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The Spectral Ewald method for singly periodic domains
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
2017 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 347, p. 341-366Article in journal (Refereed) Published
Abstract [en]

We present a fast and spectrally accurate method for efficient computation of the three dimensional Coulomb potential with periodicity in one direction. The algorithm is FFT-based and uses the so-called Ewald decomposition, which is naturally most efficient for the triply periodic case. In this paper, we show how to extend the triply periodic Spectral Ewald method to the singly periodic case, such that the cost of computing the singly periodic potential is only marginally larger than the cost of computing the potential for the corresponding triply periodic system. In the Fourier space contribution of the Ewald decomposition, a Fourier series is obtained in the periodic direction with a Fourier integral over the non-periodic directions for each discrete wave number. We show that upsampling to resolve the integral is only needed for modes with small wave numbers. For the zero wave number, this Fourier integral has a singularity. For this mode, we effectively need to solve a free-space Poisson equation in two dimensions. A very recent idea by Vico et al. makes it possible to use FFTs to solve this problem, allowing us to unify the treatment of all modes. An adaptive 3D FFT can be established to apply different upsampling rates locally. The computational cost for other parts of the algorithm is essentially unchanged as compared to the triply periodic case, in total yielding only a small increase in both computational cost and memory usage for this singly periodic case.

Place, publisher, year, edition, pages
Academic Press, 2017. Vol. 347, p. 341-366
Keyword [en]
Adaptive FFT, Coulomb potentials, Fast Ewald summation, Fast Fourier transform, Fourier integral, Single periodic, Spectral accuracy
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-212215DOI: 10.1016/j.jcp.2017.07.001ISI: 000408045500017Scopus ID: 2-s2.0-85024927535OAI: oai:DiVA.org:kth-212215DiVA, id: diva2:1134039
Funder
Swedish Research Council, 2011-3178Swedish e‐Science Research Center
Note

QC 20170817

Available from: 2017-08-17 Created: 2017-08-17 Last updated: 2017-12-12Bibliographically approved
In thesis
1. Fast methods for electrostatic calculations in molecular dynamics simulations
Open this publication in new window or tab >>Fast methods for electrostatic calculations in molecular dynamics simulations
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis deals with fast and efficient methods for electrostatic calculations with application in molecular dynamics simulations. The electrostatic calculations are often the most expensive part of MD simulations of charged particles. Therefore, fast and efficient algorithms are required to accelerate these calculations. In this thesis, two types of methods have been considered: FFT-based methods and fast multipole methods (FMM).

The major part of this thesis deals with fast N.log(N) and spectrally accurate methods for accelerating the computation of pairwise interactions with arbitrary periodicity. These methods are based on the Ewald decomposition and have been previously introduced for triply and doubly periodic problems under the name of Spectral Ewald (SE) method. We extend the method for problems with singly periodic boundary conditions, in which one of three dimensions is periodic. By introducing an adaptive fast Fourier transform, we reduce the cost of upsampling in the non periodic directions and show that the total cost of computation is comparable with the triply periodic counterpart. Using an FFT-based technique for solving free-space harmonic problems, we are able to unify the treatment of zero and nonzero Fourier modes for the doubly and singly periodic problems. Applying the same technique, we extend the SE method for cases with free-space boundary conditions, i.e. without any periodicity.

This thesis is also concerned with the fast multipole method (FMM) for electrostatic calculations. The FMM is very efficient for parallel processing but it introduces irregularities in the electrostatic potential and force, which can cause an energy drift in MD simulations. In this part of the thesis we introduce a regularized version of the FMM, useful for MD simulations, which approximately conserves energy over a long time period and even for low accuracy requirements. The method introduces a smooth transition over the boundary of boxes in the FMM tree and therefore it removes the discontinuity at the error level inherent in the FMM.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2018. p. 58
Series
TRITA-MAT-A ; 2018:02
National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-219775 (URN)978-91-7729-640-9 (ISBN)
Public defence
2018-01-26, F3, Lindstedtsvägen 26, Stockholm, 10:00 (English)
Opponent
Supervisors
Funder
Swedish e‐Science Research Center
Note

QC 20171213

Available from: 2017-12-13 Created: 2017-12-12 Last updated: 2017-12-13Bibliographically approved

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