Approximating Symplectic Realizations: A General Framework for the Construction of Poisson Integrators

TL;DR

This paper introduces a general framework for constructing Poisson integrators on Poisson manifolds, addressing the geometric complexities and providing error analysis and applications.

Contribution

It develops a novel approach for Poisson integrator construction based on symplectic realizations, filling a gap in geometric integration theory.

Findings

Provides a general method for Poisson integrator construction

Includes error analysis for the proposed methods

Demonstrates applications on example systems

Abstract

While the construction of symplectic integrators for Hamiltonian dynamics is well understood, an analogous general theory for Poisson integrators is still lacking. The main challenge lies in overcoming the singular and non-linear geometric behavior of Poisson structures, such as the presence of symplectic leaves with varying dimensions. In this paper, we propose a general approach for the construction of geometric integrators on any Poisson manifold based on independent geometric and dynamic sources of approximation. The novel geometric approximation is obtained by adapting structural results about symplectic realizations of general Poisson manifolds. We also provide an error analysis for the resulting methods and illustrative applications.