Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-26T02:42:54.739Z Has data issue: false hasContentIssue false
  • English
  • Français

High-Order Symplectic Schemes for Stochastic Hamiltonian Systems

Published online by Cambridge University Press:  03 June 2015

Jian Deng ,
Cristina Anton  and
Yau Shu Wong
Jian Deng*
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada
Cristina Anton*
Affiliation:
Department of Mathematics and Statistics, Grant MacEwan University, Edmonton, AB T5J 4S2, Canada
Yau Shu Wong*
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada
*
Corresponding author.Email:deng2@ualberta.ca
Email:popescuc@macewan.ca
Email:yaushu.wong@ualberta.ca
Get access

Abstract

The construction of symplectic numerical schemes for stochastic Hamiltonian systems is studied. An approach based on generating functions method is proposed to generate the stochastic symplectic integration of any desired order. In general the proposed symplectic schemes are fully implicit, and they become computationally expensive for mean square orders greater than two. However, for stochastic Hamiltonian systems preserving Hamiltonian functions, the high-order symplectic methods have simpler forms than the explicit Taylor expansion schemes. A theoretical analysis of the convergence and numerical simulations are reported for several symplectic integrators. The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.

Keywords

60H1065C3065P10Stochastic Hamiltonian systemssymplectic integrationmean-square convergencehigh-order schemes.
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 1 , July 2014 , pp. 169 - 200
Copyright
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]de Vogelaere, R., Methods of integration which preserve the contact transformation property of the hamiltonian equations, Tech. Rep., Report No. 4, Dept. Math., Univ. of Notre Dame, Notre Dame, Ind., (1956).Google Scholar
[2]Ruth, R., A canonical integration technique, IEEE Trans. Nuclear Sci., 30 (1983), 2669–2671.Google Scholar
[3]Feng, K., On difference schemes and symplectic geometry, in: Proceedings of the 5-th Intern, Symposium on Differential Geometry & Differential Equations, Beijing, China, 1985, 42–58.Google Scholar
[4]Liu, X., Qi, Y., He, J. and Ding, P., Recent progress in symplectic algorithms for use in quantum systems, Commun. Comput. Phys., 2(1) (2007), 1–53.Google Scholar
[5]Hong, J., Jiang, S., Li, C. and Liu, H., Explicit multi-symplectic methods for Hamiltonian wave equations, Commun. Comput. Phys., 2 (2007), 662–683.Google Scholar
[6]Milstein, G. N., Tretyakov, M. V. and Repin, Y. M., Symplectic integration of hamiltonian systems with additive noise, SIAM J. Numer. Anal., 39 (2002), 2066–2088.Google Scholar
[7]Kunita, H., Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, 1990.Google Scholar
[8]Milstein, G. N., Tretyakov, M. V. and Repin, Y. M., Numerical methods for stochastic systems preserving symplectic structure, SIAM J. Numer. Anal., 40 (2002), 1583–1604.Google Scholar
[9]Hong, J., Scherer, R. and Wang, L., Predictorcorrector methods for a linear stochastic oscillator with additive noise, Math. Comput. Model., 46(5-6) (2007), 738–764.Google Scholar
[10]Melbo, A. and Higham, D., Numerical simulation of a linear stochastic oscillator with additive noise, Appl. Numer. Math., 51(1) (2004), 89–99.CrossRefGoogle Scholar
[11]Wang, L., Hong, J., Scherer, R. and Bai, F., Dynamics and variational integrators of stochastic Hamiltonian systems, Int. J. Numer. Anal. Model., 6(4) (2009), 586–602.Google Scholar
[12]Hong, J., Wang, L. and Scherer, R., Simulation of stochastic Hamiltonian systems via generating functions, in: Proceedings IEEE 2011 4th ICCSIT, 2011.Google Scholar
[13]Hong, J., Wang, L. and Scherer, R., Symplectic numerical methods for a linear stochastic oscillator with two additive noises, in: Proceedings of the World Congress on Engineering, Vol. I, London, U.K., 2011.Google Scholar
[14]Wang, L.Variational integrators and generating functions for stochastic Hamiltonian systems, KIT Scientific Publishing, University of Karlsruhe, Germany, 2007, Dissertation, www.ksp.kit.edu.Google Scholar
[15]Bismut, J.-M. and Aleatoire, Mecanique, Lecture Notes in Mathematics, Vol. 866, Springer-Verlag, Berlin-Heidelberg-New York, 1981.Google Scholar
[16]Hairer, E., Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer, Berlin, New York, 2006.Google Scholar
[17]Feng, K., Wu, H., Qin, M. and Wang, D., Construction of canonical difference schemes for Hamiltonian formalism via generating functions, J. Comput. Math., 7 (1989), 71–96.Google Scholar
[18]Anton, C., Deng, J. and Wong, Y. S., Weak symplectic schemes for stochastic Hamiltonian equations, Electronic Transactions on Numerical Analysis, to appear.Google Scholar
[19]Kloeden, P. and Platen, E., Numerical Solutions of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992.Google Scholar
[20]Milstein, G. N., Numerical Integration of Stochastic Differential Equations, Kluwer Academic Publishers, Dordrecht, Boston, 1995.Google Scholar
[21]Anton, C., Wong, Y. S. and Deng, J., Symplectic schemes for stochastic Hamiltonian systems preserving Hamiltonian functions, Int. J. Numer. Anal. Model., 11(3) (2014), 427–451.Google Scholar