  Error estimates for the numerical solution of the master equation are
presented. Estimates are based on adjoint methods. We find that a good estimate
can often be computed without spending computational effort on a dual problem.
Estimates are applicable to both settings with time-independent, and
time-dependent propensity functions. The Finite State Projection algorithm
reduces the dimensionality of the problem and time propagation is based on an
Arnoldi exponential integrator, which in the time-dependent setting is combined
with a Magnus method. Local error estimates are devised for the truncation of
both the Magnus expansion and the Krylov subspace in the Arnoldi algorithm. An
issue with existing methods is that error estimates for truncation of the state
space depend on measuring a loss of probability mass in a way that is not
usually compatible with the approximation of the exponential. We suggest an
alternative error estimate that is compatible with a Krylov approximation to
the matrix exponential. Finally, we apply the new error estimates to develop an
adaptive simulation algorithm. Numerical examples demonstrate the benefits of
the approach.
