  Mapping the recent expansion history of the universe offers the best hope for
uncovering the characteristics of the dark energy believed to be responsible
for the acceleration of the expansion. In determining cosmological and
dark-energy parameters to the percent level, systematic uncertainties impose a
floor on the accuracy more severe than the statistical measurement precision.
We delineate the categorization, simulation, and understanding required to
bound systematics for the specific case of the Type Ia supernova method. Using
simulated data of forthcoming ground-based surveys and the proposed space-based
SNAP mission we present Monte Carlo results for the residual uncertainties on
the cosmological parameter determination. The tight systematics control with
optical and near-infrared observations and the extended redshift reach allow a
space survey to bound the systematics below 0.02 magnitudes at z=1.7. For a
typical SNAP-like supernova survey, this keeps total errors within 15% of the
statistical values and provides estimation of Omega_m to 0.03, w_0 to 0.07, and
w' to 0.3; these can be further improved by incorporating complementary data.
