  This paper focuses on the construction and analysis of explicit numerical
methods of high dimensional stochastic nonlinear Schrodinger equations
(SNLSEs). We first prove that the classical explicit numerical methods are
unstable and suffer from the numerical divergence phenomenon. Then we propose a
kind of explicit splitting numerical methods and prove that the
structure-preserving splitting strategy is able to enhance the numerical
stability. Furthermore, we establish the regularity analysis and strong
convergence analysis of the proposed schemes for SNLSEs based on two key
ingredients. One ingredient is proving new regularity estimates of SNLSEs by
constructing a logarithmic auxiliary functional and exploiting the Bourgain
space. Another one is providing a dedicated error decomposition formula and a
novel truncated stochastic Gronwall's lemma, which relies on the tail estimates
of underlying stochastic processes. In particular, our result answers the
strong convergence problem of numerical methods for 2D SNLSEs emerged from [C.
Chen, J. Hong and A. Prohl, Stoch. Partial Differ. Equ. Anal. Comput. 4 (2016),
no. 2, 274-318] and [J. Cui and J. Hong, SIAM J. Numer. Anal. 56 (2018), no. 4,
2045-2069].
