New 8-step symmetric embedded predictor–corrector (EPCM) method with vanished phase-lag and its first derivative for the numerical integration of the Schrödinger equation

In the present work, a new embedded predictor–corrector phase-fitted method with vanished phase-lag is developed for the first time in literature. It is about a multistep symmetric method built on the multistep method of Quinlan and Tremaine (Astron J 100(5):1694–1700, 1990 ) with eight steps. It can be used to solve numerically IVPs with oscillatory solutions, orbital problems and the Schrödinger equation. Initially we present a pair of embedded predictor–corrector method upon which the new method is built. We test our method and the numerical results indicate that this method is much more accurate than other well known methods including the radial Schrödinger equation.