A new six-step algorithm with improved properties for the numerical solution of second order initial and/or boundary value problems

A new four-stages symmetric six-step finite difference pair with improved properties is developed in this paper and for the first time in the literature. The new finite difference pair: (1) is a symmetric non-linear six-step method, (2) is of four stages, (3) is of fourteenth algebraic order, (4) has eliminated the phase-lag, (5) has eliminated the first derivative of the phase-lag. A numerical analysis of the new developed finite difference pair is presented. The analysis of the scheme consists of: (1) the production of the new four-stages symmetric six-step finite difference pair, (2) the calculation of the local truncation error of the new finite difference pair, (3) the comparative error analysis of the new method with other scheme of the same family which is the classical finite difference pair of the family (i.e. the finite difference pair with constant coefficients). (4) The stability and an interval of periodicity analysis and (5) finally, the study of the numerical and computational efficiency of the new method for the solution of the Schrödinger equation. The theoretical and numerical achievements for the new obtained four-stages symmetric six-step finite difference pair show its efficiency compared with other known or recently obtained schemes of the literature.