Abstract
The paper deals with numerical discretizations of separable nonlinear Hamiltonian systems with additive noise. For such problems, the expected value of the total energy, along the exact solution, drifts linearly with time. We present and analyze a time integrator having the same property for all times. Furthermore, strong and weak convergence of the numerical scheme along with efficient multilevel Monte Carlo estimators are studied. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme.
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Acknowledgements
We appreciate the referees’ comments on an earlier version of the paper. We would like to thank Gilles Vilmart (University of Geneva) for interesting discussions. The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at HPC2N, Umeå University.
Funding
Open access funding provided by Umeå University. The work of CCC is supported by the National Natural Science Foundation of China (Nos. 11871068, 11971470, 11711530017, 91630312, 11926417). The work of DC was supported by the Swedish Research Council (VR) (projects nr. 2013 − 4562 and nr. 2018 − 04443). The work of RD was supported by GNCS-INDAM project and by PRIN2017-MIUR project “Structure preserving approximation of evolutionary problems”. The author RD is member of the GNCS-INDAM group. The work of AL was supported in part by the Swedish Research Council under Reg. No. 621-2014-3995 and by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation. This work was partially supported by STINT and NSFC Joint China-Sweden Mobility programme (project nr. CH2016 − 6729).
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Communicated by: Anthony Nouy
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Chen, C., Cohen, D., D’Ambrosio, R. et al. Drift-preserving numerical integrators for stochastic Hamiltonian systems. Adv Comput Math 46, 27 (2020). https://doi.org/10.1007/s10444-020-09771-5
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DOI: https://doi.org/10.1007/s10444-020-09771-5
Keywords
- Stochastic differential equations
- Stochastic Hamiltonian systems
- Energy
- Trace formula
- Numerical schemes
- Strong convergence
- Weak convergence
- Multilevel Monte Carlo