A Trigonometrically Fitted Runge-Kutta Pair of Orders Four and Five for the Numerical Solution of the Schrödinger Equation

We are constructing a trigonometrically fitted pair of explicit Runge-Kutta methods of orders four and five. The pair has a variable step size which is determined by a specific algorithm. The algorithm is based on the error as that is expressed by the absolute difference of the values computed by each method separately. The step size control in combination with the trigonometrical fitting result in a very efficient method when compared to well known methods.