An efficient and economical high order method for the numerical approximation of the Schrödinger equation

Abstract

A new scheme is developed in this paper, for the first time in the literature. The new scheme: (1) is a symmetric two-step method, (2) is of three-stages scheme, (3) is a high order method (i.e of eight-algebraic order), (4) the approximations of the layers are taken place as follows: first layer on the point \(x_{n-1}\), second layer on the point \(x_{n}\), third layer on the point \(x_{n+1}\), (5) has vanished the phase-lag and its derivatives up to order four, (6) has good interval of periodicity properties [i.e. interval of periodicity equal to (0, 9.8)]. A detailed theoretical analysis is also presented. More specifically we present: (1) the development of the new method, (2) comparative error analysis (3) stability analysis. The effectiveness of the new scheme is tested via the solution of systems of coupled differential equations of the Schrodinger type.