  To simulate thermalizing systems at long times, the most straightforward
approach is to calculate the thermal properties at the corresponding energy. In
a quantum many-body system of size $N$, for local observables and many initial
states, this approach has an error of $O(1/N)$, which is reminiscent of the
finite-size error of the equivalence of ensembles. In this paper, we propose a
simple and efficient numerical method so that the simulation error is of higher
order in $1/N$. This finite-size error scaling is proved by assuming the
eigenstate thermalization hypothesis.
