An exponential map free implicit midpoint method for stochastic Lie-Poisson systems

TL;DR

This paper introduces a new integrator for stochastic Lie-Poisson systems that preserves key geometric structures and is scalable to high-dimensional problems, with proven convergence and conservation properties.

Contribution

The paper develops an exponential map free implicit midpoint method tailored for stochastic Lie-Poisson systems, ensuring structure preservation and scalability.

Findings

Preserves Casimir functions and coadjoint orbits almost surely.

Demonstrates strong and weak convergence rates.

Validates method on rigid body, vortex dynamics, and Euler equations.

Abstract

An integrator for a class of stochastic Lie-Poisson systems driven by Stratonovich noise is developed. The integrator is suited for Lie-Poisson systems that also admit an isospectral formulation, which enables scalability to high-dimensional systems. Its derivation follows from discrete Lie-Poisson reduction of the symplectic midpoint scheme for stochastic Hamiltonian systems. We prove almost sure preservation of Casimir functions and coadjoint orbits under the numerical flow and provide strong and weak convergence rates of the proposed method. The scalability, structure-conservation, and convergence rates are illustrated numerically for the (generalized) rigid body, point vortex dynamics, and the two-dimensional Euler equations on the sphere.