Abstract: Some practical consequences on the molecular-dynamics simulations arising from a recently published numerical algorithm are presented. The algorithm is not a finite-difference method and therefore it could represent a complementary approach to the traditional numerical integration of the equations of motion. It consists of two steps. First, an analytic form of polynomials in some formal parameter lambda (set to one in the following) is derived, which approximate the solution of the system of differential equations under consideration. Next, the numerical values of the derived polynomials in the interval, in which the difference between them and their truncated part of smaller degree does not exceed a given accuracy epsilon, become the numerical solution. The particular examples, which we have considered, represent the forced linear and nonlinear oscillator and the 2D Lennard-Jones fluid. In the latter case we have restricted our attention to the polynomials of the first degree in the formal parameter lambda.