  In order to clarify the puzzling problems in double charm production,
relativistic corrections at order $v^{2}$ to the processes $e^{+}e^{-}\to
J/\psi+\eta_{c}$ and $e^{+}e^{-}\to J/\psi+c\bar{c}$ at B factories are studied
in non-relativistic quantum chromodynamics. The short-distance parts of
production cross sections are calculated perturbatively, while the
long-distance matrix elements are estimated from $J/\psi$ and $\eta_c$ decays
up to errors of order $v^4$. Our results show that the relativistic correction
to the exclusive process $e^{+}e^{-}\to J/\psi+\eta_{c}$ is significant, which,
when combined together with the next-to-leading order $\alpha_{s}$ corrections,
could resolve the large discrepancy between theory and experiment; whereas for
the inclusive process $e^{+}e^{-}\to J/\psi+c\bar{c}$ the relativistic
correction is tiny and negligible. The physical reason for the above difference
between exclusive and inclusive processes largely lies in the fact that in the
exclusive process the relative momentum between quarks in charmonium
substantially reduces the virtuality of the gluon that converts into a charm
quark pair, but this is not the case for the inclusive process, in which the
charm quark fragmentation $c\to J/\psi+c$ is significant, and QCD radiative
corrections can be more essential.
