  Near-future cosmological observations targeted at investigations of dark
energy pose stringent requirements on the accuracy of theoretical predictions
for the clustering of matter. Currently, N-body simulations comprise the only
viable approach to this problem. In this paper we demonstrate that N-body
simulations can indeed be sufficiently controlled to fulfill these requirements
for the needs of ongoing and near-future weak lensing surveys. By performing a
large suite of cosmological simulation comparison and convergence tests we show
that results for the nonlinear matter power spectrum can be obtained at 1%
accuracy out to k~1 h/Mpc. The key components of these high accuracy
simulations are: precise initial conditions, very large simulation volumes,
sufficient mass resolution, and accurate time stepping. This paper is the first
in a series of three, with the final aim to provide a high-accuracy prediction
scheme for the nonlinear matter power spectrum.
