Comparison of some special optimized fourth-order Runge-Kutta methods for the numerical solution of the Schrödinger equation

A new fourth-order explicit Runge-Kutta method based on an approach of Simos is developed in this paper. Numerical experiments reveal that the new method is much more efficient than other special tuned RK methods and the Numerov method for the numerical solution of the radial Schrödinger equation for large energies. An error analysis is made and the asymptotic expressions of the local errors for large energies explain the numerical results in the case of the resonance problem.