An adaptive time-stepping fully discrete scheme for stochastic NLS equation: Strong convergence and numerical asymptotics

In this paper, we propose and analyze an adaptive time-stepping fully discrete scheme which possesses the optimal strong convergence order for the stochastic nonlinear Schr\"odinger equation with multiplicative noise. Based on the splitting skill and the adaptive strategy, the $H^1$-exponential integrability of the numerical solution is obtained, which is a key ingredient to derive the strong convergence order. We show that the proposed scheme converges strongly with orders $\frac12$ in time and $2$ in space. To investigate the numerical asymptotic behavior, we establish the large deviation principle for the numerical solution. This is the first result on the study of the large deviation principle for the numerical scheme of stochastic partial differential equations with superlinearly growing drift. And as a byproduct, the error of the masses between the numerical and exact solutions is finally obtained.