ORDER PROPERTIES AND CONSTRUCTION OF SYMPLECTIC RUNGE-KUTTA METHODS

The main results of this paper are as follows: (1) Suppose an s stage RungeKutta method is consistent, irreducible, non-confluent and symplectic. Then this method is of order at least 2P +(1) provided that the simplifying conditions C(P) (or D(p) with non-zero weights) and B(2P+1) hold, where 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 stability matrix M of the method has an (s-p) x (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 been designed, by means of which, the coefficients of s stage symplectic Runge-Kutta methods satisfying C(p), D(p) and B(2P+1) can be easily computed, where 1<