Family of symmetric linear six-step methods with vanished phase-lag and its derivatives and their application to the radial Schrödinger equation and related problems

A new family of implicit eighth algebraic order symmetric six-step methods is obtained in this paper. For this family, we request for the first time in the literature vanishing of the phase-lag and its derivatives. For this new family of methods, we will investigate the following: (1) the construction of the methods of the family , i.e. the definition of the coefficients of each method of the family in order the phase-lag and its derivatives to be vanished, (2) the calculation of the formula of the Local Truncation Error, (3) the comparative local truncation error analysis (i.e. the application methods of the family on a test problem and the investigation of their behaviors), (4) the stability analysis of the methods of the new proposed family, by applying the methods of the family to a scalar test equation with frequency different than the frequency of the scalar test equation for the phase-lag analysis and by investigating the obtained results i.e. by studying the interval of periodicity of the obtained methods of the family. We finally investigate the behavior the methods of the new family by applying the new methods to the numerical solution of the resonance problem of the radial Schrödinger equation. We prove the efficiency of the obtained methods of the new family by comparing it with (1) well known methods in the literature and (2) very recently obtained methods.