  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.
