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This paper focuses on the construction and analysis of explicit numericalmethods of high dimensional stochastic nonlinear Schrodinger equations(SNLSEs). We first prove that the classical explicit numerical methods areunstable and suffer from the numerical divergence phenomenon. Then we propose akind of explicit splitting numerical methods and prove that thestructure-preserving splitting strategy is able to enhance the numericalstability. Furthermore, we establish the regularity analysis and strongconvergence analysis of the proposed schemes for SNLSEs based on two keyingredients. One ingredient is proving new regularity estimates of SNLSEs byconstructing a logarithmic auxiliary functional and exploiting the Bourgainspace. Another one is providing a dedicated error decomposition formula and anovel truncated stochastic Gronwall's lemma, which relies on the tail estimatesof underlying stochastic processes. In particular, our result answers thestrong 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].

Explicit Numerical Methods for High Dimensional Stochastic Nonlinear Schr\"odinger Equation: Divergence, Regularity and Convergence