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Construction of Symplectic Runge-Kutta Methods for Stochastic Hamiltonian Systems

Part of: Stochastic analysis Numerical problems in dynamical systems Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  05 December 2016

Peng Wang ,
Jialin Hong  and
Dongsheng Xu
Peng Wang*
Affiliation:
Institute of Mathematics, Jilin University, Changchun 130012, P.R. China
Jialin Hong*
Affiliation:
State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100080 Beijing, P.R. China
Dongsheng Xu*
Affiliation:
State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100080 Beijing, P.R. China University of Chinese Academy of Sciences, P.R. China
*
*Corresponding author. Email addresses:wpemk@163.com; pwang@jlu.edu.cn (P.Wang), hjl@lsec.cc.ac.cn (J. Hong), xuds@lsec.cc.ac.cn (D. Xu)
*Corresponding author. Email addresses:wpemk@163.com; pwang@jlu.edu.cn (P.Wang), hjl@lsec.cc.ac.cn (J. Hong), xuds@lsec.cc.ac.cn (D. Xu)
*Corresponding author. Email addresses:wpemk@163.com; pwang@jlu.edu.cn (P.Wang), hjl@lsec.cc.ac.cn (J. Hong), xuds@lsec.cc.ac.cn (D. Xu)
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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 Runge-Kutta 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.

Keywords

Stochastic differential equationStochastic Hamiltonian systemsymplectic integrationRunge-Kutta methodorder condition

MSC classification

Secondary: 65C30: Stochastic differential and integral equations 60H35: Computational methods for stochastic equations 65P10: Hamiltonian systems including symplectic integrators
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 1 , January 2017 , pp. 237 - 270
Copyright
Copyright © Global-Science Press 2016 

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