  We perform an error analysis for numerical approximation methods of
continuous time Markov chain models commonly found in the chemistry and
biochemistry literature. The motivation for the analysis is to be able to
compare the accuracy of different approximation methods and, specifically,
Euler tau-leaping and midpoint tau-leaping. We perform our analysis under a
scaling in which the size of the time discretization is inversely proportional
to some (bounded) power of the norm of the state of the system. We argue that
this is a more appropriate scaling than that found in previous error analyses
in which the size of the time discretization goes to zero independent of the
rest of the model. Under the present scaling, we show that midpoint tau-leaping
achieves a higher order of accuracy, in both a weak and a strong sense, than
Euler tau-leaping; a result that is in contrast to previous analyses. We
present examples that demonstrate our findings.
