Some New Properties And Order Results Of Runge-Kutta-Nystr(?)m Methods

The main results in this paper are as follows:(1) We study the two different ways by which a Runge - Kutta(RK) methodΦgenerates a Runge - Kutta - Nystrom (RKN) methodΦ_N~*:Φ→Φ-→Φ_N~* andΦ→Φ_N→Φ_N~*,and proved thatΦ_N~* generated by the above two ways are the same. We discuss the relations amongΦ,Φ_N andΦ_N~* with symplecticity, symmetry or P-stability , and show that:①ifΦis symplectic, thenΦ_N also is symplectic;②ifΦis symmetric, thenΦ_N is symmetric;③the simplifying assumptions ofΦ_N generated fromΦcan be droved;④Φ_N~* is symplectic whenΦ_N is symplectic;⑤Φ_N andΦ_N~* have the same P-stability.By①and②, we provide a new approach to construct symplectic (or symmetric) RKN methods from symplectic (or symmetric) RK methods.(2) We investigate the order conditions and the order characterizations of symplectic RKN methods and study the order of singly diagonally symplectic RKN methods and explicit RKN methods, and prove that the highest possible order is six.(3) Some examples of the symplectic RKN methods are given from the symplectic RK methods. By using these examples, some numerical tests are presented.