ACCURATELY CLOSED NEWTON-COTES TRIGONOMETRICALLY-FITTED FORMULAE FOR THE NUMERICAL SOLUTION OF THE SCHRODINGER EQUATION
The investigation on the connection between: (1) closed Newton Cotes formulae of high-order, (2) trigonometrically-fitted differential schemes and (3) symplectic integrators is presented in this paper. In the last decades, several one step symplectic methods were obtained based on symplectic geometry (see the appropriate literature). The investigation on multistep symplectic integrators is poor. In the present paper: (1) we study a trigonometrically-fitted high-order closed Newton-Cotes formula, (2) we investigate the necessary conditions in a general eight-step differential method to be presented as symplectic multilayer integrator, (3) we present a comparative error analysis in order to show the theoretical superiority of the present method, (4) we apply it to solve the resonance problem of the radial Schrodinger equation. Finally, remarks and conclusions on the efficiency of the new developed method are given which are based on the theoretical and numerical results.
symplectic integrators numerical methods closed newton-cotes differential methods multistep methods trigonometric fitting