Mean-square Convergence of a Symplectic Local Discontinuous Galerkin Method Applied to Stochastic Linear Schroedinger Equation

TL;DR

This paper introduces a symplectic local discontinuous Galerkin method for stochastic linear Schrödinger equations, demonstrating its mean-square convergence, stability, and charge conservation properties under specific conditions.

Contribution

The paper presents a novel symplectic local discontinuous Galerkin method with proven mean-square convergence and charge conservation for stochastic Schrödinger equations.

Findings

Mean-square error depends on step-sizes and their ratio.

Method preserves discrete charge conservation law.

Convergence rate derived under specific assumptions.

Abstract

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