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Spectral accuracy in fast Ewald-based methods for particle simulations
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
2011 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 230, no 24, p. 8744-8761Article in journal (Refereed) Published
##### Abstract [en]

A spectrally accurate fast method for electrostatic calculations under periodic boundary conditions is presented. We follow the established framework of FFT-based Ewald summation, but obtain a method with an important decoupling of errors: it is shown, for the proposed method, that the error due to frequency domain truncation can be separated from the approximation error added by the fast method. This has the significance that the truncation of the underlying Ewald sum prescribes the size of the grid used in the FFT-based fast method, which clearly is the minimal grid. Both errors are of exponential-squared order, and the latter can be controlled independently of the grid size. We compare numerically to the established SPME method by Essmann et al. and see that the memory required can be reduced by orders of magnitude. We also benchmark efficiency (i.e. error as a function of computing time) against the SPME method, which indicates that our method is competitive. Analytical error estimates are proven and used to select parameters with a great degree of reliability and ease.

##### Place, publisher, year, edition, pages
2011. Vol. 230, no 24, p. 8744-8761
##### Keywords [en]
Ewald summation, FFT, Molecular dynamics, PME, Spectral accuracy, SPME
##### National Category
Computational Mathematics Other Physics Topics
##### Identifiers
ISI: 000297081700007Scopus ID: 2-s2.0-80053628839OAI: oai:DiVA.org:kth-48765DiVA, id: diva2:458547
##### Funder
Knut and Alice Wallenberg FoundationSwedish e‐Science Research Center
##### Note
QC 20111124Available from: 2011-11-23 Created: 2011-11-23 Last updated: 2017-12-08Bibliographically approved
##### In thesis
1. Spectral Accuracy in Fast Ewald Methods and Topics in Fluid Interface Simulation
Open this publication in new window or tab >>Spectral Accuracy in Fast Ewald Methods and Topics in Fluid Interface Simulation
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

This work contains two separate but related parts: one on spectrally  accurate and fast Ewald methods for electrostatics and viscous flow,  and one on micro- and complex fluid interface problems.  In Part I we are concerned with fast and spectrally accurate methods  to compute sums of slowly decaying potentials over periodic  lattices. We consider two PDEs: Laplace (electrostatics, the Coulomb  potential) and Stokes (viscous flow, the Stokeslet''  potential). Moreover, we consider both full and planar periodicity,  the latter meaning that periodicity applies in two dimensions and  the third is free''. These are major simulation tasks in current  molecular dynamics simulations and in many areas of computational  fluid mechanics involving e.g. particle suspensions.   For each of the four combinations of PDE and periodic structure, we  give spectrally accurate and O(N log N) fast methods based on  Ewald's or Ewald-like decompositions of the underlying potential  sums. In the plane-periodic cases we derive the decompositions in a  manner that lets us develop fast methods. Associated error estimates  are developed as needed throughout. All four methods can be placed  in the P3M/PME (Particle Mesh Ewald) family. We argue that they  have certain novel and attractive features: first, they are spectral  accurate; secondly, they use the minimal amount of memory possible  within the PME family; third, each has a clear and reliable view of  numerical errors, such that parameters can be chosen  wisely. Analytical and numerical results are given to support these  propositions. We benchmark accuracy and performance versus an  established (S)PME method.  Part II deals with free boundary problems, specifically numerical  methods for multiphase flow. We give an interface tracking method  based on a domain-decomposition idea that lets us split the  interface into overlapping patches. Each patch is discretized on a  uniform grid, and accurate and efficient numerical methods are given  for the equations that govern interface transport. We demonstrate  that the method is accurate and how it's used in immersed boundary,  and interface, Navier-Stokes methods, as well as in a boundary  integral Stokes setting.  Finally, we consider a problem in complex fluidics where there is a  concentration of surfactants \emph{on} the interface and the  interface itself is in contact with a solid boundary (the contact  line problem). We argue that the domain-decomposition framework is  attractive for formulating and treating complex models  (e.g. involving PDEs on a dynamic interface) and proceed with  developing various aspects of such a method.

##### Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2011. p. xv, 104
##### Series
Trita-CSC-A, ISSN 1653-5723 ; 2011:19
##### National Category
Computational Mathematics
##### Identifiers
urn:nbn:se:kth:diva-48805 (URN)978-91-7501-195-0 (ISBN)
##### Public defence
2011-12-16, Salongen, KTHB, Osquars backe 25, Stockholm, 10:00 (English)
##### Funder
Swedish e‐Science Research Center
##### Note
QC 20111125Available from: 2011-11-25 Created: 2011-11-23 Last updated: 2012-05-24Bibliographically approved

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Cite
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