A New Algorithm for the Approximation of the Schrödinger Equation

In this paper a four stages twelfth algebraic order symmetric two-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives is developed for the first time in the literature. For the new proposed method: (1) we will study the phase-lag analysis, (2) we will present the development of the new method, (3) the local truncation error (LTE) analysis will be studied. The analysis is based on a test problem which is the radial time independent Schrödinger equation, (4) the stability and the interval of periodicity analysis will be presented, (5) stepsize control technique will also be presented, (6) the examination of the accuracy and computational cost of the proposed algorithm which is based on the approximation of the Schrödinger equation