Casimir preserving stochastic Lie-Poisson integrators

TL;DR

This paper introduces Casimir-preserving stochastic integrators for Lie-Poisson systems with Stratonovich noise, ensuring invariants are conserved during long-term simulations, demonstrated on stochastic heavy top and sine-Euler equations.

Contribution

It extends Runge-Kutta Munthe-Kaas methods to stochastic Lie-Poisson equations, preserving Casimir invariants exactly during stochastic integration.

Findings

Numerical methods preserve Casimir invariants exactly.

Method applied successfully to stochastic heavy top.

Method maintains structure in stochastic sine-Euler equations.

Abstract

Casimir preserving integrators for stochastic Lie-Poisson equations with Stratonovich noise are developed extending Runge-Kutta Munthe-Kaas methods. The underlying Lie-Poisson structure is preserved along stochastic trajectories. A related stochastic differential equation on the Lie algebra is derived. The solution of this differential equation updates the evolution of the Lie-Poisson dynamics by means of the exponential map. The constructed numerical method conserves Casimir-invariants exactly, which is important for long time integration. This is illustrated numerically for the case of the stochastic heavy top and the stochastic sine-Euler equations.