  Weak lensing peak counts are a powerful statistical tool for constraining
cosmological parameters. So far, this method has been applied only to surveys
with relatively small areas, up to several hundred square degrees. As future
surveys will provide weak lensing datasets with size of thousands of square
degrees, the demand on the theoretical prediction of the peak statistics will
become heightened. In particular, large simulations of increased cosmological
volume are required. In this work, we investigate the possibility of using
simulations generated with the fast Comoving-Lagrangian acceleration (COLA)
method, coupled to the convergence map generator Ufalcon, for predicting the
peak counts. We examine the systematics introduced by the COLA method by
comparing it with a full TreePM code. We find that for a 2000 deg$^2$ survey,
the systematic error is much smaller than the statistical error. This suggests
that the COLA method is able to generate promising theoretical predictions for
weak lensing peaks. We also examine the constraining power of various
configurations of data vectors, exploring the influence of splitting the sample
into tomographic bins and combining different smoothing scales. We find the
combination of smoothing scales to have the most constraining power, improving
the constraints on the $S_8$ amplitude parameter by at least 40% compared to a
single smoothing scale, with tomography brining only limited increase in
measurement precision.
