An New High Algebraic Order Efficient Finite Difference Method for the Solution of the Schr\"odinger Equation

Abstract

The development of a new five-stages symmetric two-step method of fourteenth algebraic order with vanished phase--lag and its first, second, third and fourth derivatives is analyzed in this paper. More specifically : (1) we will present the development of the new method, (2) we will determine the local truncation error (LTE) of the new proposed method, (3) we will analyze the local truncation error based on the radial time independent Schr\"odinger equation, (4) we will study the stability and the interval of periodicity of the new proposed method based on a scalar test equation with frequency  different than the frequency of the scalar test equation used for the phase--lag analysis, (5) we will test the efficiency of the new obtained method based on its application on the coupled differential equations arising from the Schr\"odinger equation.