Casimir preserving integrators for stochastic Lie-Poisson equations with Stratonovich noise are developed extending Runge-Kutta Munthe-Kaas methods. The underlying Lie-Poisson structure is preserved along stochastic trajectories. A related stochastic differential equation on the Lie algebra is derived. The solution of this differential equation updates the evolution of the Lie-Poisson dynamics by means of the exponential map. The constructed numerical method conserves Casimir-invariants exactly, which is important for long time integration. This is illustrated numerically for the case of the stochastic heavy top and the stochastic sine-Euler equations.

翻译：针对带有Stratonovich噪声的随机Lie-Poisson方程，发展了Casimir保持的积分器，该方法推广了龙格-库塔-蒙特-卡斯方法（Runge-Kutta Munthe-Kaas methods）。底层Lie-Poisson结构沿随机轨迹得以保持。推导了李代数上的相关随机微分方程。该微分方程的解通过指数映射更新Lie-Poisson动力学的演化。所构造的数值方法精确保持Casimir不变量，这对于长时间积分至关重要。这一特性通过随机重陀螺（stochastic heavy top）和随机正弦-欧拉方程（stochastic sine-Euler equations）的数值算例得到了验证。