# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2021095
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## Numerical methods preserving multiple Hamiltonians for stochastic Poisson systems

 1 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China 2 Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA

* Corresponding author: Yanzhao Cao

Received  December 2020 Revised  July 2021 Early access August 2021

Fund Project: The first and second authors are supported by NNSFC No. 11971458, No. 11471310

In this paper, we propose a class of numerical schemes for stochastic Poisson systems with multiple invariant Hamiltonians. The method is based on the average vector field discrete gradient and an orthogonal projection technique. The proposed schemes preserve all the invariant Hamiltonians of the stochastic Poisson systems simultaneously, with possibility of achieving high convergence orders in the meantime. We also prove that our numerical schemes preserve the Casimir functions of the systems under certain conditions. Numerical experiments verify the theoretical results and illustrate the effectiveness of our schemes.

Citation: Lijin Wang, Pengjun Wang, Yanzhao Cao. Numerical methods preserving multiple Hamiltonians for stochastic Poisson systems. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021095
##### References:

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##### References:
Root mean-square convergence orders of the Milstein scheme, the Klöden scheme, the P-Milstein scheme, and the P-Klöden scheme
Evolution of $H^0(y),H^1(y)$ by the Milstein scheme and the P-Milstein scheme for system (32)
Evolution of $y^1$ by the Milstein scheme and the P-Milstein scheme
Evolution of the Casimir function by the Milstein scheme and the P-Milstein scheme
Root mean-square convergence orders of the Euler scheme, the Milstein scheme, the P-Euler scheme, and the P-Milstein scheme
Evolution of $H^0(X),\,\,H^1(X)$ by the Milstein scheme and the P-Milstein scheme for the system (34)
A sample path of $y^1$ produced by the Milstein scheme and the P-Milstein scheme for the system (34)
Root mean-square convergence orders of the Milstein scheme and the P-Milstein scheme
Evolution of $H^0(y)$ and $H^1(y)$ by the Milstein scheme and the P-Milstein scheme for system (35)
Evolution of $y^2$ by the Milstein scheme and the P-Milstein scheme for the system (35)
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