Trigonometrically Fitted Runge-Kutta Methods of Order Five for the Numerical Solution of the Schrödinger Equation

We are constructing two trigonometrically fitted methods based on a classical Runge-Kutta method. The new methods maintain the 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 Schrödinger Equation, where the high efficiency of the new methods is shown.