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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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References

[1] Abdulle, A., Cohen, D., Vilmart, G. and Zygalakis, K. C., High order weak methods for stochastic differential equations based on modified equations, SIAM J. Sci. Comput., 34(3) (2012), 1800–1823.CrossRefGoogle Scholar
[2] Burrage, K. and Burrage, P.M., General order conditions for stochastic Runge-Kutta methods for both commuting and non-commuting stochastic ordinarydiffer ential equation systems, Appl. Numer. Math., 28 (1998), 161–177.Google Scholar
[3] Burrage, K. and Burrage, P.M., High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations, Appl. Numer. Math., 22 (1996), 81–101.Google Scholar
[4] Burrage, K. and Burrage, P.M., Low rank Runge-Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise, J. Comput. Appl. Math., 236 (2012), 3920–3930.CrossRefGoogle Scholar
[5] Burrage, K. and Tian, T. H., Implicit stochastic Runge-Kutta methods for stochastic differential equations, BIT, 44 (2004), 21–39.CrossRefGoogle Scholar
[6] Burrage, P. M., Runge-Kutta Methods for Stochastic Differential Equations, Ph.D. thesis, Department of Mathematics, University of Queensland, Australia, 1999.Google Scholar
[7] Castell, F. and Gaines, J., An efficient approximation method for stochastic differential equations by means of the exponential Lie series, Math. Comput. Simulation, 38 (1995), 13–19.CrossRefGoogle Scholar
[8] Cohen, D. and Duardin, G., Energy-preserving intergrators for stochastic Poisson systems, Communications in Mathmatical Sciences, 12(8) (2014) 1523–1539.Google Scholar
[9] Cristina, A., Deng, J. and Wong, Y. S., Weak symplectic schemes for stochastic Hamiltonian equations, Electronic Transactions on Numerical Analysis, 43 (2014) 1–20.Google Scholar
[10] Cristina, A., Wong, Y. S. and Deng, J., Symplectic numerical schemes for stochastic systems preserving Hamiltonian functions, International Journal of Numerical Analysis and Modeling, 11(3) (2014) 427–451.Google Scholar
[11] Debrabant, K., Runge-Kutta methods for third order weak approximation of SDEs with multidimensional additive noise, BIT Numer. Math., 50(3) (2010), 541–558.CrossRefGoogle Scholar
[12] Deng, J., Anton, C. and Wong, Y. S., High-order symplectic schemes for stochastic Hamiltonian systems, Communications in Computational Physics, 16(1) (2014), 169–200.Google Scholar
[13] Hairer, E., Lubich, Ch. and Wanner, G., Geometric Numerical Integration, Springer-Verlag, 2002.Google Scholar
[14] Hong, J., Scherer, R. and Wang, L., Midpoint rule for a linear stochastic oscillator with additive noise, Neural Parallel Sci. Comput., 14 (2006), 1–12.Google Scholar
[15] Hong, J., Scherer, R. and Wang, L., Predictor-corrector methods for a linear stochastic oscillator with additive noise, Mathematical and Computer Modelling, 46 (2007), 738–764.Google Scholar
[16] Hong, J., Xu, D. and Wang, P., Preservation of quadratic invariants of stochastic differential equations via Runge-Kutta methods, Appl. Numer. Math., 87 (2015), 38–52.CrossRefGoogle Scholar
[17] Hong, J., Zhai, S. and Zhang, J., Discrete gradient approach to stochastic differential equations with a conserved quantity, SIAM J. Numer. Anal., 49(5) (2011), 2017–2038.CrossRefGoogle Scholar
[18] Jiang, P., Ju, X., Liu, D. and Fan, S., Exact Finite-Difference schemes for d-dimensional linear stochastic systems with constant coefficients, Journal of Applied Mathematics, (2013), Article ID 830936.Google Scholar
[19] Kloeden, P. E. and Platen, E., Numerical Solutions of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992.Google Scholar
[20] Küpper, D., Kværnø, A. and Rößler, A., A Runge-Kutta Method for Index 1 Stochastic Differential-Algebraic Equations with Scalar Noise, BIT Numer. Math., 52 (2012), 437–455.Google Scholar
[21] Lázaro-Camí, L.-A. and Ortega, J.-P., Reduction, reconstruction, and skew-product decomposition of symmetric stochastic differential equations, Stoch. Dyn., 9 (2009), 1–46.CrossRefGoogle Scholar
[22] Liao, M., Random motion of a rigid body, J. Theoret. Probab., 10 (1997), 201–211.Google Scholar
[23] Ma, Q., Ding, D. and Ding, X., Symplectic conditions and stochastic generating functions of stochastic Runge-Kutta methods for stochastic Hamiltonian systems with multiplicative noise, Applied Mathematics and Computation, 219 (2012), 635–643.CrossRefGoogle Scholar
[24] Malham, S. J. A. and Wiese, A., Stochastic Lie group integrators, SIAM J. Sci. Comput., 30(2) (2008), 597–617.Google Scholar
[25] Milstein, G. N., Repin, Yu. M. and Tretyakov, M. V., Numerical methods for stochastic systems preserving symplectic structure, SIAM J. Numer. Anal., 40 (2002), 1583–1604.Google Scholar
[26] Milstein, G. N., Repin, Yu. M. and Tretyakov, M. V., Symplectic integration of Hamiltonian systems with additive noise, SIAM J. Numer. Anal., 39(6) (2002), 2066–2088.Google Scholar
[27] Milstein, G. N. and Tretyakov, M. V., Quasi-symplectic methods for Langevin-type equations, IMA J. Numer. Anal., 23 (2003), 593–626.CrossRefGoogle Scholar
[28] Milstein, G. N. and Tretyakov, M. V., Stochastic numerics formathematical physics, Sci. Comput. Springer-Verlag, Berlin, 2004.Google Scholar
[29] Misawa, T., Energy Conservative Stochastic Difference Scheme for Stochastic Hamilton Dynamical Systems, Japan J. Indust. Appl. Math., 17 (2000), 119–128 Google Scholar
[30] Rößler, A., Runge-Kutta methods for Itô stochastic differential equations with scalar noise, BIT, 46 (2006), 97–110.CrossRefGoogle Scholar
[31] Rößler, A., Runge-Kutta methods for Stratonovich stochastic differential equation systems with commutative noise, J. Comput. Appl. Math., 164 (2004), 613–627.Google Scholar
[32] Rößler, A., Runge-Kutta methods for the numerical solution of stochastic differential equation, PhD. Thesis, Darmstadt University of Technology, Shaker Verlag, Aachen, 2003.Google Scholar
[33] Rößler, A., Runge-Kutta methods for the strong approximation of solutions of stochastic differential equations, SIAM J. Numer. Anal., 48 (2010), 922–952.CrossRefGoogle Scholar
[34] Rößler, A., Second order Runge-Kutta methods for Itô stochastic differential equations, SIAM J. Numer. Anal., 47 (2009), 1713–1738.Google Scholar
[35] Rößler, A., Second order Runge-Kutta methods for Stratonovich stochastic differential equations, BIT, 47 (2007), 657–680.Google Scholar
[36] Rümelin, W., Numerical treatment of stochastic differential equations, SIAM J. Numer. Anal., 19(3) (1982)604–613.Google Scholar
[37] Sanz-Serna, J. M. and Calvo, M. P., Numerical Hamiltonian Problems, Chapman and Hall, London, UK, 1994.Google Scholar
[38] Seesselberg, M., Breuer, H. P., Mais, H., Petruccione, F. and Honerkamp, J., Simulation of one-dimensional noisy Hamiltonian systems and their application to particle storage rings, Z. Phys. C, 62 (1994), 62–73.Google Scholar
[39] Stømmen, A.H. and Higham, D. J., Numerical simulation of a linear stochastic oscillator with additive noise, Appl. Numer. Math., 51 (2004), 89–99.Google Scholar
[40] Tretyakov, M. V. and Tret’jakov, S. V., Numerical integration of Hamiltonian systems with external noise, Phys. Lett. A, 194 (1994), 371–374.Google Scholar
[41] Wang, L., Hong, J., Scherer, R. and Bai, F., Dynamics and variational integrators of stochastic Hamiltonian systems, International Journal of Numerical Analysis And Modeling, 6(4) (2009), 586–602.Google Scholar
[42] Wang, L., Variational integrators and generating functions for stochastic Hamiltonian systems, Doctoral Thesis of the University of Karlsruhe, Universitäsverlag Karlsruhe, Germany, 2007.Google Scholar