How to mesh up Ewald sums. I. A theoretical and numerical comparison of various particle mesh routines

Abstract

Standard Ewald sums, which calculate, e.g., the electrostatic energy or the force in periodically closed systems of charged particles, can be efficiently speeded up by the use of the fast Fourier transformation ͑FFT͒. In this article we investigate three algorithms for the FFT-accelerated Ewald sum, which have attracted widespread attention, namely, the so-called particle-particle-particle mesh ͑P 3 M), particle mesh Ewald ͑PME͒, and smooth PME method. We present a unified view of the underlying techniques and the various ingredients which comprise those routines. Additionally, we offer detailed accuracy measurements, which shed some light on the influence of several tuning parameters and also show that the existing methods -although similar in spirit -exhibit remarkable differences in accuracy. We propose a set of combinations of the individual components, mostly relying on the P 3 M approach, that we regard to be the most flexible. The issue of estimating the errors connected with particle mesh routines is reserved to paper II.

Computer Simulations of Charged Systems

“…Implementing a fast Fourier transformation (FFT) for the Fourier part results in the so-called particle-mesh-Ewald formulations, which improve the efficiency to O(N log N ) [22,23,24,25], and which can also be efficiently be parallelized [26,27]. The most versatile and accurate method of all meshmethods is the oldest P3M algorithm [22,28], for which also precise error estimates exist [29]. Another way of computing Eq.…”

Effects of asymmetric salt and a cylindrical macroion on charge inversion: Electrophoresis by molecular dynamics simulations

“…The sum is calculated with the use of the PPPM algorithm [24,25]. We use (32) 3 spatial meshes for the calculation of the reciprocal space contributions to the Coulomb forces, with the choice of the Ewald parameter α ≈ 0.262 and the cutoff length 10a.…”

Pressure Profile Calculation with Mesh Ewald Methods

“…16,24 The choice of the assignment function for the discrete charge distribution is not unique, a fact that has led to the appearance of families of methods such as the Particle-Particle-Particle-Mesh (P3M) of Hockney and Eastwood, 17 and the Particle Mesh Ewald (PME) in its original variant 15 that uses Lagrange interpolation or in its "smooth" version 16 (sPME) based on cardinal Bsplines. Once a physical quantity pertaining to single particles has been calculated on the lattice, it has to be interpolated back from the lattice to the real position of the particles.…”

“…This is usually done by the same assignment function used to generate the distribution on the lattice. 24 If a function A(r p ) is known at the lattice nodes, it can be distributed back to the atomic positions using…”