Error estimate of the u-series method for molecular dynamics simulations

This paper provides an error estimate for the u-series decomposition of the Coulomb interaction in molecular dynamics simulations. We show that the number of truncated Gaussians $M$ in the u-series and the base of interpolation nodes $b$ in the bilateral serial approximation are two key parameters for the algorithm accuracy, and that the errors converge as $\mathcal{O}(b^{-M})$ for the energy and $\mathcal{O}(b^{-3M})$ for the force. Error bounds due to numerical quadrature and cutoff in both the electrostatic energy and forces are obtained. Closed-form formulae are also provided, which are useful in the parameter setup for simulations under a given accuracy. The results are verified by analyzing the errors of two practical systems.