A family of two stages tenth algebraic order symmetric six-step methods with vanished phase-lag and its first derivatives for the numerical solution of the radial Schrödinger equation and related problems

Abstract

In this paper, we introduce (first time in the literature) a family of two-stages high algebraic order symmetric six-step methods—especially one of them which is phase-fitted and another one with phase-lag and its first derivative vanished. For this family of methods we study (1) the production of the methods, i.e. the calculation of the coefficients of the members of the family so that the phase-lag and its derivative be eliminated, (2) the calculation of the formulae of the Local Truncation Error of the new introduced methods, (3) the comparative local truncation error analysis. We note here that, for the purpose of this analysis, we use, as a test equation, the time independent radial Schrodinger equation, (4) the stability analysis (interval of periodicity) of the new methods of the proposed family. We note that for the purpose of stability analysis, we use a scalar test equation with frequency other than the frequency of the scalar test equation used for the phase-lag analysis, and (5) the computational behavior of the new obtained methods by applying them to the numerical solution of the resonance problem of the radial Schrodinger equation. Based on the above, we prove the efficiency of the new developed methods by comparing them with (a) well known methods in the literature and (b) very recently obtained methods.