High order closed Newton–Cotes trigonometrically-fitted formulae for the numerical solution of the Schrödinger equation

In this paper, we investigate the connection between
• closed Newton–Cotes formulae,
• trigonometrically-fitted methods,
• symplectic integrators and
• efficient integration of the Schrödinger equation.The study of multistep symplectic integrators is very poor although in the last decades several one step symplectic integrators have been produced based on symplectic geometry (see the relevant literature and the references here). In this paper we study the closed Newton–Cotes formulae and we write them as symplectic multilayer structures. Based on the closed Newton–Cotes formulae, we also develop trigonometrically-fitted symplectic methods. An error analysis for the one-dimensional Schrödinger equation of the new developed methods and a comparison with previous developed methods is also given. We apply the new symplectic schemes to the well-known radial Schrödinger equation in order to investigate the efficiency of the proposed method to these type of problems.