Structure-Preserving Implicit Runge-Kutta Methods for Stochastic Poisson Systems with Multiple Noises

TL;DR

This paper introduces structure-preserving implicit Runge-Kutta methods for stochastic Poisson systems with multiple noises, ensuring the preservation of key geometric properties in numerical simulations.

Contribution

It develops diagonal implicit and transformed Runge-Kutta methods that preserve Poisson structures and Casimir functions for stochastic systems with constant and non-constant structure matrices.

Findings

Methods effectively preserve Poisson structure and Casimir functions.

Numerical experiments confirm the structure-preserving properties.

Proposed methods outperform traditional schemes in accuracy and stability.

Abstract

In this paper, we propose the diagonal implicit Runge-Kutta methods and transformed Runge-Kutta methods for stochastic Poisson systems with multiple noises. We prove that the first methods can preserve the Poisson structure, Casimir functions, and quadratic Hamiltonian functions in the case of constant structure matrix. Darboux-Lie theorem combined with coordinate transformation is used to construct the transformed Runge-Kutta methods for the case of non-constant structure matrix that preserve both the Poisson structure and the Casimir functions. Finally, through numerical experiments on stochastic rigid body systems and linear stochastic Poisson systems, the structure-preserving properties of the proposed two kinds of numerical methods are effectively verified.