In this article, stochastic conformal schemes of damped stochastic Klein-Gordon equation with additive noise are studied. It is shown that this equation possesses stochastic conformal multi-symplectic conservation law. Under appropriate boundary conditions, global momentum evolution law and global energy evolution law are proposed. We chiefly develop stochastic conformal Preissman scheme, stochastic conformal discrete gradient scheme and stochastic conformal Euler box scheme to preserve geometric structures of original system. Specifically, we make theoretical discussions on three proposed schemes to obtain corresponding discrete conservation law or discrete evolution law. Then the damped stochastic linear Klein-Gordon equation and the damped stochastic nonlinear Klein-Gordon equations are taken as examples to demonstrate the validity of the proposed schemes. Through numerical experiments and comparisons, the superiorities of three proposed schemes are fully shown, which are consistent with our theoretical analysis. Moreover, the mean square convergence orders of the three stochastic conformal schemes in time direction and space direction are tested numerically.

本文研究了带有加性噪声的阻尼随机Klein-Gordon方程的随机共形方案。结果表明，该方程具有随机的保形多辛守恒律。在适当的边界条件下，提出了全球动量演化定律和全球能量演化定律。我们主要开发随机保形Preissman方案，随机保形离散梯度方案和随机保形Euler盒方案，以保留原始系统的几何结构。具体来说，我们对三种方案进行了理论讨论，以获得相应的离散守恒定律或离散演化定律。然后以阻尼随机线性Klein-Gordon方程和阻尼随机非线性Klein-Gordon方程为例，验证了所提方案的有效性。通过数值实验和比较，充分表明了三种方案的优越性，这与我们的理论分析是一致的。此外，对三种随机共形方案在时间方向和空间方向上的均方收敛阶进行了数值测试。