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Volume 21, Issue 1
Lie-Poisson Numerical Method for a Class of Stochastic Lie-Poisson Systems

Qianqian Liu & Lijin Wang

DOI: 10.4208/ijnam2024-1004

Int. J. Numer. Anal. Mod., 21 (2024), pp. 104-119.

Published online: 2024-01

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  • Abstract

We propose a numerical method based on the Lie-Poisson reduction for a class of stochastic Lie-Poisson systems. Such system is transformed to SDE on the dual $\mathfrak{g}^∗$ of the Lie algebra related to the Lie group manifold where the system is located, which is also the reduced form of a stochastic Hamiltonian system on the cotangent bundle of the Lie group by momentum mapping. Stochastic Poisson integrators are obtained by discretely reducing stochastic symplectic methods on the cotangent bundle to integrators on $\mathfrak{g}^∗.$ Stochastic generating functions creating stochastic symplectic methods are used to construct the schemes. An application to the stochastic rigid body system illustrates the theory and provides numerical validation of the method.

  • Keywords

Stochastic Lie-Poisson systems, structure-preserving algorithms, Poisson integrators, Lie-Poisson reduction, Poisson structure, Casimir functions.

  • AMS Subject Headings

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

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-21-104, author = {Liu , Qianqian and Wang , Lijin}, title = {Lie-Poisson Numerical Method for a Class of Stochastic Lie-Poisson Systems }, journal = {International Journal of Numerical Analysis and Modeling}, year = {2024}, volume = {21}, number = {1}, pages = {104--119}, abstract = {

We propose a numerical method based on the Lie-Poisson reduction for a class of stochastic Lie-Poisson systems. Such system is transformed to SDE on the dual $\mathfrak{g}^∗$ of the Lie algebra related to the Lie group manifold where the system is located, which is also the reduced form of a stochastic Hamiltonian system on the cotangent bundle of the Lie group by momentum mapping. Stochastic Poisson integrators are obtained by discretely reducing stochastic symplectic methods on the cotangent bundle to integrators on $\mathfrak{g}^∗.$ Stochastic generating functions creating stochastic symplectic methods are used to construct the schemes. An application to the stochastic rigid body system illustrates the theory and provides numerical validation of the method.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2024-1004}, url = {http://global-sci.org/intro/article_detail/ijnam/22330.html} }
TY - JOUR T1 - Lie-Poisson Numerical Method for a Class of Stochastic Lie-Poisson Systems AU - Liu , Qianqian AU - Wang , Lijin JO - International Journal of Numerical Analysis and Modeling VL - 1 SP - 104 EP - 119 PY - 2024 DA - 2024/01 SN - 21 DO - http://doi.org/10.4208/ijnam2024-1004 UR - https://global-sci.org/intro/article_detail/ijnam/22330.html KW - Stochastic Lie-Poisson systems, structure-preserving algorithms, Poisson integrators, Lie-Poisson reduction, Poisson structure, Casimir functions. AB -

We propose a numerical method based on the Lie-Poisson reduction for a class of stochastic Lie-Poisson systems. Such system is transformed to SDE on the dual $\mathfrak{g}^∗$ of the Lie algebra related to the Lie group manifold where the system is located, which is also the reduced form of a stochastic Hamiltonian system on the cotangent bundle of the Lie group by momentum mapping. Stochastic Poisson integrators are obtained by discretely reducing stochastic symplectic methods on the cotangent bundle to integrators on $\mathfrak{g}^∗.$ Stochastic generating functions creating stochastic symplectic methods are used to construct the schemes. An application to the stochastic rigid body system illustrates the theory and provides numerical validation of the method.

Liu , Qianqian and Wang , Lijin. (2024). Lie-Poisson Numerical Method for a Class of Stochastic Lie-Poisson Systems . International Journal of Numerical Analysis and Modeling. 21 (1). 104-119. doi:10.4208/ijnam2024-1004
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