Closed newton-cotes trigonometrically-fitted formulae for long-time integration of orbital problems

 

 T. E. Simos¹*

 

¹ Laboratory of Computational Sciences, Department of Computer Science and Technology, Faculty of Sciences and Technology, University of Peloponnese, GR-221 00 Tripolis, Greece (tsimos@mail.ariadne-t.gr).

Received 2005 October 5
Accepted 2006 April 18

 

RESUMEN

En este trabajo se investiga la conexión entre formulas Newton-Cotes, métodos diferenciales por ajustes trigonométricos e integradores simplécticos. Se conoce, a través de la literatura, que varios integradores simplécticos de un paso han sido obtenidos basándose en geometría simpléctica. Sin embargo, la investigación de integradores simplécticos multicapa es muy pobre. Zhu et al. (1996) presentaron los conocidos métodos diferenciales Newton-Cotes abiertos como integradores simplécticos multicapa. También Chiou & Wu (1997) investigaron la construcción de integradores simplécticos multicapa basándose en los métodos de integración abierta Newton-Cotes. En este trabajo investigamos las fórmulas cerradas Newton-Cotes y las escribimos como estructuras simplécticas multicapa. Después de esto, construimos métodos simplécticos por ajustes trigonométricos, los cuales se basan en las formulas cerradas Newton-Cotes. Aplicamos los esquemas simplécticos para resolver las ecuaciones de movimiento de Hamilton que son lineales en posición y momento. Observamos que la energía hamiltoniana del sistema permanece casi constante a medida que la integración avanza.

 

ABSTRACT

The connection between closed Newton-Cotes, trigonometrically-fitted differential methods and symplectic integrators is investigated in this paper. It is known from the literature that several one step symplectic integrators have been obtained based on symplectic geometry. However, the investigation of multistep symplectic integrators is very poor. Zhu et al. (1996) presented the well known open Newton-Cotes differential methods as multilayer symplectic integrators. Also, Chiou & Wu (1997) investigated the construction of multistep symplectic integrators based on the open Newton-Cotes integration methods. In this paper we investigate the closed Newton-Cotes formulae and we write them as symplectic multilayer structures. After this we construct trigonometrically-fitted symplectic methods which are based on the closed Newton-Cotes formulae. We apply the symplectic schemes in order to solve Hamilton's equations of motion which are linear in position and momentum. We observe that the Hamiltonian energy of the system remains almost constant as integration procceeds.

Key Words: CELESTIAL MECHANICS - METHODS: NUMERICAL.

 

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NOTES:

* Active Member of the European Academy of Sciences and Arts, Corresponding Member of the European Academy of Sciences, and Corresponding Member of European Academy of Arts, Sciences, and Humanities.