A high algebraic order multistage explicit four-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives for the numerical solution of the Schrödinger equation

A new Multistage high algebraic order four-step method is obtained in this paper. It is the first time in the literature that a method of this category is developed and has vanishing of the phase-lag and its first, second, third, fourth and fifth derivatives. We study this new method by investigating: (1) the development of the new method, i.e. the calculation of the coefficients of the method in order the phase-lag and its first, second, third, fourth and fifth derivatives of the phase-lag to be vanished, (2) the determination of the formula of the Local Truncation Error, (3) the comparative analysis of the Local Truncation Error (with this we mean the application of the new method and similar methods on a test problem and the analysis of their behavior), (4) the stability of the new method, by applying the new obtained method to a scalar test equation with frequency different than the frequency of the scalar test equation for the phase-lag analysis and by studying the results of this application i.e. by investigating the interval of periodicity of the new obtained method. We finally study the computational behavior the new developed method by using the application of the new method to the approximate solution of the resonance problem of the radial Schrödinger equation. We prove the effectiveness of the new obtained method by comparing it with (1) well known methods of the literature and (2) very recently obtained methods.