Taiwanese Journal of Mathematics

Arbitrary High-order EQUIP Methods for Stochastic Canonical Hamiltonian Systems

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

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1533866419

Digital Object Identifier
doi:10.11650/tjm/180803

Mathematical Reviews number (MathSciNet)
MR3952248

Zentralblatt MATH identifier
07068571

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

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

Citation

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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References

  • C. A. Anton, Y. S. Wong and J. Deng, Symplectic schemes for stochastic Hamiltonian systems preserving Hamiltonian functions, Int. J. Numer. Anal. Model. 11 (2014), no. 3, 427–451.
  • N. Bou-Rabee and H. Owhadi, Stochastic variational integrators, IMA J. Numer. Anal. 29 (2009), no. 2, 421–443.
  • L. Brugnano, G. F. Caccia and F. Iavernaro, Line integral formulation of energy and quadratic invariants preserving (EQUIP) methods for Hamiltonian systems, AIP Conference Proceedings 1738 (2016), 100002.
  • L. Brugnano, G. Gurioli and F. Iavernaro, Analysis of energy and quadratic invariant preserving (EQUIP) methods, J. Comput. Appl. Math. 335 (2018), 51–73.
  • L. Brugnano and F. Iavernaro, Line Integral Methods for Conservative Problems, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2016.
  • L. Brugnano, F. Iavernaro and D. Trigiante, Energy and quadratic invariants preserving integrators of Gaussian type, AIP Conference Proceedings 1281 (2010), 227–230.
  • ––––, Energy- and quadratic invariants–preserving integrators based upon Gauss collocation formulae, SIAM J. Numer. Anal. 50 (2012), no. 6, 2897–2916.
  • L. Brugnano and Y. Sun, Multiple invariants conserving Runge-Kutta type methods for Hamiltonian problems, Numer. Algorithms 65 (2014), no. 3, 611–632.
  • K. Burrage and P. M. Burrage, High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations, Appl. Numer. Math. 22 (1996), no. 1-3, 81–101.
  • ––––, Low rank Runge-Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise, J. Comput. Appl. Math. 236 (2012), no. 16, 3920–3930.
  • J. C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, Chichester, 2003.
  • E. Celledoni, R. I. McLachlan, D. I. McLaren, B. Owren, G. R. W. Quispel and W. M. Wright, Energy-preserving Runge-Kutta methods, M2AN Math. Model. Numer. Anal. 43 (2009), no. 4, 645–649.
  • D. Cohen and G. Dujardin, Energy-preserving integrators for stochastic Poisson systems, Commun. Math. Sci. 12 (2014), no. 8, 1523–1539.
  • K. Debrabant and A. Kværnø, Cheap arbitrary high order methods for single integrand SDEs, BIT 57 (2017), no. 1, 153–168.
  • K. Feng and M. Qin, Symplectic Geometric Algorithms for Hamiltonian Systems, Zhejiang Science and Technology, Hangzhou, Springer, Heidelberg, 2010.
  • O. Gonzalez, Time integration and discrete Hamiltonian systems, J. Nonlinear Sci. 6 (1996), no. 5, 449–467.
  • E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-preserving algorithms for ordinary differential equations, Second edition, Springer Series in Computational Mathematics 31, Springer-Verlag, Berlin, 2006.
  • E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and differential-algebraic problems, Second edition, Springer Series in Computational Mathematics 14, Springer-Verlag, Berlin, 1996.
  • J. Hong, S. Zhai and J. Zhang, Discrete gradient approach to stochastic differential equations with a conserved quantity, SIAM J. Numer. Anal. 49 (2011), no. 5, 2017–2038.
  • C. Huang, Exponential mean square stability of numerical methods for systems of stochastic differential equations, J. Comput. Appl. Math. 236 (2012), no. 16, 4016–4026.
  • P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics (New York) 23, Springer-Verlag, Berlin, 1992.
  • Q. Ma and X. Ding, Stochastic symplectic partitioned Runge-Kutta methods for stochastic Hamiltonian systems with multiplicative noise, Appl. Math. Comput. 252 (2015), 520–534.
  • Q. Ma, D. Ding and X. Ding, Symplectic conditions and stochastic generating functions of stochastic Runge-Kutta methods for stochastic Hamiltonian systems with multiplicative noise, Appl. Math. Comput. 219 (2012), no. 2, 635–643.
  • X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Series in Mathematics & Applications, Horwood, Chichester, 1997.
  • G. N. Milstein, Numerical Integration of Stochastic Differential Equations, Mathematics and its Applications 313, Kluwer Academic Publishers Group, Dordrecht, 1995.
  • G. N. Milstein, Y. M. Repin and M. V. Tretyakov, Symplectic integration of Hamiltonian systems with additive noise, SIAM J. Numer. Anal. 39 (2002), no. 6, 2066–2088.
  • ––––, Numerical methods for stochastic systems preserving symplectic structure, SIAM J. Numer. Anal. 40 (2002), no. 4, 1583–1604.
  • T. Misawa, Energy conservative stochastic difference scheme for stochastic Hamilton dynamical systems, Japan. J. Indust. Appl. Math. 17 (2000), no. 1, 119–128.
  • ––––, Symplectic integrators to stochastic Hamiltonian dynamical systems derived from composition methods, Math. Probl. Eng. 2010 (2010), Art. ID 384937, 12 pp.
  • G. Sun, Construction of high order symplectic PRK methods, J. Comput. Math. 13 (1995), no. 1, 40–50.
  • L. Wang and J. Hong, Generating functions for stochastic symplectic methods, Discrete Contin. Dyn. Syst. 34 (2014), no. 3, 1211–1228.
  • D. Wang, A. Xiao and X. Li, Parametric symplectic partitioned Runge-Kutta methods with energy-preserving properties for Hamiltonian systems, Comput. Phys. Commun. 184 (2013), no. 2, 303–310.
  • A. Xiao and X. Tang, High strong order stochastic Runge-Kutta methods for Stratonovich stochastic differential equations with scalar noise, Numer. Algorithms 72 (2016), no. 2, 259–296.
  • G. Zhong and J. E. Marsden, Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators, Phys. Lett. A 133 (1988), no. 3, 134–139.
  • W. Zhou, L. Zhang, J. Hong and S. Song, Projection methods for stochastic differential equations with conserved quantities, BIT 56 (2016), no. 4, 1497–1518.
  • W. Zhou, J. Zhang, J. Hong and S. Song, Stochastic symplectic Runge-Kutta methods for the strong approximation of Hamiltonian systems with additive noise, J. Comput. Appl. Math. 325 (2017), 134–148.