Holm Darryl D, Tyranowski Tomasz M
1Mathematics Department, Imperial College London, London, SW7 2AZ UK.
2Max-Planck-Institut für Plasmaphysik, Boltzmannstraße 2, 85748 Garching, Germany.
BIT Numer Math. 2018;58(4):1009-1048. doi: 10.1007/s10543-018-0720-2. Epub 2018 Aug 16.
Variational integrators are derived for structure-preserving simulation of stochastic Hamiltonian systems with a certain type of multiplicative noise arising in geometric mechanics. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and its corresponding variational principle. Our approach permits to recast in a unified framework a number of integrators previously studied in the literature, and presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators are symplectic; they preserve integrals of motion related to Lie group symmetries; and they include stochastic symplectic Runge-Kutta methods as a special case. Several new low-stage stochastic symplectic methods of mean-square order 1.0 derived using this approach are presented and tested numerically to demonstrate their superior long-time numerical stability and energy behavior compared to nonsymplectic methods.
变分积分器是为具有几何力学中出现的某种类型乘性噪声的随机哈密顿系统的保结构模拟而推导出来的。该推导基于一个随机离散哈密顿量，它近似于哈密顿系统随机流的II型随机生成函数。通过引入适当的随机作用泛函及其相应的变分原理来获得生成函数。我们的方法允许在一个统一的框架中重铸文献中先前研究的许多积分器，并提出了一种推导新的保结构数值格式的通用方法。所得的积分器是辛的；它们保留与李群对称性相关的运动积分；并且它们包括随机辛龙格 - 库塔方法作为特殊情况。使用这种方法推导并数值测试了几种新的均方阶为1.0的低阶随机辛方法，以证明它们与非辛方法相比具有卓越的长期数值稳定性和能量特性。
