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Structure-preserving Runge-Kutta methods for stochastic Hamiltonian equations with additive noise

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Abstract

There has been considerable recent work on the development of energy conserving one-step methods that are not symplectic. Here we extend these ideas to stochastic Hamiltonian problems with additive noise and show that there are classes of Runge-Kutta methods that are very effective in preserving the expectation of the Hamiltonian, but care has to be taken in how the Wiener increments are sampled at each timestep. Some numerical simulations illustrate the performance of these methods.

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References

  1. Brugnano, L., Iavernaro, F., Trigiante, D.: Analysis of Hamiltonian Boundary Value Methods (HBVMs): a class of energy-preserving Runge-Kutta methods for the numerical solution of polynomial Hamiltonian dynamical systems, submitted. arXiv:09095659 (2009)

  2. Brugnano, L., Iavernaro, F., Trigiante, D.: Hamiltonian boundary value methods (energy preserving discrete line integral methods). J. Numer. Anal. Industr. Appl. Math. 5(1–2), 17–37 (2010). arXiv:09103621

    MathSciNet  Google Scholar 

  3. Brugnano, L., Iavernaro, F., Trigiante, D.: The lack of continuity and the role of infinite and infinitesimal in numerical methods for ODEs: the case of symplecticity. Appl. Math. Comput. 218, 8053–8063 (2012)

    Google Scholar 

  4. Brugnano, L., Iavernaro, F., Trigiante, D.: A simple framework for the derivation and analysis of effective classes of one-step methods for ODEs. Appl. Math. Comput. 218, 8475–8485 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. Brugnano, L., Iavernaro, F., Trigiante, D.: A note on the efficient implementation of Hamiltonian BVMs. J. Comput. Appl. Math. 236, 375–383 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. Burrage, K.: Parallel and Sequential Methods for Ordinary Differential Equations. Oxford University Press, Oxford (1995)

    MATH  Google Scholar 

  7. Burrage, K., Burrage, P.M.: Low rank Runge-Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise. J. Comput. Appl. Math. 236, 3920–3930 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  8. Burrage, K., Butcher, J.C.: Stability criteria for implicit Runge-Kutta methods. SIAM J. Numer. Anal. 16, 46–57 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  9. Burrage, K., Lenane, I., Lythe, G.: Numerical methods for second-order stochastic differential equations. SIAM J. Sci. Comput. 29(1), 245–264 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Burrage, K., Lythe, G.: Accurate stationary densities with partitioned numerical methods for stochastic differential equations. SIAM J. Numer. Anal. 47, 1601–1618 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Celledoni, E., McLachlan, R., Owren, B., Quispel, G.R.W.: Geometric properties of Kahan’s method. J. Phys. A: Math. Theor. 46, 025201 (2013)

    Article  MathSciNet  Google Scholar 

  12. Chartier, P., Faou, E., Murua, A.: An algebraic approach to invariant preserving integrators: the case of quadratic and Hamiltonian invariants. Numer. Math. 103, 575–590 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Crouzeix, M.: Sur la B-stabilité des méthodes de Runge-Kutta. Numer. Math. 32, 75–82 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  14. Faou, E., Hairer, E., Pham, T.-L.: Energy conservation with non-symplectic methods: examples and counter-examples. BIT Numer. Math. 44, 699–709 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hairer, E., Lubich, C.: The life span of backward error analysis for numerical integrators. Numer. Math. 76, 441–462 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration: structure-preserving algorithms for ordinary differential equations. Springer, Berlin (2006)

    Google Scholar 

  17. Kahan, W., Li, R.C.: Unconventional schemes for a class of ordinary differential equations - with applications to the Korteweg-de Vries equation. J. Comput. Phys. 134, 316–331 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kloeden, P.E., Platen, E.: Numerical solution of stochastic differential equations. Springer, Berlin (1999)

    Google Scholar 

  19. Mattingly, J.C., Stuart, A.M., Higham, D.J.: Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise. Stoch. Proc. Appl. 101(2), 185–232 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Melbo, A.H.S., Higham, D.J.: Numerical simulation of a linear oscillator with additive noise. Appl. Numer. Math. 51, 89 (2004)

    Article  MathSciNet  Google Scholar 

  21. Quispel, G.R.W., McLaren, D.I.: A new class of energy-preserving numerical integration methods. J. Phys. A. Math. Theor. 41, 045206 (2008)

    Article  MathSciNet  Google Scholar 

  22. Sanz-Serna, J.M., Calvo, M.P.: Numerical Hamiltonian Problems. Chapman & Hall, London (1994)

    Book  MATH  Google Scholar 

  23. Schurz, H.: The invariance of asymptotic laws of linear stochastic systems under discretization. Z. Angew. Math. Mech. 6, 375–382 (1999)

    Article  Google Scholar 

  24. Soize, C.: Exact steady-state solution of FKP equation in higher dimension for a class of nonlinear Hamiltonian dissipative dynamical systems excited by Gaussian white noise. Probabilistic Methods in Applied Physics. In: Krée, P., Wedig, W. (eds.) Lecture Notes in Physics 451, 284–309 (1995)

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Authors and Affiliations

  1. Mathematical Sciences School, Queensland University of Technology, Brisbane, 4001, Australia

    Pamela M. Burrage & Kevin Burrage

  2. Department of Computer Science, University of Oxford, Oxford, OX13QD, UK

    Kevin Burrage

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  1. Pamela M. Burrage
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  2. Kevin Burrage
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Correspondence to Pamela M. Burrage.

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Burrage, P.M., Burrage, K. Structure-preserving Runge-Kutta methods for stochastic Hamiltonian equations with additive noise. Numer Algor 65, 519–532 (2014). https://doi.org/10.1007/s11075-013-9796-6

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  • DOI: https://doi.org/10.1007/s11075-013-9796-6

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