A family of four-step trigonometrically-fitted methods and its application to the schrödinger equation

In this article we present a singularly almost P-stable exponentially-fitted four-step method for the approximate solution of the one-dimensional Schrödinger equation. More specifically we present a method that is singularly almost P-stable (a concept later introduced in this article) and also integrates exactly any linear combination of the functions {1, x, exp ( ±I v x) , x exp ( ±I v x) , x2 exp ( ±I v x)}. The numerical experimentation showed that our method is considerably more efficient compared to well known methods used for the approximate solution of resonance problem of the radial Schrödinger equation.