High-order closed Newton-Cotes trigonometrically-fitted formulae for long-time integration of orbital problems

The connection between closed Newton-Cotes, trigonometrically-tted dierential 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 NewtonCotes dieren tial 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-tted 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.

[1]  A. Hinchliffe,et al.  Chemical Modelling: Applications and Theory , 2008 .

[2]  Xinsheng Zhao,et al.  Numerical methods with a high order of accuracy applied in the quantum system , 1996 .

[3]  J. M. Sanz-Serna,et al.  Numerical Hamiltonian Problems , 1994 .

[4]  S. Tremaine,et al.  Symmetric Multistep Methods for the Numerical Integration of Planetary Orbits , 1990 .

[5]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[6]  G. Vanden Berghe,et al.  Extended One‐Step Methods: An Exponential Fitting Approach , 2004 .

[7]  E. Hairer,et al.  Solving Ordinary Differential Equations II , 2010 .

[8]  T. E. Simos,et al.  SIMPLE AND ACCURATE EXPLICIT BESSEL AND NEUMANN FITTED METHODS FOR THE NUMERICAL SOLUTION OF THE SCHRÖDINGER EQUATION , 2000 .

[9]  Tom E. Simos.  Exponentially-fitted and trigonometrically-fitted methods for long-term integration of orbital problems , 2002 .

[10]  T. E. Simos.  P-stable Four-Step Exponentially-Fitted Method for the Numerical Integration of the Schr¨odinger Equation , 2005 .

[11]  Zacharias A. Anastassi,et al.  A dispersive-fitted and dissipative-fitted explicit Runge–Kutta method for the numerical solution of orbital problems , 2004 .

[12]  Tom E. Simos,et al.  Trigonometrically-fitted partitioned multistep methods for the integration of orbital problems , 2004 .

[13]  G. Psihoyios.  A Block Implicit Advanced Step-point (BIAS) Algorithm for Stiff Differential Systems , 2006 .

[14]  Shuen-De Wu,et al.  Open Newton–Cotes differential methods as multilayer symplectic integrators , 1997 .

[15]  G. Vanden Berghe,et al.  Exponentially-fitted algorithms: fixed or frequency dependent knot points? , 2004 .

[16]  Tom E. Simos.  Dissipative trigonometrically fitted methods for the numerical solution of orbital problems , 2004 .

[17]  T. E. Simos.  Exponentially-fitted and trigonometrically-fitted methods for the numerical solution of orbital problems , 2003 .

[18]  Tom E. Simos,et al.  An Eighth-Order Method With Minimal Phase-Lag For Accurate Computations For The Elastic Scattering Phase-Shift Problem , 1996 .

[19]  D. G. Bettis,et al.  Stabilization of Cowell's method , 1969 .

[20]  Jesús Vigo-Aguiar,et al.  An exponentially-fitted high order method for long-term integration of periodic initial-value problems , 2001 .

[21]  Zacharias A. Anastassi,et al.  A trigonometrically fitted Runge–Kutta method for the numerical solution of orbital problems , 2005 .

[22]  Georgios Psihoyios,et al.  Trigonometrically-fitted symmetric multistep methods for the approximate solution of orbital problems , 2003 .