(untitled) (Journal of Computational Mathematics, 2021)
The main results of this paper are as follows: (1) Suppose an s stage Runge-Kutta method is consistent, irreducible, non-confluent and symplectic. Then this method is of order at least 2p+l(1 p s-1) provided that the simplifying conditions C(p) (or D(p) with non-zero weights) and B(2p+l) hold, where l=0,1,2. (2) Suppose an s stage Runge-Kutta method is consistent, irreducible and non-confluent, and satisfies the simplifying conditions C(p) and D(p) with 0 ＜ p s. Then this method is symplectic if and only if either p = s or the nonlinear stablility matrix M of the method has an (s-p)×(s-p) chief submatrix M=0. (3) Using the results (1) and (2) as bases, we present a general approach for the construction of symplectic Runge-Kutta methods, and a software has benn designed, by means of which, the coefficients of s stage symplectic Runge-Kutta methods satisfying C(p), D(p) and B(2p+l) can be easily computed, where 1 p s, 0 l 2, s 2p+l 2s.