  General relativistic corrections to the expansion rate of the Universe arise
when the Einstein equations are averaged over a spatial volume in a locally
inhomogeneous cosmology. It has been suggested that they may contribute to the
observed cosmic acceleration. In this paper, we propose a new scheme that
utilizes numerical simulations to make a realistic estimate of the magnitude of
these corrections for general inhomogeneities in (3+1) spacetime. We then
quantitatively calculate the volume averaged expansion rate using N-body
large-scale structure simulations and compare it with the expansion rate in a
standard FRW cosmology. We find that in the weak gravitational field limit, the
converged corrections are slightly larger than the previous claimed 10^{-5}
level, but not large enough nor even of the correct sign to drive the current
cosmic acceleration. Nevertheless, the question of whether the cumulative
effect can significantly change the expansion history of the Universe needs to
be further investigated with strong-field relativity.
