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Lie-Poisson Methods for Isospectral Flows

Journal article
Authors Klas Modin
Milo Viviani
Published in Foundations of Computational Mathematics
ISSN 1615-3375
Publication year 2019
Published at Department of Mathematical Sciences
Language en
Links doi.org/10.1007/s10208-019-09428-w
Keywords Bloch–Iserles flow, Chu’s flow, Euler equations, Generalized rigid body, Isospectral flow, Lie–Poisson integrator, Point vortices, Symplectic Runge–Kutta methods, Toda flow
Subject categories Computational Mathematics

Abstract

© 2019, The Author(s). 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 spectrum 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.

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