Advertisement

Calcolo

, 56:5 | Cite as

Structure-preserving stochastic conformal exponential integrator for linearly damped stochastic differential equations

  • Guoguo Yang
  • Qiang MaEmail author
  • Xuliang Li
  • Xiaohua Ding
Article
  • 40 Downloads

Abstract

In this paper, we study the linearly damped stochastic differential equations, which have the invariants satisfying a linear differential equation whose coefficients are linear constant or time-dependent. A stochastic exponential integrator is proposed for linearly damped stochastic differential equations to preserve their intrinsic properties. Then, the conformal symplecticity of stochastic Hamiltonian systems with linearly damped term is studied. For linearly damped stochastic Hamiltonian systems, it is shown that the stochastic exponential integrator can exactly preserve conformal quadratic invariant and conformal symplecticity. The mean-square convergence order of the method is analyzed. Numerical tests present the good performance of the proposed stochastic exponential integrator in structure-preserving.

Keywords

Damped stochastic differential equations Conformal invariant Conformal symplectic Linear damping Stochastic exponential integrator 

Mathematics Subject Classification

65C30 65C20 

Notes

Acknowledgements

This work is supported by the National Key R&D Program of China (No. 2017YFC1405600), the National Natural Science Foundation of China (Nos. 11501150 and 11701124) and the Natural Science Foundation of Shandong Province of China (No. ZR2017PA006). The authors would like to express their appreciation to the anonymous referees for their many valuable suggestions and for carefully correcting the preliminary version of the manuscript.

References

  1. 1.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Springer, Berlin (2002)CrossRefGoogle Scholar
  2. 2.
    Feng, K., Qin, M.: Symplectic Geometric Algorithms for Hamiltonian Systems. Zhejiang Science and Technology Press, Hangzhou (2003)Google Scholar
  3. 3.
    Wang, B., Yang, H., Meng, F.: Sixth-order symplectic and symmetric explicit ERKN schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations. Calcolo 54, 1–24 (2016)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Milstein, G.N., Repinand, YuM, Tretyakov, M.V.: Numerical methods for stochastic systems preserving symplectic structure. SIAM J. Numer. Anal. 40, 1583–1604 (2002)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Milstein, G.N., Repinand, YuM, Tretyakov, M.V.: Symplectic integration of Hamiltonian systems with additive noise. SIAM J. Numer. Anal. 39, 2066–2088 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ma, Q., Ding, D., Ding, X.: Symplectic conditions and stochastic generating functions of stochastic Runge–Kutta methods for stochastic Hamiltonian systems with multiplicative noise. Appl. Math. Comput. 219, 635–643 (2012)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Misawa, T.: Energy conservative stochastic difference scheme for stochastic Hamilton dynamical systems. Jpn. J. Ind. Appl. Math. 17(1), 119–128 (2000)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Li, X., Zhang, C., Ma, Q., Ding, X.: Discrete gradient methods and linear projection methods for preserving a conserved quantity of stochastic differential equations. Int. J. Comput. Math. 95(12), 2511–2524 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    McLachlan, R.I., Perlmutter, M.: Conformal Hamiltonian systems. J. Geom. Phys. 39, 276–300 (2001)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Moore, B.E.: Conformal multi-symplectic integration methods for forced-damped semi-linear wave equations. Math. Comput. Simul. 80, 20–28 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Moore, B.E., Norena, L., Schober, C.M.: Conformal conservation laws and geometric integration for damped Hamiltonian partial differential equations. J. Comput. Phys. 232, 214–233 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chen, C., Hong, J., Ji, L.: Stochastic conformal multi-symplectic method for damped stochastic nonlinear Schrödinger equation. arXiv:1803.10885 (2018)
  13. 13.
    Bhatt, A., Moore, B.E.: Structure-preserving exponential Runge–Kutta methods. SIAM J. Sci. Comput. 39, 593–612 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Komori, Y., Cohen, D., Burrage, K.: Weak second order explicit exponential Runge–Kutta methods for stochastic differential equations. SIAM J. Sci. Comput. 39, 2857–2878 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Arara, A.A., Debrabant, K., Kværnø, A.: Stochastic B-series and order conditions for exponential integrators. arXiv:1801.02051 (2018)
  16. 16.
    Milstein, G.N.: Numerical Integration of Stochastic Differential Equations, vol. 313. Springer, Dordrecht (1994)zbMATHGoogle Scholar
  17. 17.
    Jentzen, A., Kloeden, P.: Taylor Approximations for Stochastic Partial Differential Equations. Society for Idustrial and Applied Mathematics, Philadelphia (2011)CrossRefGoogle Scholar

Copyright information

© Istituto di Informatica e Telematica (IIT) 2019

Authors and Affiliations

  1. 1.Department of MathematicsHarbin Institute of TechnologyWeihaiChina
  2. 2.Department of MathematicsQingdao UniversityQingdaoChina

Personalised recommendations