Explicit Symplectic Runge–Kutta–Nyström Methods Based on Roots of Shifted Legendre Polynomial

Abstract

To date, all explicit symplectic Runge–Kutta–Nyström 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–Nyström 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.

Keywords

• explicit symplectic Runge–Kutta–Nyström methods
• the shifted Legendre polynomials
• order conditions
• five-stage fourth-order
• seven-stage fifth-order