Lie-Poisson methods for isospectral flows. (English) Zbl 1450.37073
Lie-Poisson systems and isospectral flows are two well-known classes of dynamical systems. Lie-Poisson systems appear as Poisson reductions of Hamiltonian systems. The isospectral flows appear as Lax formulations of integrable systems.
In this paper, spectral preserving numerical methods of arbitrary order for isospectral flows are developed. They preserve in the case of Hamiltonian isospectral flows the Lie-Poisson structure. The methods are simply constructed and avoid the use of constraints or exponential maps as for other numerical methods for isospectral flows. The authors show through the framework of Poisson reduction, that the methods are directly related to classical symplectic Runge-Kutta methods and partitioned symplectic Runge-Kutta methods. Therefore they are named as Isospectral Symplectic Runge-Kutta (IsoSyRK) methods. The methods are applied to Hamiltonian systems such as the generalized rigid body, the (periodic) Toda lattice, the Euler equations on a sphere, point vortices on a sphere, the Heisenberg spin chain, and the Bloch-Iserles flow. They are also applied to non-Hamiltonian systems as for example the Toeplitz inverse eigenvalue problem, Chu’s flow on symmetric real matrices, and the Brockett flow. Through numerical examples, the efficiency and high-order accuracy of the methods is demonstrated. The MATLAB codes are available at https://bitbucket.org/Milo_Viviani/iso-runge-kutta.
In this paper, spectral preserving numerical methods of arbitrary order for isospectral flows are developed. They preserve in the case of Hamiltonian isospectral flows the Lie-Poisson structure. The methods are simply constructed and avoid the use of constraints or exponential maps as for other numerical methods for isospectral flows. The authors show through the framework of Poisson reduction, that the methods are directly related to classical symplectic Runge-Kutta methods and partitioned symplectic Runge-Kutta methods. Therefore they are named as Isospectral Symplectic Runge-Kutta (IsoSyRK) methods. The methods are applied to Hamiltonian systems such as the generalized rigid body, the (periodic) Toda lattice, the Euler equations on a sphere, point vortices on a sphere, the Heisenberg spin chain, and the Bloch-Iserles flow. They are also applied to non-Hamiltonian systems as for example the Toeplitz inverse eigenvalue problem, Chu’s flow on symmetric real matrices, and the Brockett flow. Through numerical examples, the efficiency and high-order accuracy of the methods is demonstrated. The MATLAB codes are available at https://bitbucket.org/Milo_Viviani/iso-runge-kutta.
Reviewer: Bülent Karasözen (Ankara)
MSC:
| 37M15 | Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems |
| 65P10 | Numerical methods for Hamiltonian systems including symplectic integrators |
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 37J39 | Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.) |
| 53D20 | Momentum maps; symplectic reduction |
Keywords:
isospectral flow; Lie-Poisson integrator; symplectic Runge-Kutta methods; Toda flow; generalized rigid body; Chu’s flow; Bloch-Iserles flow; Euler equations; point vorticesReferences:
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