Optimal Error Estimates of Conservative Local Discontinuous Galerkin   Method for Nonlinear Schr\"odinger Equation
  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. 
Jialin Hong and Lihai Ji and Zhihui Liu
math.NA