Structure-preserving numerical method for a class of stochastic Poisson systems

doi:10.7523/j.ucas.2022.015

Journal of University of Chinese Academy of Sciences ›› 2024, Vol. 41 ›› Issue (3): 298-305.DOI: 10.7523/j.ucas.2022.015

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Structure-preserving numerical method for a class of stochastic Poisson systems

LIU Qianqian, WANG Lijin

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Received:2021-11-24 Revised:2022-02-28 Online:2024-05-15
Supported by:
Supported by the National Natural Science Foundation of China (11971458, 11471310, 11071251)

Abstract

Abstract: In this paper, we consider the structure-preserving numerical simulation of a class of stochastic Poisson systems, i.e. the stochastic Lotka-Volterra systems. We propose a stochastic Poisson integrator for the systems which can preserve the Poisson structure and the Casimir functions of the systems, and prove that the numerical integrator has root mean-square convergence order 1. Numerical experiments are performed to verify the theoretical results.

Key words: stochastic Poisson systems, Lotka-Volterra systems, Stratonovich SDEs, Poisson structure, Casimir functions

CLC Number:

O241.8

LIU Qianqian, WANG Lijin. Structure-preserving numerical method for a class of stochastic Poisson systems[J]. Journal of University of Chinese Academy of Sciences, 2024, 41(3): 298-305.

References

[1] Hong J L, Ruan J L, Sun L Y, et al. Structure-preserving numerical methods for stochastic Poisson systems[J]. Communications in Computational Physics, 2021, 29(3): 802-830. DOI:10.4208/cicp.oa-2019-0084.
[2] Hairer E, Wanner G, Lubich C. Geometric numerical integration[M]. Berlin, Heidelberg: Springer Berlin Heidelberg, 2002. DOI:10.1007/978-3-662-05018-7.
[3] Arnold V I. Mathematical methods of classical mechanics [M]. 2nd ed. New York, NY: Springer New York, 1989. DOI:10.1007/978-1-4757-2063-1.
[4] Milstein G N, Repin Y M, Tretyakov M V. Symplectic integration of Hamiltonian systems with additive noise[J]. SIAM Journal on Numerical Analysis, 2002,39(6): 2066-2088. DOI:10.1137/S0036142901387440.
[5] Cohen D, Dujardin G. Energy-preserving integrators for stochastic Poisson systems[J]. Communications in Mathematical Sciences, 2014, 12(8): 1523-1539. DOI:10.4310/CMS.2014.v12.n8.a7.
[6] Li X Y, Ma Q, Ding X H. High-order energy-preserving methods for stochastic Poisson systems[J]. East Asian Journal on Applied Mathematics, 2019, 9(3): 465-484. DOI:10.4208/eajam.290518.310718.
[7] Hong J L, Ji L H, Wang X, et al. Stochastic K-symplectic integrators for stochastic non-canonical Hamiltonian systems and applications to the Lotka-Volterra model[EB/OL]. 2017: arXiv: 1711.03258[math.NA]. https://arxiv.org/abs/1711.03258.
[8] Wang Y C, Wang L J. Analysis of the solutions of a class of stochastic Poisson systems[J]. Journal of University of Chinese Academy of Sciences, 2020, to appear. DOI: 10.7523/j.ucas.2020.0057.
[9] Milstein G N, Repin Y M, Tretyakov M V. Numerical methods for stochastic systems preserving symplectic structure[J]. SIAM Journal on Numerical Analysis, 2002, 40(4): 1583-1604. DOI:10.1137/S0036142901395588.
[10] Deng J, Anton C, Wong Y S. High-order symplectic schemes for stochastic Hamiltonian systems[J]. Communications in Computational Physics, 2014, 16(1): 169-200. DOI:10.4208/cicp.311012.191113a.
[11] Ruan J L, Wang L J, Wang P J. Exponential discrete gradient schemes for a class of stochastic differential equations[J]. Journal of Computational and Applied Mathematics, 2022, 402: 113797. DOI:10.1016/j.cam.2021.113797.
[12] Milstein G N. Numerical integration of stochastic differential equations[M]. Dordrecht: Springer, 1995. DOI:10.1007/978-94-015-8455-5.

Structure-preserving numerical method for a class of stochastic Poisson systems

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[1]	WANG Yuchao, WANG Lijin. Solutions of a class of stochastic Poisson systems [J]. Journal of University of Chinese Academy of Sciences, 2022, 39(6): 721-726.
[2]	WANG Pengjun, WANG Lijin. Stochastic Poisson integrators based on Padé approximations for linear stochastic Poisson systems [J]. Journal of University of Chinese Academy of Sciences, 2021, 38(2): 160-170.