Modified explicit Runge-Kutta methods for the numerical solution of the Schrödinger equation

A new procedure for constructing Runge-Kutta methods for an efficient integration of the radial Schrödinger equation is developed in this paper. The new modified Runge-Kutta methods are of algebraically order four. The asymptotic expressions of the local errors for large energies are discussed. Numerical results obtained for the widely used Woods-Saxon potential show the efficiency of the new methods compared with other special optimized fourth-order Runge-Kutta methods. The error analysis will be clearly confirmed by the resonance problem.