Abstract
A unified treatment for the fast and spectrally accurate evaluation of electrostatic potentials with periodic boundary conditions in any or none of the three spatial dimensions is presented. Ewald decomposition is used to split the problem into real-space and Fourier-space parts, and the Fast Fourier Transform (FFT)-based Spectral Ewald (SE) method is used to accelerate computation of the latter, yielding the total runtime O(Nlog(N)) for N sources. A key component is a new FFT-based solution technique for the free-space Poisson problem. The computational cost is further reduced by a new adaptive FFT for the doubly and singly periodic cases, allowing for different local upsampling factors. The SE method is most efficient in the triply periodic case where the cost of computing FFTs is the lowest, whereas the rest of the algorithm is essentially independent of periodicity. We show that removing periodic boundary conditions from one or two directions out of three will only moderately increase the total runtime, and in the free-space case, the runtime is around four times that of the triply periodic case. The Gaussian window function previously used in the SE method is compared with a new piecewise polynomial approximation of the Kaiser-Bessel window, which further reduces the runtime. We present error estimates and a parameter selection scheme for all parameters of the method, including a new estimate for the shape parameter of the Kaiser-Bessel window. Finally, we consider methods for force computation and compare the runtime of the SE method with that of the fast multipole method.
Highlights
The task of computing interactions in an N-body problem is the most demanding part of various numerical simulations such as electrostatics in molecular dynamics, gravitational fields in cosmological formation of galaxies, and potentials in Stokes flow simulations
These methods belong to a family of Particle–Mesh–Ewald (PME) methods, which, applied to a system of N particles, reduce the computational complexity from O(N2) to O(N log(N)) with a prefactor depending on the required accuracy
The methods introduced in Ref. 9 are based on regularization and periodic extension of such functions to enable the use of Fast Fourier Transform (FFT)
Summary
The task of computing interactions in an N-body problem is the most demanding part of various numerical simulations such as electrostatics in molecular dynamics, gravitational fields in cosmological formation of galaxies, and potentials in Stokes flow simulations. There exist several methods that utilize this decomposition together with the Fast Fourier Transform (FFT) in order to accelerate the calculation of the Fourier space sum.2–5 These methods belong to a family of Particle–Mesh–Ewald (PME) methods, which, applied to a system of N particles, reduce the computational complexity from O(N2) to O(N log(N)) with a prefactor depending on the required accuracy. The methods introduced in Ref. 9 are based on regularization and periodic extension of such functions to enable the use of FFTs. In this paper, we present a different approach that works directly with the numerical discretization of the integrals in Fourier space. The Spectral Ewald (SE) method has been developed over the last decade in order to provide a fast and spectrally accurate approach for evaluating electrostatics problems with different periodicities.
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