Abstract
We consider a stochastic autonomous Hamiltonian system for which the flow preserves the symplectic structure. Numerical simulations show that for stochastic Hamiltonian systems symplectic schemes produce more accurate results for long term simulations than non-sysmplectic numerical schemes. We study the approximation error corresponding to a symplectic weak scheme of order one. A backward error analysis is done at the level of the Kolmogorov equation associated with the initial stochastic Hamiltonian system. We obtain an expansion of the error in terms of powers of the discretization step size and the solutions of the modified Kolmogorov equation.
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References
Abdulle, A., Cohen, D., Vilmart, G., Zygalakis, K.: High weak order methods for stochastic differential equations based on modified equations. SIAM J. Sci. Comput. 34(3), 1800–1823 (2012)
Anton, C., Wong, Y., Deng, J.: On global error of symplectic schemes for stochastic Hamiltonian systems. Int. J. Numer. Anal. Model. Ser. B 4(1), 80–93 (2013)
Debussche, A., Faou, E.: Weak backward error analysis for SDEs. SIAM J. Numer. Anal. 50(3), 1735–1752 (2012)
Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer, Berlin (2006)
Hong, J., Sun, L., Wang, X.: High order conformal symplectic and ergodic schemes for the stochastic Langevin equation via generating functions. SIAM J. Numer. Anal. 55(6), 3006–3029 (2017)
Kopec, M.: Weak backward error analysis for Langevin process. BIT Numer. Math. 55(4), 1057–1103 (2015)
Kopec, M.: Weak backward error analysis for overdamped Langevin processes. IMA J Numer. Anal. 35(2), 583–614 (2015)
Mattingly, J., Stuart, A., Higham, D.J.: Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise. Stochast. Process. Appl. 2(101), 185–232 (2002)
Milstein, G.N., Tretyakov, M.V.: Stochastic Numerics for Mathematical Physics. Springer, Berlin (2004)
Shardlow, T.: Modified equations for stochastic differential equations. BIT Numer. Math. 46(1), 111–125 (2006)
Sun, L., Wang, L.: Stochastic symplectic methods based on the Pade approximations for linear stochastic Hamiltonian systems. J. Comp. Appl. Math. 311, 439–456 (2017)
Talay, D.: Second order discretization schemes of stochastic differential systems for the computation of the invariant law. Stochast. Stochast. Rep. 29(1), 13–36 (1990)
Talay, D., Tubaro, L.: Expansion of the global error for numerical schemes solving stochastic differential equations. Stochast. Anal. Appl. 8(4), 483–509 (1990)
Wang, L., Hong, J., Sun, L.: Modified equations for weakly convergent stochastic symplectic schemes via their generating functions. BIT Numer. Math. 56, 1131–1162 (2016)
Acknowledgements
This work is supported by the NSERC grant DDG-2015-00041.
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Anton, C. (2018). Error Expansion for a Symplectic Scheme for Stochastic Hamiltonian Systems. In: Kilgour, D., Kunze, H., Makarov, R., Melnik, R., Wang, X. (eds) Recent Advances in Mathematical and Statistical Methods . AMMCS 2017. Springer Proceedings in Mathematics & Statistics, vol 259. Springer, Cham. https://doi.org/10.1007/978-3-319-99719-3_51
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DOI: https://doi.org/10.1007/978-3-319-99719-3_51
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