  In modern cosmology, the precision of the theoretical prediction is
increasingly required. In cosmological $N$-body simulations, the effect of
higher-order Lagrangian perturbation on the initial conditions appears in terms
of statistical quantities of matter density field. We have considered the
effect of third-order Lagrangian perturbation (3LPT) on the initial conditions,
which can be applied to Gadget-2 code. Then, as statistical quantities,
non-Gaussianity of matter density field has been compared between cases of
different order perturbations for the initial conditions. Then, we demonstrate
the validity of the initial conditions with second-order Lagrangian
perturbation (2LPT).
