Fast and spectrally accurate Ewald summation for 2-periodic electrostatic systems

Abstract

A new method for Ewald summation in planar/slablike geometry, i.e. systems where periodicity applies in two dimensions and the last dimension is "free" (2P), is presented. We employ a spectral representation in terms of both Fourier series and integrals. This allows us to concisely derive both the 2P Ewald sum and a fast PME-type method suitable for large-scale computations. The primary results are: (i) close and illuminating connections between the 2P problem and the standard Ewald sum and associated fast methods for full periodicity; (ii) a fast, O(N log N), and spectrally accurate PME-type method for the 2P k-space Ewald sum that uses vastly less memory than traditional PME methods; (iii) errors that decouple, such that parameter selection is simplified. We give analytical and numerical results to support this.

Fast Ewald summation based on NFFT with mixed periodicity

“…We remark that this method is used in [35]. As pointed out in [35, page 12] this approach is limited to functions that decay sufficiently fast in the interval [−h/2, h/2).…”

“…We will later refer again to [35], which is the latest development for the 2d-periodic case, in order to discuss the differences to our method, see Section 4.1. Another approach for long range interactions on surfaces is proposed in [37].…”

“…In contrast to the case of 3d-periodic boundary conditions, the application of the Ewald formulas for mixed periodic systems does not straightforwardly lead to fast algorithms. Some Fourier based algorithms, like MMM2D, MMM1D or ELC, see [6,8,7] and the fast and spectrally accurate Ewald summation in slab geometry [35], already exist, see also [51,52,11,10] for algorithms with higher complexity.…”

“…Furthermore, we present numerical results in Section 4.3, which show its efficiency. In order to rate the very good performance of the new algorithm, we compare the method to the particle-particle NFFT (P 2 NFFT) method for 3d-periodic systems [42] as well as to the method proposed in [35] by considering similar numerical examples. Note that the P 2 NFFT algorithm is highly optimized and recently compared with other methods, such as the particle-particle particle-mesh (P 3 M) method, the fast multipole method or multigrid based methods, see [5].…”

A Green's Function Numerical Method for Solving Parabolic Partial Differential Equations

“…The fundamental idea behind Ewald summation is to break up a single divergent series into the sum of two series, one that is summable in the Fourier domain, and one that is summable with rapidly decaying terms in the real domain. This allows the energy of an infinite system to be approximated accurately and efficiently with truncated sums, which in turn leads to highly efficient numerical algorithms that take advantage of this rapid convergence [5].…”