Drift-preserving numerical integrators for stochastic Poisson systems

TL;DR

This paper introduces a novel numerical integrator for stochastic Poisson systems that preserves key invariants like energy and Casimirs over long times, with proven convergence properties and demonstrated effectiveness.

Contribution

It proposes a drift-preserving splitting scheme that maintains invariants exactly and achieves high-order convergence, advancing numerical methods for stochastic Hamiltonian and Poisson systems.

Findings

The scheme preserves energy and Casimirs exactly.

Achieves mean-square order of convergence one.

Achieves weak order of convergence two.

Abstract

We perform a numerical analysis of a class of randomly perturbed {H}amiltonian systems and {P}oisson systems. For the considered additive noise perturbation of such systems, we show the long time behavior of the energy and quadratic Casimirs for the exact solution. We then propose and analyze a drift-preserving splitting scheme for such problems with the following properties: exact drift preservation of energy and quadratic Casimirs, mean-square order of convergence one, weak order of convergence two. These properties are illustrated with numerical experiments.