Numerical integration of Lie-Poisson systems while preserving coadjoint orbits and energy. (English) Zbl 0993.65142
New numerical integrators of Runge-Kutta Munthe-Kaas type [cf. H. Muntha-Kaas, Appl. Numer. Math. 29, No. 1, 115-127 (1999; Zbl 0934.65077)] for the solution of Lie-Poisson systems are proposed. After examining several geometric properties associated to the flow of Lie-Poisson systems, the authors propose a family of algorithms which preserve coadjoint orbits and Casimirs of the Lie-Poisson equation.
Further, they show that by using the discrete derivative of the Hamiltonian function it is possible to construct implicit methods that preserve not only the above properties but also the energy of the system for a constant step size implementation. A suitable modified Newton iteration is proposed to solve the implicit equations in the Lie algebra. Finally some numerical experiments with the Lie-Poisson equations of the rigid body and the sine-Euler equations are presented.
A comparison of the behaviour of the classical Runge-Kutta method of order four, the Lie trapezoidal rule as well as other integrators of D. Lewis and J. C. Simo [J. Nonlinear Sci. 4, No. 3, 253-299 (1994; Zbl 0799.58069)] and O. Gonzalez [ibid. 6, No. 5, 449-467 (1996; Zbl 0866.58030)] are displayed.
Further, they show that by using the discrete derivative of the Hamiltonian function it is possible to construct implicit methods that preserve not only the above properties but also the energy of the system for a constant step size implementation. A suitable modified Newton iteration is proposed to solve the implicit equations in the Lie algebra. Finally some numerical experiments with the Lie-Poisson equations of the rigid body and the sine-Euler equations are presented.
A comparison of the behaviour of the classical Runge-Kutta method of order four, the Lie trapezoidal rule as well as other integrators of D. Lewis and J. C. Simo [J. Nonlinear Sci. 4, No. 3, 253-299 (1994; Zbl 0799.58069)] and O. Gonzalez [ibid. 6, No. 5, 449-467 (1996; Zbl 0866.58030)] are displayed.
Reviewer: Manuel Calvo (Zaragoza)
MSC:
| 65P10 | Numerical methods for Hamiltonian systems including symplectic integrators |
| 34A26 | Geometric methods in ordinary differential equations |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 70H05 | Hamilton’s equations |
| 37M15 | Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems |
| 37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |
| 37J05 | Relations of dynamical systems with symplectic geometry and topology (MSC2010) |
