Symplectic and symmetric methods for the numerical solution of some mathematical models of celestial objects

In the last years, the theory of numerical methods for system of non-stiff and stiff ordinary differential equations has reached a certain maturity. So, there are many excellent codes which are based on Runge–Kutta methods, linear multistep methods, Obreshkov methods, hybrid methods or general linear methods. Although these methods have good accuracy and desirable stability properties such as A-stability and L-stability, they are not suitable for the numerical solution of special classes of problems arising from different research areas, for example the mathematical models of celestial objects which are Hamiltonian systems. Since the solution of such problems has special geometric property such as symplecticity and usually reversibility. Therefore, it is natural to search for numerical methods that share this property. It is the purpose of this paper to design high order symplectic and symmetric methods. Efficiency and accuracy of the constructed methods are confirmed by implementing on well-known Hamiltonian problems of the motions of celestial objects.