Is a symmetric hybrid six-step method,

For the first time in the literature we develop in this paper a three stages symmetric six-step scheme with twelfth algebraic and eliminated phase-lag and its first derivative. An additional characteristic of the new scheme is that the first and the second layer denote the approximation of the function on the point \(x_{n+3}\). We also present a local truncation error analysis and a stability and interval of periodicity analysis and we compared the new scheme with the classical scheme (i.e. scheme with constant coefficients). Additionally, we examine in details the accuracy and computational efficiency of the new developed scheme on the numerical solution of the Schrödinger equation. The study and investigation which are presented in this paper, lead to the conclusion that the new obtained scheme is more effective than other known or recently developed methods of the literature.