Efficient low computational cost hybrid explicit four-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical integration of the Schrodinger equation

Based on an optimized explicit four-step method, a new hybrid high algebraic order four-step method is introduced in this paper. For this new hybrid method, we investigate the procedure of vanishing of the phase-lag and its first, second, third and fourth derivatives. More specifically, we investigate: (1) the construction of the new method, i.e. the computation of the coefficients of the method in order its phase-lag and first, second, third and fourth derivatives of the phase-lag to be eliminated, (2) the definition of the local truncation error, (3) the analysis of the local truncation error, (4) 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, we investigate computationally the new obtained method by applying it to the numerical solution of the resonance problem of the radial Schrodinger equation. The efficiency of the new developed method is tested comparing this method with well known methods of the literature but also using very recently developed methods.