High order closed Newton-Cotes exponentially and trigonometrically fitted formulae as multilayer symplectic integrators and their application to the radial Schrödinger equation

Abstract

In this paper we study the connection between: (i) closed Newton-Cotes formulae of high order, (ii) trigonometrically-fitted and exponentially-fitted differential methods, (iii) symplectic integrators. Several one step symplectic integrators have been produced based on symplectic geometry during the last decades (see the relevant literature and the references here). However, the study of multistep symplectic integrators is very poor. In this paper we investigate the High Order Closed Newton-Cotes Formulae and we write them as symplectic multilayer structures. We develop trigonometrically-fitted and exponentially-fitted symplectic methods which are based on the closed Newton-Cotes formulae. We apply the symplectic schemes in order to solve the resonance problem of the radial Schrödinger equation. Based on the theoretical and numerical results, conclusions on the efficiency of the new obtained methods are given.Highly Cited

Accurately Closed Newton–cotes Trigonometrically-Fitted Formulae for the Numerical Solution of the Schrödinger Equation

“…5, the method integrates exactly the above functions with P ¼ 5 and K ¼ 1 while in the present paper the method integrates exactly the above functions with P ¼ 3 and K ¼ 2. Taking into account that the crucial aspect in the development of the trigonometrically-¯tted methods is the method to integrate exactly the above functions with largest possible K (see the error analysis presented in paragraph 4), the contribution of the present paper is obvious.…”

Derivative-Based Trapezoid Rule for the Riemann-Stieltjes Integral

“…Zhao and Li have proposed midpoint derivative-based closed Newton-Cotes quadrature [13] and numerical superiority has been shown. Recently, Simos and his partners have made a contribution to the Newton-Cotes formula for the Riemann integral and its applications [14][15][16][17][18][19][20][21], especially the connection between closed Newton-Cotes, trigonometrically fitted differential methods, symplectic integrators, and efficient solution of the Schrodinger equation [17][18][19][20][21].…”