  We present a simulated cosmology analysis using the second and third moments
of the weak lensing mass (convergence) maps. The analysis is geared towards the
third year (Y3) data from the Dark Energy Survey (DES), but the methodology can
be applied to other weak lensing data sets. The second moment, or variances, of
the convergence as a function of smoothing scale contains information similar
to standard shear 2-point statistics. The third moment, or the skewness,
contains additional non-Gaussian information. We present the formalism for
obtaining the convergence maps from the measured shear and for obtaining the
second and third moments of these maps given partial sky coverage. We estimate
the covariance matrix from a large suite of numerical simulations. We test our
pipeline through a simulated likelihood analyses varying 5 cosmological
parameters and 10 nuisance parameters. Our forecast shows that the combination
of second and third moments provides a 1.5 percent constraint on $S_8 \equiv
\sigma_8 (\Omega_{\rm m}/0.3)^{0.5}$ for DES Y3 data. This is 20 percent better
than an analysis using a simulated DES Y3 shear 2-point statistics, owing to
the non-Gaussian information captured by the inclusion of higher-order
statistics. The methodology developed here can be applied to current and future
weak lensing datasets to make use of information from non-Gaussianity in the
cosmic density field and to improve constraints on cosmological parameters.
