Taiwanese Journal of Mathematics

Arbitrary High-order EQUIP Methods for Stochastic Canonical Hamiltonian Systems

Xiuyan Li, Chiping Zhang, Qiang Ma, and Xiaohua Ding

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This paper is concerned with arbitrary high-order energy-preserving numerical methods for stochastic canonical Hamiltonian systems. Energy and quadratic invariants-preserving (EQUIP) methods for deterministic Hamiltonian systems are applied to stochastic canonical Hamiltonian systems and analyzed accordingly. A class of stochastic parametric Runge-Kutta methods with a truncation technique of random variables are obtained. Increments of Wiener processes are replaced by some truncated random variables. We prove the replacement doesn't change the convergence order under some conditions. The methods turn out to be symplectic for any given parameter. It is shown that there exists a parameter $\alpha_{n}^{*}$ at each step such that the energy-preserving property holds, and the energy-preserving methods retain the order of the underlying stochastic Gauss Runge-Kutta methods. Numerical results illustrate the effectiveness of EQUIP methods when applied to stochastic canonical Hamiltonian systems.

Article information

Taiwanese J. Math., Volume 23, Number 3 (2019), 703-725.

Received: 27 December 2017
Revised: 21 May 2018
Accepted: 1 August 2018
First available in Project Euclid: 10 August 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 37N30: Dynamical systems in numerical analysis 65P10: Hamiltonian systems including symplectic integrators

stochastic canonical Hamiltonian systems EQUIP methods symplectic methods energy-preserving methods mean-square convergence


Li, Xiuyan; Zhang, Chiping; Ma, Qiang; Ding, Xiaohua. Arbitrary High-order EQUIP Methods for Stochastic Canonical Hamiltonian Systems. Taiwanese J. Math. 23 (2019), no. 3, 703--725. doi:10.11650/tjm/180803. https://projecteuclid.org/euclid.twjm/1533866419

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