Explicit higher-order schemes for molecular dynamics problems

The Runge–Kutta–Nyström (RKN) explicit symplectic difference schemes are considered with a number of stages from 1 to 5 for the numerical solution of molecular dynamics problems described by systems with separable Hamiltonians. For the numbers of stages 2 and 3, the parameters of the RKN schemes are obtained using the Gröbner basis technique. For the number of stages 4 and 5, new schemes were found using the Nelder–Mead numerical optimization method. In particular, four new schemes are obtained for the number of stages 4. For the number of stages 5, three new schemes are obtained in addition to the four schemes, which are well-known in the literature. For each specific number of stages, a scheme is found being the best in terms of the minimum of the leading term of the approximation error. Verification of the schemes is carried out on a problem that has an exact solution. It is shown that the symplectic five-stage RKN scheme provides a more accurate conservation of the total energy balance of the particle system than schemes of lower orders of accuracy. The stability studies of the schemes were performed using the Mathematica software package.