Four-Stages Twelfth Algebraic Order Two-Step Method with Vanished Phase-Lag and its First and Second Derivatives for the Numerical Solution of the Schrodinger Equation

In this paper we introduce, for the fist time in the literature, a new four-stages symmetric two-step method of twelfth algebraic order. For the new family of methods we request the vanishing of the phase-lag and its first and second derivatives. We investigate how the vanishing of the phase-lag and its derivatives affect on the effectiveness of the finally produced new four stages symmetric two-step method. We will study the following: the construction of the method, the computation of the local truncation error (LTE) of the new four-stages symmetric two-step method the analysis of the LTE when the produced method is applied to a test problem (which is the radial Schrodinger equation) the comparison of the asymptotic formula of the LTE of the developed method (which is produced by the application of the new obtained four-stages symmetric two-step method to the test problem mentioned above) with the asymptotic formulae of the LTEs of other similar methods in the literature (comparative local truncation error analysis), the stability (interval of periodicity) of the new obtained four-stages symmetric two-step method. We mention that for the investigation of the stability of the new produced method we use a scalar test equation with frequency different than the frequency of the scalar test equation used for the phase-lag analysis (stability analysis), the examination of the effectiveness of the new obtained method applying it to two problems of the literature: (i) the resonance problem of the Schrodinger equation and (ii) the coupled differential equations arising from the Schrodinger equation. Finally, it will be proved that this new introduced family of methods is very efficient for the numerical solution of the Schrodinger equation and related initial-value or boundary-value problems with periodical and/or oscillating solutions. We mention here that the proposed method is an improvement of the recent developed methods in [1], [2] and [3].