A note on pseudo-symplectic Runge-Kutta methods (Q1279703)

In a previous paper of the authors [ibid. 38, No. 3, 439-461 (1998; Zbl 0916.65084)] they introduced the so-called pseudo-symplectic Runge-Kutta methods of order \(r\) as methods of this type which preserve the symplectic form within \({\mathcal O}(h^r)\) terms. Further they proved that Runge-Kutta methods with algebraic order \(p\) and pseudo-symplectic order \(2p\), termed as methods of type \((p,2p)\), show a linear growth of the global error, and therefore can be useful for long term integrations. In this note the authors complete this research proving that there exist methods of the above type with \(p=4\) and six explicit stages and with \(p=5\) and eight explicit stages.