Mean-square Convergence of a Symplectic Local Discontinuous Galerkin   Method Applied to Stochastic Linear Schroedinger Equation
  In this paper, we investigate the mean-square convergence of a novel symplectic local discontinuous Galerkin method in L^2-norm for stochastic linear Schroedinger equation with multiplicative noise. It is shown that the mean-square error is bounded not only by the temporal and spatial step-sizes, but also by their ratio. The mean-square convergence rate with respect to the computational cost is derived under appropriate assumptions for initial data and noise. Meanwhile, we show that the method preserves the discrete charge conservation law which implies an L^2-stability 
Chuchu Chen, Jialin Hong and Lihai Ji
math.NA