A minimal-variable symplectic method for isospectral flows
Artikel i vetenskaplig tidskrift, 2020

Isospectral flows are abundant in mathematical physics; the rigid body, the the Toda lattice, the Brockett flow, the Heisenberg spin chain, and point vortex dynamics, to mention but a few. Their connection on the one hand with integrable systems and, on the other, with Lie–Poisson systems motivates the research for optimal numerical schemes to solve them. Several works about numerical methods to integrate isospectral flows have produced a large varieties of solutions to this problem. However, many of these algorithms are not intrinsically defined in the space where the equations take place and/or rely on computationally heavy transformations. In the literature, only few examples of numerical methods avoiding these issues are known, for instance, the spherical midpoint method on {{\mathfrak {s}}}{{\mathfrak {o}}}(3). In this paper we introduce a new minimal-variable, second order, numerical integrator for isospectral flows intrinsically defined on quadratic Lie algebras and symmetric matrices. The algorithm is isospectral for general isospectral flows and Lie–Poisson preserving when the isospectral flow is Hamiltonian. The simplicity of the scheme, together with its structure-preserving properties, makes it a competitive alternative to those already present in literature.

symplectic runge–kutta methods

point-vortex on the hyperbolic plane

heisenberg spin chain

brockett flow

isospectral flow

generalized rigid body

lie–poisson integrator

Författare

[Person a357d643-93c5-496c-8170-334d33dfd94c not found]

Chalmers, Matematiska vetenskaper, Tillämpad matematik och statistik

BIT Numerical Mathematics

0006-3835 (ISSN) 1572-9125 (eISSN)

Vol. 60 3 741-758

Ämneskategorier

Beräkningsmatematik

Reglerteknik

Datavetenskap (datalogi)

DOI

10.1007/s10543-019-00792-1

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2021-02-11