We investigate stochastic modified equations to explain the mathematical mechanism of symplectic methods applied to rough Hamiltonian systems. The contribution of this paper is threefold. First, we construct a new type of stochastic modified equation. For symplectic methods applied to rough Hamiltonian systems, the associated stochastic modified equations are proved to have Hamiltonian formulations. Secondly, the pathwise convergence order of the truncated modified equation to the numerical method is obtained by techniques in rough path theory. Thirdly, if increments of noises are simulated by truncated random variables, we show that the error can be made exponentially small with respect to the time step size.

Publication:

IMA Journal of Numerical Analysis, drae019

https://doi.org/10.1093/imanum/drae019

Author:

Chuchu Chen

LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China

Email: chenchuchu@lsec.cc.ac.cn

Jialin Hong

LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China

Email: hjl@lsec.cc.ac.cn

Chuying Huang

School of Mathematics and Statistics, Key Laboratory of Analytical Mathematics and Applications (Ministry of Education), Fujian Key Laboratory of Analytical Mathematics and Applications (FJKLAMA) and Center for Applied Mathematics of Fujian Province (FJNU), Fujian Normal University, Fuzhou, China