A new eight algebraic order embedded explicit six-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the Schrödinger equation

Abstract

A new embedded symmetric six-step method with vanished phase-lag and its first, second, third and fourth derivatives is obtained for the first time in the literature in this paper. More precisely, for this new method we will investigate: -The development of the methods, i.e., the definition of the coefficients of the new method in order the phase-lag and its first, second, third and fourth derivatives of the phase-lag to be vanished, -The computation of the formulae of the Local Truncation Error (LTE) of the methods of the new family of methods, -The computation of the asymptotic forms of the LTE by applying of the new developed method on a scalar test problem (which is the time independent radial Schrödinger equation), -The computation of the asymptotic forms of the LTE of other methods of the same family and the comparative local truncation error analysis of all the methods of the family, -The stability (interval of periodicity) analysis of the new obtained method. To study the stability, we will use a scalar test equation with frequency different than the frequency of the scalar test equation for the phase-lag analysis.123 J Math Chem -The behavior of the new produced method by applying them to the numerical solution of the resonance problem of the radial Schrödinger equation. We will prove its efficiency by comparing it with (1) well known methods of the literature and (2) very recently obtained methods.