New optimized explicit modified RKN methods for the numerical solution of the Schrödinger equation

Abstract

This paper investigates a family of modified Runge-Kutta-Nystrom (RKN) methods for the integration of second-order ordinary differential equations with oscillatory solutions. The order conditions for up to order five are presented. Two new optimized explicit four-stage modified RKN methods are derived by nullifying their dispersions and the dissipations in two different ways, respectively. These methods are checked to be of algebraic order five and both are dispersive of order six and dissipative of order five. The stability is examined and the error formulas are analyzed to show that advantages of the new methods compared with some highly efficient integrators from the recent literature. The high accuracy of the second new method is explained by its comparatively small dispersion and dissipation constants. In the integration of the resonance problem and the bound-states problem of the radial Schrodinger equation with the Woods-Saxon potential, the numerical results show the effectiveness and robustness of the new methods.