An implicit symmetric linear six-step methods with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the radial Schrodinger equation and related problems

In this paper we develop a new implicit eighth algebraic order symmetric six-step method. For thismethod we request for the first time in the literature vanishing of the phase-lag and its first, second, third and fourth derivatives. The investigation of the new method consists of the following: the development of the method, i.e. the production of the coefficients of the method in order the phase-lag and the derivatives of the phase-lag to be vanished, the computation of the formula of the Local Truncation Error, the application of the new obtained method to a test problem (the radial Schrodinger equation) and the production of the asymptotic form of the local truncation for this test problem the comparison of the asymptotic forms of the local truncation error of known similar methods with the asymptotic form local truncation error of of the new developed method (comparative local truncation error analysis) the stability investigation of the new produced method. This investigation is taken place by using a scalar test equation with frequency different than the frequency of the scalar test equation for the phase-lag analysis and by studying the produced results i.e. by studying the interval of periodicity of the developed method. Finally we will apply the new developed method to the approximate solution of the resonance problem of the radial Schrodinger equation. The above mentioned application will help us on the study of the efficiency of the new obtained method. We will test the efficiency of the produced method by comparing it with (1) well known methods of the literature and (2) very recently obtained methods.