Efficient symplectic Runge-Kutta methods

The present paper continues the research in [H.Y. Liu, G. Sun, Symplectic RK methods and symplectic PRK methods with real eigenvalues, J. Comput. Math. 22(5) (2004) 769-776] on symplectic Runge-Kutta (RK) methods with real eigenvalues. In a general setting, a new but simple proof of the main result in [H.Y. Liu, G. Sun, Symplectic RK methods and symplectic PRK methods with real eigenvalues, J. Comput. Math. 22(5) (2004) 769-776] is given that an s-stage, pth order such method must have that p ⩽ s + 1 when s is odd, and p ⩽ s when s is even. Then it is shown that in case s is odd, the maximum order is reachable. However, in comparison with composition method, the latter is superior in consideration of efficiency in high order. Some theoretically interesting properties of such methods are included.