Lie–Poisson Methods for Isospectral Flows
Artikel i vetenskaplig tidskrift, 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

Författare

Klas Modin

Chalmers, Matematiska vetenskaper, Tillämpad matematik och statistik

Milo Viviani

Chalmers, Matematiska vetenskaper, Tillämpad matematik och statistik

Foundations of Computational Mathematics

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

Ämneskategorier

Beräkningsmatematik

Reglerteknik

Matematisk analys

DOI

10.1007/s10208-019-09428-w

Mer information

Senast uppdaterat

2019-11-08