A multistep method with optimal properties for second order differential equations

In the present paper an effective symmetric hybrid six-step method for the approximation of the solution of the Schrödinger equation and related problems is developed. More specifically, we will produced a method with the following properties: (1) is a symmetric hybrid six-step method, (2) is of twelfth algebraic order, (3) has three stages, (4) has eliminated phase-lag, (5) has eliminated the derivatives of the phase-lag up to order three. This numerical pair is obtained for the first time in the literature. We present a detailed analysis for the new obtained numerical scheme. More specifically we present:

• the construction of the new pair
• the computation of the local truncation error of the new numerical pair
• the comparison of the asymptotic form of the local truncation error of the new numerical scheme with the the classical pair of the family (i.e. scheme with constant coefficients), the recently developed scheme of the family with vanished phase-lag and its first derivative and the recently developed algorithm of the family with vanished phase-lag and its first and second derivatives
• the stability and interval of periodicity analysis and
• finally, the accuracy and computational efficiency of the new numerical pair for the solution of the Schrödinger equation.

The theoretical and numerical achievements which are presented in this paper, show the efficiency of the new numerical pair compared with other known or recently developed pairs of the literature.