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A minimal-variable symplectic method for isospectral flows. (English) Zbl 1448.37107

A new minimal-variable second-order implicit midpoint method is introduced for general isospectral flows and isospectral Lie-Poisson systems. The technique avoids computationally heavy transformations and it belongs to the class of Isospectral Symplectic Rung-Kutta (IsoSyRK) methods (see [K. Modin and M. Viviani, Found. Compute. Math. 20, No. 4, 889–921 (2020; Zbl 1450.37073)]).
The method is here derived reducing the number of unknowns up to minimality and revealing an intrinsic relation between the implicit midpoint method and the Cayley transform. When the isospectral flow is Lie-Poisson, it preserves the coadjoint orbits. The method is compared numerically with the spherical midpoint method, which is another minimal-variable Lie-Poisson integrator on \(\mathbb{R}^3\). Structure-preserving properties of the isospectral minimal midpoint method are stated for the generalized rigid body, the Brockett flow, Lie-Poisson systems on \((\mathbb{R}^3,\mathsf{x})\) and on \(\mathrm{sl}(2,\mathbb{R})\). The Casimir functions are conserved up to round-off errors, and for isospectral Hamilton flows, near conservation of the Hamiltonians are shown. Furthermore, the authors present the isospectral minimal midpoint on \(\mathrm{sl}(2,\mathbb{R})\), applying it to the point vortex equations on the hyperbolic plane.

MSC:

37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37J70 Completely integrable discrete dynamical systems
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
53D20 Momentum maps; symplectic reduction
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics

Citations:

Zbl 1450.37073
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References:

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