Structure-Preserving Numerical Methods for Stochastic Poisson Systems (Communications in Computational Physics, 2021)
We propose a numerical integration methodology for stochastic Poisson systems (SPSs) of arbitrary dimensions and multiple noises with different Hamiltoniansin diffusion coefficients, which can provide numerical schemes preserving both thePoisson structure and the Casimir functions of the SPSs, based on the Darboux-Lie theorem. We first transform the SPSs to their canonical form, the generalized stochasticHamiltonian systems (SHSs), via canonical coordinate transformations found by solving certain PDEs defined by the Poisson brackets of the SPSs. An $α$-generating function approach with $α∈[0,1]$ is then constructed and used to create symplectic schemesfor the SHSs, which are then transformed back by the inverse coordinate transformation to become stochastic Poisson integrators of the original SPSs. Numerical tests on athree-dimensional stochastic rigid body system illustrate the efficiency of the proposedmethods.
Keywords: Stochastic Poisson systems; Poisson structure; Casimir functions; Poisson integrators; symplectic integrators; generating functions; stochastic rigid body system.