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.
翻译:Casimir用Stratonovich噪声来保护含有Stratonovich的静脉利皮色方程的Casimir保护集成器,正在开发扩展龙格-Kutta Munthe-Kaas方法。基底的利皮森结构在随机轨迹上保持。从利代数上得出一个相关的随机差分方程。这一差异方程的解决方案通过指数图更新了利皮-普瓦森动态的演变过程。构建的数字方法完全保存了Casimir-异性物,这对于长时间的整合很重要。这是用数字方式为蒸汽重物顶部的情况说明的。