  In this paper, we establish the optimal convergence result of a second order
exponential-type integrator from (136, Numer. Math., 2017) for solving the KdV
equation under rough initial data. The scheme is explicit and efficient to
implement. By rigorous error analysis, we show that the scheme provides the
second order accuracy in $H^\gamma$ for initial data in $H^{\gamma+4}$ for any
$\gamma\geq0$, where the regularity requirement is lower than the classical
methods. The result is confirmed by numerical experiments and comparisons are
made with the Strang splitting scheme.
