TheENUFmethod—Ewald summation based onnonuniformfast Fourier transform: Implementation, parallelization, and application

Abstract

Computer simulations of model systems are widely used to explore striking phenomena in promising applications spanning from physics, chemistry, biology, to materials science and engineering. The long range electrostatic interactions between charged particles constitute a prominent factor in determining structures and states of model systems. How to efficiently calculate electrostatic interactions in simulation systems subjected to partial or full periodic boundary conditions has been a grand challenging task. In the past decades, a large variety of computational schemes has been proposed, among which the Ewald summation method is the most reliable route to accurately deal with electrostatic interactions between charged particles in simulation systems. In addition, extensive efforts have been done to improve computational efficiencies of the Ewald summation based methods. Representative examples are approaches based on cutoffs, reaction fields, multi‐poles, multi‐grids, and particle‐mesh schemes. We sketched an ENUF method, an abbreviation for the Ewald summation method based on the nonuniform fast Fourier transform technique, and have implemented this method in particle‐based simulation packages to calculate electrostatic energies and forces at micro‐ and mesoscopic levels. Extensive computational studies of conformational properties of polyelectrolytes, dendrimer‐membrane complexes, and ionic fluids demonstrated that the ENUF method and its derivatives conserve both energy and momentum to floating point accuracy, and exhibit a computational complexity of scriptO)(NlogN with optimal physical parameters. These ENUF based methods are attractive alternatives in molecular simulations where high accuracy and efficiency of simulation methods are needed to accelerate calculations of electrostatic interactions at extended spatiotemporal scales.

A formula and numerical study on Ewald 1D summation

“…The exponential function exp x ð Þ is truncated at x ¼ À25 for its magnitude being exp À25 ð Þ≈ 1:39 Â 10 À11 . The reference value is determined by smaller mesh size (ζ ¼ 0:001) when discretizing Equations ( 25), (26), and (27).…”

“…[22][23][24] Some methods also strictly control the numerical errors by implementing nonuniform fast Fourier transform (NFFT) technique while keeping the scaling at O NlogN ð Þas well. [25][26][27] A pairwise formula suggested by Hu 21 utilizes the neutrality of the system, which is beneficial for analyzing the interactions between particles:…”

“…Combined with FFT and interpolation strategy, a PME‐like algorithm can be designed for large systems 22–24 . Some methods also strictly control the numerical errors by implementing nonuniform fast Fourier transform (NFFT) technique while keeping the scaling at as well 25–27 …”