  Up to the next-leading order (NLO) of quantum chromodynamics (QCD), the
process $e^+e^-\to J/\psi +X$ with the center-of-mass (CM) energy range from
3.7 to 10.6 GeV is calculated. At 10.6 GeV, the results is consistent with the
experiment results at the Belle. However, the predictions are much smaller than
the measurement at BESIII at low CM energy range from 3.7 to 4.6 GeV. This
indicates that the convergence of QCD perturbative expansion becomes worse as
the CM energy becomes lower and closer to the inclusive $J/\psi$ production
threshold. For a further study of the QCD mechanism on $J/\psi$ production at
$e^+e^-$ collider with different CM energy, the initial state radiation effect
of $e^+e^-\to J/\psi+gg$ and $e^+e^-\to J/\psi+c \bar{c}$ are calculated at the
QCD NLO. The results are plotted and the numbers of events for different CM
energy bins are provided for the designed SuperKEKB. This provides a method to
precisely test the validity of perturbative prediction on $J/\psi$ production
in future measurements.
