Trigonometrically fitted fifth-order runge-kutta methods for the numerical solution of the schrِdinger equation

Two trigonometrically fitted methods based on a classical Runge-Kutta method of Kutta-Nystrِm are being constructed. The new methods maintain the fifth algebraic order of the classical one but also have some other significant properties. The most important one is that in the local truncation error of the new methods the powers of the energy are lower and that keeps the error at lower values, especially at high values of energy. The error analysis justifies the actual results when integrating the radial Schrِdinger equation, where the high efficiency of the new methods is shown.