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Symplectic Numerical Schemes for Stochastic Systems Preserving Hamiltonian Functions

  • Cristina Anton
  • Yau Shu Wong
  • Jian Deng
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8236)

Abstract

We present high-order symplectic schemes for stochastic Hamiltonian systems preserving Hamiltonian functions. The approach is based on the generating function method, and we show that for the stochastic Hamiltonian systems, the coefficients of the generating function are invariant under permutations. As a consequence, the high-order symplectic schemes have a simpler form than the explicit Taylor expansion schemes with the same order. Moreover, we demonstrate numerically that the symplectic schemes are effective for long time simulations.

Keywords

stochastic Hamiltonian systems symplectic integration mean-square convergence high-order numerical schemes 

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References

  1. 1.
    Anton, C., Deng, J., Wong, Y.S.: Symplectic schemes for stochastic Hamiltonian systems preserving Hamiltonian functions. Int. J. Num. Anal. Mod. (submitted)Google Scholar
  2. 2.
    Deng, J., Anton, C., Wong, Y.S.: High-order symplectic schemes for stochastic Hamiltonian systems. Comm. Comp. Physics (submitted)Google Scholar
  3. 3.
    Hairer, E.: Geometric numerical integration: structure-preserving algorithms for ordinary differential equations. Springer, Berlin (2006)Google Scholar
  4. 4.
    Hong, J., Wang, L., Scherer, R.: Simulation of stochastic Hamiltonian systems via generating functions. In: Proceedings IEEE 2011 4th ICCSIT (2011)Google Scholar
  5. 5.
    Kloeden, P., Platen, E.: Numerical solutions of stochastic differential equations. Springer, Berlin (1992)CrossRefGoogle Scholar
  6. 6.
    Milstein, G.N., Tretyakov, M.V., Repin, Y.M.: Symplectic integration of Hamiltonian systems with additive noise. SIAM J. Num. Anal. 39, 2066–2088 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Milstein, G.N., Tretyakov, M.V., Repin, Y.M.: Numerical methods for stochastic systems preserving symplectic structure. SIAM J. Num. Anal. 40, 1583–1604 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Wang, L.: Variational integrators and generating functions for stochastic Hamiltonian systems. Dissertation, University of Karlsruhe, Germany, KIT Scientific Publishing (2007), http://www.ksp.kit.edu

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Cristina Anton
    • 1
  • Yau Shu Wong
    • 2
  • Jian Deng
    • 2
  1. 1.Grant MacEwan UniversityEdmontonCanada
  2. 2.University of AlbertaEdmontonCanada

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