Mean-square convergence of a symplectic local discontinuous Galerkin method applied to stochastic linear Schrödinger equation

In this article, we investigate the mean-square convergence of a novel symplectic local discontinuous Galerkin method in L2-norm for stochastic linear Schrödinger 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 L2-stability.