Analysis of A Splitting Scheme for Damped Stochastic Nonlinear Schr\"odinger Equation with Multiplicative Noise

In this paper, we investigate the damped stochastic nonlinear Schr"odinger(NLS) equation with multiplicative noise and its splitting-based approximation. When the damped effect is large enough, we prove that the solutions of the damped stochastic NLS equation and the splitting scheme are exponential stable and possess some exponential integrability. These properties lead that the strong order of the scheme is $\frac{1}{2}$ and independent of time. Meanwhile, we analyze the regularity of the Kolmogorov equation with respect to the equation. As a consequence, the weak order of the scheme is shown to be twice the strong order and independent of time.