Explicit Symplectic Runge-Kutta-Nystrom Methods Based on Roots of Shifted Legendre Polynomial

To date, all explicit symplectic Runge-Kutta-Nystrom methods of order five or above are derived by numerical solutions of order condition equations and symplectic condition. In this paper, we derive 124 sets of seven-stage fifth-order explicit symplectic Runge-Kutta-Nystrom methods with closed-form coefficients in the Butcher tableau using the roots of a degree-3 shifted Legendre polynomial. One method is analyzed and its P-stable interval is derived. Numerical tests on the two newly discovered methods are performed, showing their long-time stability and large step size stability over some existing methods.