Lie-Poisson methods for isospectral flows

TL;DR

This paper introduces simple, high-order numerical methods for isospectral flows that preserve spectra and Lie--Poisson structures, improving accuracy and structure preservation in discretizations of integrable systems.

Contribution

The authors develop a class of arbitrary order methods that preserve both spectra and Lie--Poisson structures without complex constraints or exponential maps.

Findings

Methods preserve spectra and Lie--Poisson structure.

Near conservation of Hamiltonian function observed.

Applied successfully to classical isospectral flows.

Abstract

The theory of isospectral flows comprises a large class of continuous dynamical systems, particularly integrable systems and Lie--Poisson systems. Their discretization is a classical problem in numerical analysis. Preserving the spectra in the discrete flow requires the conservation of high order polynomials, which is hard to come by. Existing methods achieving this are complicated and usually fail to preserve the underlying Lie--Poisson structure. Here we present a class of numerical methods of arbitrary order for Hamiltonian and non-Hamiltonian isospectral flows, which preserve both the spectra and the Lie--Poisson structure. The methods are surprisingly simple, and avoid the use of constraints or exponential maps. Furthermore, due to preservation of the Lie--Poisson structure, they exhibit near conservation of the Hamiltonian function. As an illustration, we apply the methods to several classical isospectral flows.