A Family of Exponentially-fitted Runge–Kutta Methods with Exponential Order Up to Three for the Numerical Solution of the Schrödinger Equation

Abstract

We have constructed three Runge–Kutta methods based on a classical method of Fehlberg with eight stages and sixth algebraic order. These methods have exponential order one, two and three. We show through the error analysis of the methods that by increasing the exponential order, the maximum power of the energy in the error expression decreases. So the higher the exponential order the smaller the local truncation error of the method compared to the corresponding classical method. The difference is higher for higher values of energy. The results confirm this, when integrating the resonance problem of the one-dimensional time-independent Schrodinger equation.