New Three- and Five-Stage Symplectic Schemes in the Forest–Ruth Family

The explicit symplectic difference schemes with a number of stages 3 and 5 are considered for the numerical solution of molecular dynamics problems described by systems with separable Hamiltonians. The parameters of the new Forest–Ruth (FR) schemes are obtained using the Gröbner basis technique and the numerical optimization methods. In the case of the three-stage FR schemes, three new schemes have been found in the analytic form and seven schemes were found by the numerical optimization. In the case of the five-stage FR schemes, five cases of the vanishing Vandermonde-type determinant were considered and 92 new five-stage fourth-order schemes were found with the aid of numerical optimization. The accuracy of new schemes was demonstrated by the examples of the numerical solution of two test problems which have the exact solutions. It has been shown that all new schemes provide a higher accuracy of the particle energy conservation than the well-known second-order Verlet scheme.