Abstract
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.
Highlights
The numerical integration of isospectral flows is a classical subject of study in numerical analysis [7,10]
The interest in this problem is motivated by the numerical simulation of integrable systems, which are deeply related to isospectral flows via the Lax pair formulation
In the Lax pair formulation, some of these first integrals can be presented as a linear combination of the eigenvalues of the dynamical variable
Summary
The numerical integration of isospectral flows is a classical subject of study in numerical analysis [7,10]. As a special case, Lie–Poisson systems on the dual of reductive Lie algebras can be seen as isospectral flows [18]. Any real matrix Lie algebra which is closed under conjugate transpose is reductive [12, Prop. Lie–Poisson systems on the dual of a reductive Lie algebra can be equivalently seen as isospectral flows of the form (1.2) below. Throughout the paper, we identify g∗ with g, via the Frobenius inner product A, B = Tr( A† B), where † is the conjugate transpose Via this identifications, Lie–Poisson systems on the dual of a reductive Lie algebra g take the form:. In the last section of this paper, we show some numerical examples of our scheme and we compare it with the spherical midpoint method, which is another minimal-variable Lie–Poisson integrator on R3. We show how our scheme looks on sl(2, R), defining what we call the hyperbolic midpoint method
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