New three-stages symmetric six-step finite difference method with vanished phase-lag and its derivatives up to sixth derivative for second order initial and/or boundary value problems with periodical and/or oscillating solutions

In this paper, for the first time in the literature, we develop a symmetric three-stages six-step method with the following characteristics; the methodis a symmetric hybrid (multistages) six-step method,is of three-stages,is of twelfth algebraic order,has vanished the phase-lag andhas vanished the derivatives of the phase-lag up to order six. A detailed theoretical, numerical and computational analysis is also presented. The above analyses consist of:the construction of the new six-step pair,the presentation of the computed local truncation error of the new six-step pair,the comparative error analysis of the new six-step pair with other six-step pairs of the same family which are:the classical six-step pair of the family (i.e. the six-step pair with constant coefficients),the recently proposed six-step pair of the same family with vanished phase-lag and its first derivative,the recently proposed six-step pair of the same family with vanished phase-lag and its first and second derivatives,the recently proposed six-step pair of the same family with vanished phase-lag and its first, second and third derivatives,the recently proposed six-step pair of the same family with vanished phase-lag and its first, second, third and fourth derivatives and finally,the recently proposed six-step pair of the same family with vanished phase-lag and its first, second, third, fourth and fifth derivativesthe stability and the interval of periodicity analysis for the new obtained six-step pair and finallythe investigation of the accuracy and computational efficiency of the new developed six-step pair for the solution of the Schrödinger equation. The theoretical, numerical and computational achievements lead to the conclusion that the new produced three-stages symmetric six-step pair is more efficient than other known or recently developed finite difference pairs of the literature.