A high algebraic order predictor-corrector explicit method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the Schrodinger equation and related problems

In this paper an eighth algebraic order predictor-corrector explicit four-step method is studied. The main scope of this paper is to study the consequences of (1) the vanishing of the phase-lag and its first, second, third and fourth derivatives and (2) the high algebraic order on the efficiency of the new developed method. A theoretical and computational study of the obtained method is also presented. More specifically, the theoretical study of the new predictor-corrector method consists of: The development of the new predictor-corrector method, i.e. the definition of the coefficients of the method in order its phase-lag and phase-lag's first, second, third and fourth derivatives to be vanished The computation of the local truncation error The comparative local truncation error analysis The stability (interval of periodicity) analysis, using scalar test equation with frequency different than the frequency of the scalar test equation for the phase-lag analysis. Finally, the computational study of the new predictor-corrector method consists of the application of the new produced predictor-corrector explicit four-step method to the numerical solution of the resonance problem of the radial time independent Schrodinger equation.