  In this work we investigate the behavior of the microcanonical and canonical
averages of the two-dimensional Ising model during the Wang-Landau simulation.
The simulations were carried out using conventional Wang-Landau sampling and
the $1/t$ scheme. Our findings reveal that the microcanonical average should
not be accumulated during the initial modification factors \textit{f} and
outline a criterion to define this limit. We show that updating the density of
states only after every $L^2$ spin-flip trials leads to a much better
precision. We present a mechanism to determine for the given model up to what
final modification factor the simulations should be carried out. Altogether
these small adjustments lead to an improved procedure for simulations with much
more reliable results. We compare our results with $1/t$ simulations. We also
present an application of the procedure to a self-avoiding homopolymer.
