Lie–Poisson Methods for 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.

A minimal-variable symplectic method for isospectral flows

“…Applying these to (8) we get, after some computations, the scheme (5). In [18,Thm. 3], it has been proven that when B(W ) = ∇H(W ) † , for some functions H : gl(n, C) * → R, the method is a Lie-Poisson integrator in gl(n, C) * .…”

“…The example we consider is the Heisenberg spin chain on R 3N . For this one has to extend both the isospectral minimal midpoint (14) and the spherical midpoint (15) to direct products of R 3 (see [17] and [18]).…”

“…Therefore, the preservation of the spectrum of the dynamical variable is a key feature of a numerical scheme applied to isospectral flows, in order to expect the right qualitative behaviour of the discrete approximate solutions [6]. Furthermore, as a special case, Lie-Poisson systems on the dual of reductive Lie algebras can be seen as isospectral flows [18]. A reductive Lie algebra is defined as the direct sum of a semisimple Lie algebra and an abelian Lie algebra.…”

“…In this paper, we focus on the IsoSyRK associated to the implicit midpoint method, which turns out to have a specially nice structure. On the one hand, we provide a simpler proof (avoiding the use of the B-series theory) that for the implicit midpoint method, the respective IsoSyRK defined in [18] is isospectral for any B = B(W ). On the other hand, we derive a simpler scheme reducing the number of unknowns up to minimality, revealing an intrinsic relation between the implicit midpoint method and the Cayley transform.…”

A Casimir preserving scheme for long-time simulation of spherical ideal hydrodynamics

“…To discretize (4) in time we apply a Lie-Poisson preserving isospectral symplectic Runge-Kutta (IsoSRK) integrator as developed in [31]. These numerical methods exactly conserve (i.e.…”