Volume 29, Issue 3
Structure-Preserving Numerical Methods for Stochastic Poisson Systems

Commun. Comput. Phys., 29 (2021), pp. 802-830.

Published online: 2021-01

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• Abstract

We propose a numerical integration methodology for stochastic Poisson systems (SPSs) of arbitrary dimensions and multiple noises with different Hamiltonians in diffusion coefficients, which can provide numerical schemes preserving both the Poisson 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 stochastic Hamiltonian 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 schemes for the SHSs, which are then transformed back by the inverse coordinate transformation to become stochastic Poisson integrators of the original SPSs. Numerical tests on a three-dimensional stochastic rigid body system illustrate the efficiency of the proposed methods.

• Keywords

Stochastic Poisson systems, Poisson structure, Casimir functions, Poisson integrators, symplectic integrators, generating functions, stochastic rigid body system.

60H35, 60H15, 65C30, 60H10, 65D30

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@Article{CiCP-29-802, author = {Hong , Jialin and Ruan , Jialin and Sun , Liying and Wang , Lijin}, title = {Structure-Preserving Numerical Methods for Stochastic Poisson Systems}, journal = {Communications in Computational Physics}, year = {2021}, volume = {29}, number = {3}, pages = {802--830}, abstract = {

We propose a numerical integration methodology for stochastic Poisson systems (SPSs) of arbitrary dimensions and multiple noises with different Hamiltonians in diffusion coefficients, which can provide numerical schemes preserving both the Poisson 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 stochastic Hamiltonian 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 schemes for the SHSs, which are then transformed back by the inverse coordinate transformation to become stochastic Poisson integrators of the original SPSs. Numerical tests on a three-dimensional stochastic rigid body system illustrate the efficiency of the proposed methods.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2019-0084}, url = {http://global-sci.org/intro/article_detail/cicp/18567.html} }
TY - JOUR T1 - Structure-Preserving Numerical Methods for Stochastic Poisson Systems AU - Hong , Jialin AU - Ruan , Jialin AU - Sun , Liying AU - Wang , Lijin JO - Communications in Computational Physics VL - 3 SP - 802 EP - 830 PY - 2021 DA - 2021/01 SN - 29 DO - http://doi.org/10.4208/cicp.OA-2019-0084 UR - https://global-sci.org/intro/article_detail/cicp/18567.html KW - Stochastic Poisson systems, Poisson structure, Casimir functions, Poisson integrators, symplectic integrators, generating functions, stochastic rigid body system. AB -

We propose a numerical integration methodology for stochastic Poisson systems (SPSs) of arbitrary dimensions and multiple noises with different Hamiltonians in diffusion coefficients, which can provide numerical schemes preserving both the Poisson 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 stochastic Hamiltonian 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 schemes for the SHSs, which are then transformed back by the inverse coordinate transformation to become stochastic Poisson integrators of the original SPSs. Numerical tests on a three-dimensional stochastic rigid body system illustrate the efficiency of the proposed methods.

Jialin Hong, Jialin Ruan, Liying Sun & Lijin Wang. (2021). Structure-Preserving Numerical Methods for Stochastic Poisson Systems. Communications in Computational Physics. 29 (3). 802-830. doi:10.4208/cicp.OA-2019-0084
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