Symplectic Runge-Kutta Methods for Hamiltonian Systems Driven by Gaussian Rough Paths

Jialin Hong; Chuying Huang; Xu Wang

doi:10.1016/j.apnum.2018.03.006

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arXiv · 1704.04144

Symplectic Runge-Kutta Methods for Hamiltonian Systems Driven by Gaussian Rough Paths

Jialin Hong·Chuying Huang·Xu Wang

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Abstract

We consider Hamiltonian systems driven by multi-dimensional Gaussian processes in rough path sense, which include fractional Brownian motions with Hurst parameter $H\in(1/4,1/2]$. We indicate that the phase flow preserves the symplectic structure almost surely and this property could be inherited by symplectic Runge--Kutta methods, which are implicit methods in general. If the vector fields belong to $Lip^{\gamma}$, we obtain the solvability of Runge--Kutta methods and the pathwise convergence rates. For linear and skew symmetric vector fields, we focus on the midpoint scheme to give corresponding results. Numerical experiments verify our theoretical analysis.

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BibTeX RIS

Jialin Hong, Chuying Huang, Xu Wang. 2017-04-13. Symplectic Runge-Kutta Methods for Hamiltonian Systems Driven by Gaussian Rough Paths. https://doi.org/10.1016/j.apnum.2018.03.006

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