  The second-order formula of Minkowski functionals in weakly non-Gaussian
fields is compared with the numerical $N$-body simulations. Recently, weakly
non-Gaussian formula of Minkowski functionals is extended to include the
second-order effects of non-Gaussianity in general dimensions. We apply this
formula to the three-dimensional density field in the large-scale structure of
the Universe. The parameters of the second-order formula include several kinds
of skewness and kurtosis parameters. We apply the tree-level nonlinear
perturbation theory to estimate these parameters. First we compare the
theoretical values with those of numerical simulations on the basis of
parameter values, and next we test the performance of the analytic formula
combined with the perturbation theory. The second-order formula outperforms the
first-order formula in general. The performance of the perturbation theory
depends on the smoothing radius applied in defining the Minkowski functionals.
The quantitative comparisons are presented in detail.
