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Structure-preserving numerical methods for a class of stochastic Poisson systems. (English) Zbl 1490.60210

Summary: We propose a type of numerical methods for a class of stochastic Poisson systems with invariant energy. The proposed numerical methods preserve both the energy and the Casimir functions of the systems. In addition, we provide a new approach of constructing stochastic Poisson integrators which respect the Poisson structure and the Casimir functions of stochastic Poisson systems based on coordinate transformations on the midpoint method. Numerical tests are performed to demonstrate our theoretical analysis.

MSC:

60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
65C30 Numerical solutions to stochastic differential and integral equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
65D30 Numerical integration
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[1] M. Beck and M.J. Gander. On the positivity of Poisson integrators for the Lotka-Volterra equations. BIT, 55:319-340, 2014. · Zbl 1321.65184
[2] D. Cohen and G. Dujardin. Energy-preserving integrators for stochastic Poisson systems. Commun. Math. Sci., 12:1523-1539, 2014. · Zbl 1310.60074
[3] D. Cohen and E. Hairer. Linear energy-preserving integrators for Poisson systems. BIT, 51:91-101, 2011. · Zbl 1216.65175
[4] F. Diele and C. Marangi. Positive symplectic integrators for Predator-Prey dynamics. Dis-crete Cont. Dyn.-A, 23:2661-2678, 2018. · Zbl 1407.37112
[5] K. Feng and D. L. Wang. Symplectic difference schemes for Hamiltonian systems in general symplectic structure. J. Comput. Math., 9:86-96, 1991. · Zbl 0738.65058
[6] O. Gonzalez. Time integration and discrete Hamiltonian systems. J. Nonlinear Sci., 6:449-467, 1996. · Zbl 0866.58030
[7] E. Hairer, C. Lubich, and G. Wanner. Geometric numerical integration: Structure-preserving algorithms for ordinary differential equations. Springer Science & Business Media, Berlin, 2nd edition, 2006. · Zbl 1094.65125
[8] J.L. Hong, L.H. Ji, and X. Wang. Stochastic K-symplectic integrators for stochastic non-canonical Hamiltonian systems and applications to the Lotka-Volterra model. arX-iv:1711.03258v1 [math.NA] 9 Nov 2017.
[9] J.L. Hong, J.L. Ruan, L.Y. Sun, and L.J. Wang. Structure-preserving numerical methods for stochastic Poisson systems. Commun. Comput. Phys., 29:802-830, 2021. · Zbl 1481.65261
[10] B. Karasözen. Poisson integrators. Math. Comput. Model., 40:1225-1244, 2004. · Zbl 1074.65145
[11] P.E. Klöden and E. Platen. Numerical solution of stochastic differential equations. Stochastic Modelling and Applied Probability. Springer, 1995.
[12] X.Y. Li, Q. Ma, and X.H. Ding. High-order energy-preserving methods for stochastic Poisson systems. East. Asia. J. Appl. Math., 9:465-484, 2019. · Zbl 1459.60128
[13] S. Lie. Zur Theorie der Transformationsgruppen. Christ. Forh. Aar. 1888, Nr. 13, Christiania 1888;
[14] Gesammelte Abh., 5:553-557, 1888.
[15] X.R. Mao, G. Marion, and E. Renshaw. Environmental Brownian noise suppresses explosions in population dynamics. Stoch. Proc. Appl., 97:95-110, 2002. · Zbl 1058.60046
[16] R.I. McLachlan. Explicit Lie-Poisson integration and the Euler equations. Phys. Rev. Lett., 71:3043-3046, 1993. · Zbl 0972.65509
[17] R.I. McLachlan and G.R.W. Quispel. Splitting methods. Acta Numer., 11:341-434, 2002. · Zbl 1105.65341
[18] G.N. Milstein, Y.M. Repin, and M.V. Tretyakov. Mean-square symplectic methods for Hamil-tonian systems with multiplicative noise. WIAS preprint No. 670, 2001.
[19] G.N. Milstein, Y.M. Repin, and M.V. Tretyakov. Numerical methods for stochastic systems preserving symplectic structure. SIAM J. Numer. Anal., 40:1583-1604, 2002. · Zbl 1028.60064
[20] G.N. Milstein, Y.M. Repin, and M.V. Tretyakov. Symplectic integration of Hamiltonian sys-tems with additive noise. SIAM J. Numer. Anal., 39:2066-2088, 2002. · Zbl 1019.60056
[21] M. Plank. Hamiltonian structures for the n-dimensional Lotka-Volterra equations. J. Math. Phys., 36:3520-3534, 1995. · Zbl 0842.34012
[22] G.R.W. Quispel and D.I. Mclaren. A new class of energy-preserving numerical integration methods. J. Phys. A: Math. Theor., 41:045206, 2008. · Zbl 1132.65065
[23] S. Reich. Numerical integration of the generalized Euler equations. Techn. Report 93-20, Dept. Comput. Sci., Univ. of British Columbia, 1993.
[24] J.M. Sanz-Serna. An unconventional symplectic integrator of W. Kahan. Appl. Numer. Math., 16:245-250, 1994. · Zbl 0815.65090
[25] Y.F. Tang, V.M. Pérez-García, and L. Vázquez. Symplectic methods for the Ablowitz-Ladik model. Appl. Math. Comput., 82:17-38, 1997. · Zbl 0870.65115
[26] L.J. Wang, P.J. Wang, and Y.Z. Cao. Numerical methods preserving multiple Hamiltonians for stochastic Poisson systems. Discrete Cont. Dyn.-S, doi: 10.3934/dcdss.2021095, 2021. · Zbl 1492.60210 · doi:10.3934/dcdss.2021095
[27] P.J. Wang and L.J. Wang. Stochastic Poisson integrators based on Padé approximations for linear stochastic Poisson systems. J. U. Chinese Acad. Sci., 38(2):160-170, 2021.
[28] Y.C. Wang and L.J. Wang. Analysis of the solutions of a class of stochastic Poisson systems. J. U. Chinese Acad. Sci., accepted, 2020.
[29] R.L. Zhang, J. Liu, Y.F. Tang, H. Qin, and B.B. Zhu. Canonicalization and symplectic simulation of the gyrocenter dynamics in time-independent magnetic fields. Phys. Plasmas, 21:032504, 2014.
[30] G. Zhong and J.E. Marsden. Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators. Phys. Lett. A, 133(3):134-139, 1988. · Zbl 1369.70038
[31] B.B. Zhu, R.L. Zhang, Y.F. Tang, X.B. Tu, and Y. Zhao. Splitting K-symplectic methods for non-canonical separable Hamiltonian problems. J. Comput. Phys., 322:387-399, 2016. · Zbl 1352.65649
[32] W.J. Zhu and M.Z. Qin. Poisson schemes for Hamiltonian systems on Poisson manifolds. Comput. Math. Appl., 27(12):7-16, 1994. · Zbl 0807.58018
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