate or control
this error in a systematic manner. This makes the solvers hard to use for
practitioners since they must guess an appropriate timestep. It also makes the
solvers potentially less efficient than if the timesteps are adapted to control
the error. Here, we derive estimates of the local error and propose a strategy
to adaptively select the timestep when the RDME is simulated via a first order
operator splitting. While the strategy is general and applicable to a wide
range of approximate and hybrid methods, we exemplify it here by extending a
previously published approximate method, the Diffusive Finite-State Projection
(DFSP) method, to incorporate temporal adaptivity.
