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 less than or equal to p...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 less than or equal to p less than or equal to s - 1) provided that the simplifying conditions C(p) (or D(p) with non-zero weights) and B(2p + l) hold, where 1 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 <less than or equal to> 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) over cap = 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 + l) can be easily computed, where 1 I P less than or equal to s, 0 less than or equal to l less than or equal to 2, s less than or equal to 2p + l less than or equal to 2s.展开更多
基金the National Natural Science Foundation of China.
文摘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 less than or equal to p less than or equal to s - 1) provided that the simplifying conditions C(p) (or D(p) with non-zero weights) and B(2p + l) hold, where 1 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 <less than or equal to> 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) over cap = 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 + l) can be easily computed, where 1 I P less than or equal to s, 0 less than or equal to l less than or equal to 2, s less than or equal to 2p + l less than or equal to 2s.
