  In this paper, we propose a conservative local discontinuous Galerkin method
for one-dimensional nonlinear Schr\"odinger equation. By using special
upwind-biased numerical fluxes, we establish the optimal rate of convergence
$\mathcal O(h^{k+1})$, with polynomial of degree $k$ and grid size $h$.
Meanwhile, we show that this method preserves the charge conservation law and
thus we call it a conservative local discontinuous Galerkin method. Numerical
experiments verify our theoretical result.
