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Computation of the Coulomb sums is usually the most time-consuming part in the molecular-dynamic simulation of the long-range-interacting systems, such as the strongly-coupled plasmas. Here, we are going to discuss a new method based on using Green function in Fourier representation on a multi-dimensional torus. It was derived by exploiting the recent mathematical results from the theory of Pontryagin invariants and Milnor¢s linking numbers [D. DeTurck, et al. arXiv:1101.3374]. As follows from our first numerical tests, the new algorithm provides a good convergence at large interparticle distances as well as avoids the problem of "close encounters" at small distances.