In this manuscript, we propose efficient stochastic semi-explicit symplectic schemes tailored for nonseparable stochastic Hamiltonian systems (SHSs). These semi-explicit symplectic schemes are constructed by introducing augmented Hamiltonians and using symmetric projection. In the case of the artificial restraint in augmented Hamiltonians being zero, the proposed schemes also preserve quadratic invariants, making them suitable for developing semi-explicit charge-preserved multi-symplectic schemes for stochastic cubic Schr\"odinger equations with multiplicative noise. Through numerical experiments that validate theoretical results, we demonstrate that the proposed stochastic semi-explicit symplectic scheme, which features a straightforward Newton iteration solver, outperforms the traditional stochastic midpoint scheme in terms of effectiveness and accuracy.

翻译：本文提出了一类针对不可分离随机哈密顿系统的高效随机半显式辛格式。这些半显式辛格式通过引入增广哈密顿量并采用对称投影方法构造而成。当增广哈密顿量中的人工约束项为零时，所提格式还能保持二次守恒量，这使其适用于为具有乘性噪声的随机三次薛定谔方程构建半显式电荷守恒多辛格式。通过数值实验验证理论结果，我们证明所提出的随机半显式辛格式（其牛顿迭代求解器结构简洁）在计算效率与精度方面均优于传统的随机中点格式。