Symplectic RK Methods and Symplectic PRK Methods with Real Eigenvalues

Abstract

Properties of symplectic Runge-Kutta (RK) methods and symplectic partitioned Runge- Kutta (PRK) methods with real eigenvalues are discussed in this paper. It is shown that an $s$ stage such method can't reach order more than $s + 1$. Particularly, we prove that no symplectic RK method with real eigenvalues exists in stage $s$ of order $s + 1$ when $s$ is even. But an example constructed by using the W-transformation shows that PRK method of this type does not necessarily meet this order barrier. Another useful way other than W-transformation to construct symplectic PRK method with real eigenvalues is then presented. Finally, a class of efficient symplectic methods is recommended.