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Projection methods for stochastic differential equations with conserved quantities

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Abstract

In this paper, we consider the numerical methods preserving single or multiple conserved quantities, and these methods are able to reach high order of strong convergence simultaneously based on some kinds of projection methods. The mean-square convergence orders of these methods under certain conditions are given, which can reach order 1.5 or even 2 according to the supporting methods embedded in the projection step. Finally, three numerical experiments are taken into account to show the superiority of the projection methods.

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References

  1. Anton, C., Wong, Y.S., Deng, J.: Symplectic schemes for stochastic Hamiltonian systems preserving Hamiltonian functions. Int. J. Numer. Anal. Model. 11(3), 427–451 (2014)

    MathSciNet  Google Scholar 

  2. Burrage, K., Burrage, P., Tian, T.: Numerical methods for strong solutions of stochastic differential equations: an overview. Proc. R. Soc. Lond. Ser. A 460(2041), 373–402 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, C., Cohen, D., Hong, J.: Conservative methods for stochastic differential equations with a conserved quantity. Int. J. Numer. Anal. Model. (2016). arXiv:1411.1819

  4. Cohen, D.: On the numerical discretisation of stochastic oscillators. Math. Comput. Simul. 82(8), 1478–1495 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cohen, D., Dujardin, G.: Energy-preserving integrators for stochastic Poisson systems. Commun. Math. Sci. 12(8), 1523–1539 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dennis Jr., J.E., Moré, J.J.: Quasi-Newton methods, motivation and theory. SIAM Rev. 19(1), 46–89 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer, Berlin (2002)

    Book  MATH  Google Scholar 

  8. Hong, J., Scherer, R., Wang, L.: Predictor-corrector methods for a linear stochastic oscillator with additive noise. Math. Comput. Model. 46(5), 738–764 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hong, J., Xu, D., Wang, P.: Preservation of quadratic invariants of stochastic differential equations via Runge-Kutta methods. Appl. Numer. Math. 87, 38–52 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hong, J., Zhai, S., Zhang, J.: Discrete gradient approach to stochastic differential equations with a conserved quantity. SIAM J. Numer. Anal. 49(5), 2017–2038 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  12. Malham, S.J.A., Wiese, A.: Stochastic Lie group integrators. SIAM J. Sci. Comput. 30(2), 20 (2007)

    MathSciNet  MATH  Google Scholar 

  13. Milstein, G., Repin, Y.M., Tretyakov, M.: Numerical methods for stochastic systems preserving symplectic structure. SIAM J. Numer. Anal. 40(4), 1583–1604 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Milstein, G., Repin, Y.M., Tretyakov, M.V.: Symplectic integration of Hamiltonian systems with additive noise. SIAM J. Numer. Anal. 39(6), 2066–2088 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Milstein, G.N., Tretyakov, M.V.: Stochastic Numerics for Mathematical Physics. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  16. Misawa, T.: Conserved quantities and symmetries related to stochastic dynamical systems. Ann. Inst. Stat. Math. 51(4), 779–802 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Misawa, T.: Energy conservative stochastic difference scheme for stochastic Hamilton dynamical systems. Jpn. J. Ind. Appl. Math. 17(1), 119–128 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Misawa, T.: Symplectic integrators to stochastic Hamiltonian dynamical systems derived from composition methods. Math. Probl. Eng. (2010)

  19. Norton, R.A., McLaren, D.I., Quispel, G.R.W., Stern, A., Zanna, A.: Projection methods and discrete gradient methods for preserving first integrals of ODEs. Discret. Contin. Dyn. Syst. 35(5), 2079–2098 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors are thankful to the anonymous referees for their comments which helped improve the paper.

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Authors and Affiliations

  1. College of Science, National University of Defense Technology, Changsha, 410073, China

    Weien Zhou & Songhe Song

  2. School of Mathematical Science, China University of Mining and Technology, Beijing, 100083, China

    Liying Zhang

  3. Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijing, 100190, China

    Jialin Hong

  4. State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha, 410073, China

    Songhe Song

Authors
  1. Weien Zhou
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  2. Liying Zhang
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  3. Jialin Hong
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  4. Songhe Song
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Corresponding author

Correspondence to Weien Zhou.

Additional information

Communicated by David Cohen.

This work was supported by National Natural Science Foundation of China (No. 91130003) and found from HPCL.

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Zhou, W., Zhang, L., Hong, J. et al. Projection methods for stochastic differential equations with conserved quantities. Bit Numer Math 56, 1497–1518 (2016). https://doi.org/10.1007/s10543-016-0614-0

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  • DOI: https://doi.org/10.1007/s10543-016-0614-0

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