A new three-stages six-step finite difference pair with optimal phase properties for second order initial and/or boundary value problems with periodical and/or oscillating solutions

A new symmetric six-step method with improved phase properties is introduced, for the first time in the literature, in this paper. The proposed method: (1) is a symmetric nonlinear six-step method, (2) is of three-stages, (3) is of twelfth algebraic order, (4) has vanished the phase-lag and (5) has vanished the derivatives of the phase-lag up to order five. For the new symmetric six-step method we present a detailed theoretical, numerical and computational analysis which consists of: (a) the development of the new three-stages symmetric six-step method, (b) the computation of the local truncation error of the new scheme, (c) the comparative error analysis of the new algorithm with other schemes of the same family: (i) the classical algorithm of the family (i.e. the method with constant coefficients), (ii) the recently proposed method of the same family with vanished phase-lag and its first derivative, (iii) the recently proposed method of the same family with vanished phase-lag and its first and second derivatives, (iv) the recently proposed method of the same family with vanished phase-lag and its first, second and third derivatives and finally (v) the recently proposed method of the same family with vanished phase-lag and its first, second, third and fourth derivatives, (d) the stability and the interval of periodicity analysis for the new developed method and finally (e) the study of the accuracy and computational effectiveness of the new proposed method for the solution of the Schrödinger equation. The theoretical, numerical and computational analysis of the new obtained three-stages symmetric six-step method show the efficiency of the new algorithm compared with other known or recently developed schemes of the literature.