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 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.
Highlights
Lie–Poisson systems and isospectral flows are two well-studied classes of dynamical systems
We develop spectral preserving numerical methods for flows of form (1) which, in the case of Hamiltonian isospectral flows (5), preserves the Lie–Poisson structure
For i, j = 1, . . . , s, the corresponding Runge–Kutta method is symplectic when applied to canonical Hamiltonian systems on R2n [27]
Summary
Lie–Poisson systems and isospectral flows are two well-studied classes of dynamical systems. One can, in some cases, use collective symplectic integrators, which rely on Clebsch variables originating from a Hamiltonian action of G on a symplectic vector space (see [18,19] for details) Compared to these methods, our approach is: (i) simpler since the algorithms are formulated directly on the algebra g ⊂ gl(n, C); (ii) free of constraints; (iii) free of algebra-to-group maps, such as the exponential or Cayley map; (iv) generic as they apply to any isospectral Hamiltonian flow. Definition 2 (IsoSyPRK) Given two Butcher tableaux (9) fulfilling the symplectic conditions (10), the corresponding Isospectral Symplectic Partitioned Runge–Kutta method for flow (1) is the map h : gl(n, C) Wk −→ Wk+1 ∈ gl(n, C). Proof The theorem is a combination of results proved in Theorem 3, Corollary 1, Theorems 5 and 6
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