Closed Newton–Cotes trigonometrically-fitted formulae of high order for the numerical integration of the Schrödinger equation

In this paper we investigate the connection between (i) closed Newton–Cotes formulae, (ii) trigonometrically-fitted differential methods, (iii) symplectic integrators and (iv) efficient solution of the Schrödinger equation. In the last decades several one step symplectic integrators have been produced based on symplectic geometry, (see the relevant literature and the references here). However, the study of multistep symplectic integrators is very poor. In this paper we investigate the closed Newton–Cotes formulae and we write them as symplectic multilayer structures. We also develop trigonometrically-fitted symplectic methods which are based on the closed Newton–Cotes formulae. We apply the symplectic schemes to the well known radial Schrödinger equation in order to investigate the efficiency of the proposed method to these type of problems.