In this paper, we propose a new class of splitting methods to solve the stochasticLangevin equation, which can simultaneously preserve the ergodicity and exponential integrabilityof the original equation. The central idea is to extract a stochastic subsystem that possesses thestrict dissipation from the original equation, which is inspired by the inheritance of the Lyapunovstructure for obtaining the ergodicity. We prove that the exponential moment of the numerical so-lution is bounded, thus validating the exponential integrability of the proposed methods. Further,we show that under moderate verifiable conditions, the methods have the first-order convergence inboth strong and weak senses, and we present several concrete splitting schemes based on the meth-ods. The splitting strategy of methods can be readily extended to construct conformal symplecticmethods and high-order methods that preserve both the ergodicity and the exponential integrability,as demonstrated in numerical experiments. Our numerical experiments also show that the proposedmethods have good performance in the long-time simulation.

Publication:

SIAM Journal on Numerical AnalysisVolume 63, Issue 2

http://dx.doi.org/10.1137/24M1687686

Author:

CHUCHU CHEN

LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People's Republic of China and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, People's Republic of China

chenchuchu@lsec.cc.ac.cn

TONGHE DANG

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190,People's Republic of China, and Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China

tonghe.dang@polyu.edu.hk

JIALIN HONG

LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People's Republic of China and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, People's Republic of China

hjl@lsec.cc.ac.cn

FENGSHAN ZHANG

Corresponding author

LSEC, ICMSEC, Academy of Mathematics and Systems Science, ChineseAcademy of Sciences, Beijing 100190, People's Republic of China

zhangfs23@lsec.cc.ac.cn