A new explicit four-step method with vanished phase-lag and its first and second derivatives

A study on the vanishing of the phase-lag and its first and second derivatives for a family of explicit four-step methods first introduced by Anastassi and Simos (J Comput Appl Math 236:3880–3889, 2012) is presented in this paper. The methods investigated in this paper belongs to the category of methods with frequency dependent coefficients. For these methods we will investigate the procedure of vanishing of the phase-lag and its first and second derivatives. For the new proposed methods we will define the local truncation error and we will study an local truncation error analysis. Finally we will compare the results of the error analysis with other known methods of the literature. We will study also the stability analysis of the new proposed method. We will apply the new produced methods on the resonance problem of the Schrödinger equation in order to investigate their efficiency.