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Volume 21, Issue 1
Construction of Symplectic Runge-Kutta Methods for Stochastic Hamiltonian Systems

Peng Wang, Jialin Hong & Dongsheng Xu

DOI: 10.4208/cicp.261014.230616a

Commun. Comput. Phys., 21 (2017), pp. 237-270.

Published online: 2018-04

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  • Abstract

We study the construction of symplectic Runge-Kutta methods for stochastic Hamiltonian systems (SHS). Three types of systems, SHS with multiplicative noise, special separable Hamiltonians and multiple additive noise, respectively, are considered in this paper. Stochastic Runge-Kutta (SRK) methods for these systems are investigated, and the corresponding conditions for SRK methods to preserve the symplectic property are given. Based on the weak/strong order and symplectic conditions, some effective schemes are derived. In particular, using the algebraic computation, we obtained two classes of high weak order symplectic Runge-Kutta methods for SHS with a single multiplicative noise, and two classes of high strong order symplectic RungeKutta methods for SHS with multiple multiplicative and additive noise, respectively. The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.

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COPYRIGHT: © Global Science Press

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@Article{CiCP-21-237, author = {Peng Wang, Jialin Hong and Dongsheng Xu}, title = {Construction of Symplectic Runge-Kutta Methods for Stochastic Hamiltonian Systems}, journal = {Communications in Computational Physics}, year = {2018}, volume = {21}, number = {1}, pages = {237--270}, abstract = {

We study the construction of symplectic Runge-Kutta methods for stochastic Hamiltonian systems (SHS). Three types of systems, SHS with multiplicative noise, special separable Hamiltonians and multiple additive noise, respectively, are considered in this paper. Stochastic Runge-Kutta (SRK) methods for these systems are investigated, and the corresponding conditions for SRK methods to preserve the symplectic property are given. Based on the weak/strong order and symplectic conditions, some effective schemes are derived. In particular, using the algebraic computation, we obtained two classes of high weak order symplectic Runge-Kutta methods for SHS with a single multiplicative noise, and two classes of high strong order symplectic RungeKutta methods for SHS with multiple multiplicative and additive noise, respectively. The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.261014.230616a}, url = {http://global-sci.org/intro/article_detail/cicp/11239.html} }
TY - JOUR T1 - Construction of Symplectic Runge-Kutta Methods for Stochastic Hamiltonian Systems AU - Peng Wang, Jialin Hong & Dongsheng Xu JO - Communications in Computational Physics VL - 1 SP - 237 EP - 270 PY - 2018 DA - 2018/04 SN - 21 DO - http://doi.org/10.4208/cicp.261014.230616a UR - https://global-sci.org/intro/article_detail/cicp/11239.html KW - AB -

We study the construction of symplectic Runge-Kutta methods for stochastic Hamiltonian systems (SHS). Three types of systems, SHS with multiplicative noise, special separable Hamiltonians and multiple additive noise, respectively, are considered in this paper. Stochastic Runge-Kutta (SRK) methods for these systems are investigated, and the corresponding conditions for SRK methods to preserve the symplectic property are given. Based on the weak/strong order and symplectic conditions, some effective schemes are derived. In particular, using the algebraic computation, we obtained two classes of high weak order symplectic Runge-Kutta methods for SHS with a single multiplicative noise, and two classes of high strong order symplectic RungeKutta methods for SHS with multiple multiplicative and additive noise, respectively. The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.

Peng Wang, Jialin Hong and Dongsheng Xu. (2018). Construction of Symplectic Runge-Kutta Methods for Stochastic Hamiltonian Systems. Communications in Computational Physics. 21 (1). 237-270. doi:10.4208/cicp.261014.230616a
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