Non-finite-difference algorithm for integrating Newton's motion equations

Abstract

We have presented some practical consequences on the molecular-dynamics simulations arising from the numerical algorithm published recently in paper Int. J. Mod. Phys. C 16, 413 (2005). The algorithm is not a finite-difference method and therefore it could be complementary to the traditional numerical integrating of the motion equations. It consists of two steps. First, an analytic form of polynomials in some formal parameter $\lambda$ (we put $\lambda=1$ after all) 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 to the polynomials of the first degree in formal parameter $\lambda$. The computer simulations play very important role in modeling materials with unusual properties being contradictictory to our intuition. The particular example could be the auxetic materials. In this case, the accuracy of the applied numerical algorithms as well as various side-effects, which might change the physical reality, could become important for the properties of the simulated material.Comment: 11 page