Stochastic Klein--Gordon--Schr\"odinger (KGS) equations are important mathematical models and describe the interaction between scalar nucleons and neutral scalar mesons in the stochastic environment. In this paper, we propose novel structure-preserving schemes to numerically solve stochastic KGS equations with additive noise, which preserve averaged charge evolution law, averaged energy evolution law, symplecticity, and multi-symplecticity. By applying central difference, sine pseudo-spectral method, or finite element method in space and modifying finite difference in time, we present some charge and energy preserved fully-discrete scheme for the original system. In addition, combining the symplectic Runge-Kutta method in time and finite difference in space, we propose a class of multi-symplectic discretizations preserving the geometric structure of the stochastic KGS equation. Finally, numerical experiments confirm theoretical findings.

翻译：随机Klein–Gordon–Schrödinger (KGS)方程是重要的数学模型，描述了随机环境中标量核子与中性标量介子之间的相互作用。本文提出新型保结构格式以数值求解带加性噪声的随机KGS方程，这些格式能够保持平均电荷演化律、平均能量演化律、辛性与多辛性。通过空间上的中心差分、正弦拟谱法或有限元方法，并结合时间上的修正有限差分，我们为原始系统构造了若干电荷和能量守恒的全离散格式。此外，将时间上的辛Runge-Kutta方法与空间上的有限差分相结合，我们提出了一类保持随机KGS方程几何结构的多辛离散格式。最后，数值实验验证了理论结果。