Lie–Poisson Methods for Isospectral Flows
Journal article, 2019

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.

Isospectral flow

Symplectic Runge–Kutta methods

Point vortices

Generalized rigid body

Bloch–Iserles flow

Euler equations

Toda flow

Chu’s flow

Lie–Poisson integrator

Author

Klas Modin

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Milo Viviani

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Foundations of Computational Mathematics

1615-3375 (ISSN) 1615-3383 (eISSN)

Subject Categories

Computational Mathematics

Control Engineering

Mathematical Analysis

DOI

10.1007/s10208-019-09428-w

More information

Latest update

11/8/2019