Construction of Symplectic Runge-Kutta Methods for Stochastic Hamiltonian Systems

被引：19

作者：

Wang, Peng ^{[1
]}

Hong, Jialin ^{[2
]}

Xu, Dongsheng ^{[2
,3
]}

机构：

[1] Jilin Univ, Inst Math, Changchun 130012, Peoples R China

[2] Chinese Acad Sci, Acad Math & Syst Sci, Inst Computat Math & Sci Engn Comp, State Key Lab Sci & Engn Comp, Beijing 100080, Peoples R China

[3] Univ Chinese Acad Sci, Beijing, Peoples R China

来源：

COMMUNICATIONS IN COMPUTATIONAL PHYSICS | 2017年 / 21卷 / 01期

关键词：

Stochastic differential equation; Stochastic Hamiltonian system; symplectic integration; Runge-Kutta method; order condition; DIFFERENTIAL-EQUATIONS; APPROXIMATION; INTEGRATORS;

D O I：

10.4208/cicp.261014.230616a

中图分类号：

O4 [物理学];

学科分类号：

0702 ;

摘要：

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.

引用

页码：237 / 270

页数：34

共 42 条

[11] Implicit stochastic Runge-Kutta methods for stochastic differential equations [J].

Burrage, K ;

Tian, T .

BIT NUMERICAL MATHEMATICS, 2004, 44 (01) :21-39

[12] Low rank Runge-Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise [J].

Burrage, Kevin ;

Burrage, Pamela M. .

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 2012, 236 (16) :3920-3930

[13]

Burrage P. M., 1999, THESIS

[14] AN EFFICIENT APPROXIMATION METHOD FOR STOCHASTIC DIFFERENTIAL-EQUATIONS BY MEANS OF THE EXPONENTIAL LIE SERIES [J].

CASTELL, F ;

GAINES, J .

MATHEMATICS AND COMPUTERS IN SIMULATION, 1995, 38 (1-3) :13-19

[15] ENERGY-PRESERVING INTEGRATORS FOR STOCHASTIC POISSON SYSTEMS [J].

Cohen, David ;

Dujardin, Guillaume .

COMMUNICATIONS IN MATHEMATICAL SCIENCES, 2014, 12 (08) :1523-1539

[16] Runge-Kutta methods for third order weak approximation of SDEs with multidimensional additive noise [J].

Debrabant, Kristian .

BIT NUMERICAL MATHEMATICS, 2010, 50 (03) :541-558

[17] High-Order Symplectic Schemes for Stochastic Hamiltonian Systems [J].

Deng, Jian ;

Anton, Cristina ;

Wong, Yau Shu .

COMMUNICATIONS IN COMPUTATIONAL PHYSICS, 2014, 16 (01) :169-200

[18] Predictor-corrector methods for a linear stochastic oscillator with additive noise [J].

Hong, Jialin ;

Scherer, Rudolf ;

Wang, Lijin .

MATHEMATICAL AND COMPUTER MODELLING, 2007, 46 (5-6) :738-764

[19] Preservation of quadratic invariants ofstochastic differential equations via Runge-Kutta methods [J].

Hong, Jialin ;

Xu, Dongsheng ;

Wang, Peng .

APPLIED NUMERICAL MATHEMATICS, 2015, 87 :38-52

[20] DISCRETE GRADIENT APPROACH TO STOCHASTIC DIFFERENTIAL EQUATIONS WITH A CONSERVED QUANTITY [J].

Hong, Jialin ;

Zhai, Shuxing ;

Zhang, Jingjing .

SIAM JOURNAL ON NUMERICAL ANALYSIS, 2011, 49 (05) :2017-2038

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