Comparative Study of the Accuracy of Higher-Order Difference Schemes for Molecular Dynamics Problems Using the Computer Algebra Means

Abstract

The Runge–Kutta–Nyström (RKN) explicit symplectic difference schemes with the number of stages from 1 to 5 for the numerical solution of molecular dynamics problems described by the systems with separable Hamiltonians have been considered. All schemes have been compared in terms of the accuracy and stability with the use of Gröbner bases. For each specific number of stages, the schemes are found, which are the best in terms of accuracy and stability. The efficiency parameter of RKN schemes has been introduced by analogy with the efficiency parameter for Runge–Kutta schemes and the values of this parameter have been computed for all considered schemes. The verification of schemes has been done by solving a problem having the exact solution. It has been shown that the symplectic five-stage RKN scheme ensures a more accurate conservation of the total energy of a system of particles than the schemes of lower accuracy orders. All investigations of the accuracy and stability of schemes have been carried out in the analytic form with the aid of the computer algebra system (CAS) Mathematica.