A family of two-stage two-step methods for the numerical integration of the Schrödinger equation and related IVPs with oscillating solution

We develop a family of six methods for the numerical integration of the Schrödinger equation and related initial value problems with oscillating solution. Three of the methods are constructed so that they are P-stable, using the methodology of Wang (Comp Phys Comm 171(3):162–174, 2005). Also two of these three methods are trigonometrically fitted with trigonometric orders one and two. The other three methods are constructed so that they are trigonometrically fitted with orders one, two and three. We show that there is an equivalence between the three pairs of methods, as if the property of P-stability can be substituted by an extra trigonometric order, that is the P-stable method is equivalent to the method with trigonometric order one, the P-stable method with trigonometric order one is equivalent to the method with order two, and the P-stable method with order two is equivalent to the method with order three. There is a condition that we choose the same frequency for the P-stability test problem y′′ = −θ2 y and the functions that the method has to integrate exactly, in order to be trigonometrically fitted: {cos(ωx), sin(ωx), x cos(ωx), x sin(ωx), x2 cos(ωx), x2 sin(ωx)}. A stability analysis and a local truncation error analysis are performed on the methods and also the v–s diagrams are produced, where v = ω h and s = θ h. Finally the methods are applied to IVPs with oscillating solutions, such as the one-dimensional time independent Schrödinger equation and the nonlinear problem.