April  2022, 15(4): 819-836. doi: 10.3934/dcdss.2021095

Numerical methods preserving multiple Hamiltonians for stochastic Poisson systems

1. 

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

2. 

Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA

* Corresponding author: Yanzhao Cao

Received  December 2020 Revised  July 2021 Published  April 2022 Early access  August 2021

Fund Project: The first and second authors are supported by NNSFC No. 11971458, No. 11471310

In this paper, we propose a class of numerical schemes for stochastic Poisson systems with multiple invariant Hamiltonians. The method is based on the average vector field discrete gradient and an orthogonal projection technique. The proposed schemes preserve all the invariant Hamiltonians of the stochastic Poisson systems simultaneously, with possibility of achieving high convergence orders in the meantime. We also prove that our numerical schemes preserve the Casimir functions of the systems under certain conditions. Numerical experiments verify the theoretical results and illustrate the effectiveness of our schemes.

Citation: Lijin Wang, Pengjun Wang, Yanzhao Cao. Numerical methods preserving multiple Hamiltonians for stochastic Poisson systems. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 819-836. doi: 10.3934/dcdss.2021095
References:
[1]

L. BrugnanoM. CalvoJ. I. Montijano and L. Rández, Energy-preserving methods for Poisson systems, J. Comput. Appl. Math., 236 (2012), 3890-3904.  doi: 10.1016/j.cam.2012.02.033.

[2]

D. Cohen and G. Dujardin, Energy-preserving integrators for stochastic Poisson systems, Commun. Math. Sci., 12 (2014), 1523-1539.  doi: 10.4310/CMS.2014.v12.n8.a7.

[3]

D. Cohen and E. Hairer, Linear energy-preserving integrators for Poisson systems, BIT Numer. Math., 51 (2011), 91-101.  doi: 10.1007/s10543-011-0310-z.

[4]

M. Dahlby, B. Owren and T. Yaguchi, Preserving multiple first integrals by discrete gradients, J. Phys. A: Math. Theor., 44 (2011), 305205. doi: 10.1088/1751-8113/44/30/305205.

[5]

K. Engø and S. Faltinsen, Numerical integration of Lie-Poisson systems while preserving coadjoint orbits and energy, SIAM J. Numer. Anal., 39 (2001), 128-145.  doi: 10.1137/S0036142999364212.

[6]

E. Faou and T. Lelièvre, Conservative stochastic differential equations: Mathematical and numerical analysis, Math. Comput., 78 (2009), 2047-2074.  doi: 10.1090/S0025-5718-09-02220-0.

[7]

K. Feng and M. Qin, Symplectic Geometric Algorithms for Hamiltonian Systems, Springer-Verlag, Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-01777-3.

[8]

I. A. Garcia and B. Hernandez–Bermejo, Perturbed Euler top and bifurcation of limit cycles on invariant Casimir surfaces, Physica D, 239 (2010), 1665-1669.  doi: 10.1016/j.physd.2010.04.013.

[9]

O. Gonzalez, Time integration and discrete Hamiltonian systems, J. Nonlinear Sci., 6 (1996), 449-467.  doi: 10.1007/BF02440162.

[10]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, Second Edition, Springer-Verlag Berlin Heidelberg, 2006.

[11]

B. Hernández-Bermejo, Characterization, global analysis and integrability of a family of Poisson structures, Phys. Lett. A, 372 (2008), 1009-1017.  doi: 10.1016/j.physleta.2007.08.052.

[12]

B. Hernández-Bermejo, Generalization of solutions of the Jacobi PDEs associated to time reparametrizations of Poisson systems, J. Math. Anal. Appl., 344 (2008), 655-666.  doi: 10.1016/j.jmaa.2008.02.061.

[13]

J. Hietarinta, A search for integrable two-dimensional Hamiltonian systems with polynomial potential, Phys. Lett. A, 96 (1983), 273-278.  doi: 10.1016/0375-9601(83)90178-0.

[14]

J. Hong, L. Ji, X. Wang and J. Zhang, Stochastic K-symplectic integrators for stochastic non-canonical Hamiltonian systems and applications to the Lotka-Volterra model, arXiv preprint arXiv: 1711.03258, (2017).

[15]

J. HongJ. RuanL. Sun and L. Wang, Structure-preserving numerical methods for stochastic Poisson systems, Commun. Comput. Phys., 29 (2021), 802-830.  doi: 10.4208/cicp.OA-2019-0084.

[16]

J. HongS. Zhai and J. Zhang, Discrete gradient approach to stochastic differential equations with a conserved quantity, SIAM J. Numer. Anal., 49 (2011), 2017-2038.  doi: 10.1137/090771880.

[17]

P. E. Klöden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.

[18]

P. E. Klöden, E. Platen and H. Schurz, Numerical Solution of SDE Through Computer Experiments, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-57913-4.

[19]

X. LiQ. Ma and X. Ding, High-order energy-preserving methods for stochastic Poisson systems, East Asian J. Appl. Math., 9 (2019), 465-484.  doi: 10.4208/eajam.290518.310718.

[20]

X. LiC. ZhangQ. Ma and X. Ding, Discrete gradient methods and linear projection methods for preserving a conserved quantity of stochastic differential equations, Int. J. Comput. Math., 95 (2018), 2511-2524.  doi: 10.1080/00207160.2017.1408803.

[21]

X. LiC. ZhangQ. Ma and X. Ding, Arbitrary high-order EQUIP methods for stochastic canonical Hamiltonian systems, Taiwan. J. Math., 23 (2019), 703-725.  doi: 10.11650/tjm/180803.

[22]

S. Lie, Zur Theorie der Transformationsgruppen, Christ. Forth. Aar. 1888, Nr. 13, Christiania 1888; Gesammelte Abh., 5,553–557.

[23]

R. I. McLachlanG. R. W. Quispel and N. Robidoux, Geometric integration using discrete gradients, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357 (1999), 1021-1045.  doi: 10.1098/rsta.1999.0363.

[24]

L. MeiL. Huang and S. Huang, Exponential integrators with quadratic energy preservation for linear Poisson systems, J. Comput. Phys., 387 (2019), 446-454.  doi: 10.1016/j.jcp.2019.03.005.

[25]

G. N. Milstein, Numerical Integration of Stochastic Differential Equations, Kluwer Academic Publishers Group, Dordrecht, 1995 doi: 10.1007/978-94-015-8455-5.

[26]

T. Misawa, Energy conservative stochastic difference scheme for stochastic Hamilton dynamical systems, Jap. J. Ind. Appl. Math., 17 (2000), 119. doi: 10.1007/BF03167340.

[27]

Y. Miyatake, A derivation of energy-preserving exponentially-fitted integrators for Poisson systems, Comput. Phys. Commun., 187 (2015), 156-161.  doi: 10.1016/j.cpc.2014.11.003.

[28]

Y. Miyatake, A fourth-order energy-preserving exponentially-fitted integrator for Poisson systems, AIP Conf. Proc., AIP Publishing LLC, 1648 (2015), 180004. doi: 10.1063/1.4912467.

[29]

G. R. W. Quispel and R. I. McLachlan, Special issue on geometric numerical integration of differential equations, J. Phys. A: Math. Gen., 39 (2006), 5251-5651. 

[30]

B. Wang and X. Wu, Functionally-fitted energy-preserving integrators for Poisson systems, J. Comput. Phys., 364 (2018), 137-152.  doi: 10.1016/j.jcp.2018.03.015.

[31]

W. Zhu and M. Qin, Poisson schemes for Hamiltonian systems on Poisson manifolds, Computers. Math. Applic., 27 (1994), 7-16.  doi: 10.1016/0898-1221(94)90081-7.

show all references

References:
[1]

L. BrugnanoM. CalvoJ. I. Montijano and L. Rández, Energy-preserving methods for Poisson systems, J. Comput. Appl. Math., 236 (2012), 3890-3904.  doi: 10.1016/j.cam.2012.02.033.

[2]

D. Cohen and G. Dujardin, Energy-preserving integrators for stochastic Poisson systems, Commun. Math. Sci., 12 (2014), 1523-1539.  doi: 10.4310/CMS.2014.v12.n8.a7.

[3]

D. Cohen and E. Hairer, Linear energy-preserving integrators for Poisson systems, BIT Numer. Math., 51 (2011), 91-101.  doi: 10.1007/s10543-011-0310-z.

[4]

M. Dahlby, B. Owren and T. Yaguchi, Preserving multiple first integrals by discrete gradients, J. Phys. A: Math. Theor., 44 (2011), 305205. doi: 10.1088/1751-8113/44/30/305205.

[5]

K. Engø and S. Faltinsen, Numerical integration of Lie-Poisson systems while preserving coadjoint orbits and energy, SIAM J. Numer. Anal., 39 (2001), 128-145.  doi: 10.1137/S0036142999364212.

[6]

E. Faou and T. Lelièvre, Conservative stochastic differential equations: Mathematical and numerical analysis, Math. Comput., 78 (2009), 2047-2074.  doi: 10.1090/S0025-5718-09-02220-0.

[7]

K. Feng and M. Qin, Symplectic Geometric Algorithms for Hamiltonian Systems, Springer-Verlag, Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-01777-3.

[8]

I. A. Garcia and B. Hernandez–Bermejo, Perturbed Euler top and bifurcation of limit cycles on invariant Casimir surfaces, Physica D, 239 (2010), 1665-1669.  doi: 10.1016/j.physd.2010.04.013.

[9]

O. Gonzalez, Time integration and discrete Hamiltonian systems, J. Nonlinear Sci., 6 (1996), 449-467.  doi: 10.1007/BF02440162.

[10]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, Second Edition, Springer-Verlag Berlin Heidelberg, 2006.

[11]

B. Hernández-Bermejo, Characterization, global analysis and integrability of a family of Poisson structures, Phys. Lett. A, 372 (2008), 1009-1017.  doi: 10.1016/j.physleta.2007.08.052.

[12]

B. Hernández-Bermejo, Generalization of solutions of the Jacobi PDEs associated to time reparametrizations of Poisson systems, J. Math. Anal. Appl., 344 (2008), 655-666.  doi: 10.1016/j.jmaa.2008.02.061.

[13]

J. Hietarinta, A search for integrable two-dimensional Hamiltonian systems with polynomial potential, Phys. Lett. A, 96 (1983), 273-278.  doi: 10.1016/0375-9601(83)90178-0.

[14]

J. Hong, L. Ji, X. Wang and J. Zhang, Stochastic K-symplectic integrators for stochastic non-canonical Hamiltonian systems and applications to the Lotka-Volterra model, arXiv preprint arXiv: 1711.03258, (2017).

[15]

J. HongJ. RuanL. Sun and L. Wang, Structure-preserving numerical methods for stochastic Poisson systems, Commun. Comput. Phys., 29 (2021), 802-830.  doi: 10.4208/cicp.OA-2019-0084.

[16]

J. HongS. Zhai and J. Zhang, Discrete gradient approach to stochastic differential equations with a conserved quantity, SIAM J. Numer. Anal., 49 (2011), 2017-2038.  doi: 10.1137/090771880.

[17]

P. E. Klöden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.

[18]

P. E. Klöden, E. Platen and H. Schurz, Numerical Solution of SDE Through Computer Experiments, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-57913-4.

[19]

X. LiQ. Ma and X. Ding, High-order energy-preserving methods for stochastic Poisson systems, East Asian J. Appl. Math., 9 (2019), 465-484.  doi: 10.4208/eajam.290518.310718.

[20]

X. LiC. ZhangQ. Ma and X. Ding, Discrete gradient methods and linear projection methods for preserving a conserved quantity of stochastic differential equations, Int. J. Comput. Math., 95 (2018), 2511-2524.  doi: 10.1080/00207160.2017.1408803.

[21]

X. LiC. ZhangQ. Ma and X. Ding, Arbitrary high-order EQUIP methods for stochastic canonical Hamiltonian systems, Taiwan. J. Math., 23 (2019), 703-725.  doi: 10.11650/tjm/180803.

[22]

S. Lie, Zur Theorie der Transformationsgruppen, Christ. Forth. Aar. 1888, Nr. 13, Christiania 1888; Gesammelte Abh., 5,553–557.

[23]

R. I. McLachlanG. R. W. Quispel and N. Robidoux, Geometric integration using discrete gradients, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357 (1999), 1021-1045.  doi: 10.1098/rsta.1999.0363.

[24]

L. MeiL. Huang and S. Huang, Exponential integrators with quadratic energy preservation for linear Poisson systems, J. Comput. Phys., 387 (2019), 446-454.  doi: 10.1016/j.jcp.2019.03.005.

[25]

G. N. Milstein, Numerical Integration of Stochastic Differential Equations, Kluwer Academic Publishers Group, Dordrecht, 1995 doi: 10.1007/978-94-015-8455-5.

[26]

T. Misawa, Energy conservative stochastic difference scheme for stochastic Hamilton dynamical systems, Jap. J. Ind. Appl. Math., 17 (2000), 119. doi: 10.1007/BF03167340.

[27]

Y. Miyatake, A derivation of energy-preserving exponentially-fitted integrators for Poisson systems, Comput. Phys. Commun., 187 (2015), 156-161.  doi: 10.1016/j.cpc.2014.11.003.

[28]

Y. Miyatake, A fourth-order energy-preserving exponentially-fitted integrator for Poisson systems, AIP Conf. Proc., AIP Publishing LLC, 1648 (2015), 180004. doi: 10.1063/1.4912467.

[29]

G. R. W. Quispel and R. I. McLachlan, Special issue on geometric numerical integration of differential equations, J. Phys. A: Math. Gen., 39 (2006), 5251-5651. 

[30]

B. Wang and X. Wu, Functionally-fitted energy-preserving integrators for Poisson systems, J. Comput. Phys., 364 (2018), 137-152.  doi: 10.1016/j.jcp.2018.03.015.

[31]

W. Zhu and M. Qin, Poisson schemes for Hamiltonian systems on Poisson manifolds, Computers. Math. Applic., 27 (1994), 7-16.  doi: 10.1016/0898-1221(94)90081-7.

Figure 1.  Root mean-square convergence orders of the Milstein scheme, the Klöden scheme, the P-Milstein scheme, and the P-Klöden scheme
Figure 2.  Evolution of $ H^0(y),H^1(y) $ by the Milstein scheme and the P-Milstein scheme for system (32)
Figure 3.  Evolution of $ y^1 $ by the Milstein scheme and the P-Milstein scheme
Figure 4.  Evolution of the Casimir function by the Milstein scheme and the P-Milstein scheme
Figure 5.  Root mean-square convergence orders of the Euler scheme, the Milstein scheme, the P-Euler scheme, and the P-Milstein scheme
Figure 6.  Evolution of $ H^0(X),\,\,H^1(X) $ by the Milstein scheme and the P-Milstein scheme for the system (34)
Figure 7.  A sample path of $ y^1 $ produced by the Milstein scheme and the P-Milstein scheme for the system (34)
Figure 8.  Root mean-square convergence orders of the Milstein scheme and the P-Milstein scheme
Figure 9.  Evolution of $ H^0(y) $ and $ H^1(y) $ by the Milstein scheme and the P-Milstein scheme for system (35)
Figure 10.  Evolution of $ y^2 $ by the Milstein scheme and the P-Milstein scheme for the system (35)
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