On the Rokhlin-Greengard Method with Vortex Blobs for Problems Posed in All Space or Periodic in One Direction

Hamilton, John T.; Majda, George

doi:10.1006/jcph.1995.1177

Cited by 14 publications

(23 citation statements)

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“…As example, let us take the experiments presented in Figure 7 for lower values of p (say p = [5,7,9,11,13,15]). In these plots, a distinction between different areas of high and low error can be appreciated, product of the differences in the spatial decomposition for different evaluation points.…”

Section: Discussion Of Error Type IImentioning

confidence: 99%

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Characterization of the accuracy of the fast multipole method in particle simulations

Cruz¹,

Barba²

2009

Numerical Meth Engineering

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SUMMARYThe fast multipole method (FMM) is a fast summation algorithm capable of accelerating pairwise interaction calculations, known as N -body problems, from an algorithmic complexity of O(N 2 ) to O(N ) for N particles. The algorithm has brought a dramatic increase in the capability of particle simulations in many application areas, such as electrostatics, particle formulations of fluid mechanics, and others. Although the literature on the subject provides theoretical error bounds for the FMM approximation, there are not many reports on the measured errors in a suite of computational experiments that characterize the accuracy of the method in relation with the different parameters available to the user. We have performed such an experimental investigation, and summarized the results of about 1500 calculations using the FMM algorithm, applied to the 2D vortex particle method. In addition to the more standard diagnostic of the maximum error, we supply illustrations of the spatial distribution of the errors, offering visual evidence of all the contributing factors to the overall approximation accuracy: multipole expansion, local expansion, hierarchical spatial decomposition (interaction lists, local domain, far domain). This presentation is a contribution to any researcher wishing to incorporate the FMM acceleration to their application code, as it aids in understanding where accuracy is gained or compromised.

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Section: Discussion Of Error Type IImentioning

confidence: 99%

“…The first type of error occurs from the substitution of the Biot-Savart kernel (6) for all far-field interactions by a singular kernel (as in [5]). Thus, the original kernel has been replaced by one for which it is simpler to construct the MEs.…”

Section: Error Type Imentioning

confidence: 99%

Characterization of the accuracy of the fast multipole method in particle simulations

Cruz¹,

Barba²

2009

Numerical Meth Engineering

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“…Nevertheless, several open-source and commercial packages implementing the FMM for standard kernels in free space have become available. The FMM is used less often in practice to compute periodic sums, even though several methods to handle periodic boundary conditions using extensions to the basic FMM have been proposed, starting from the first publication of the algorithm [6,7,8,9,10,11,12,13,14,15,16]. Some authors (e.g.…”

Section: Introductionmentioning

confidence: 99%

A method to compute periodic sums

Gumerov¹,

Duraiswami²

2014

Journal of Computational Physics

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In a number of problems in computational physics, a finite sum of kernel functions centered at N particle locations located in a box in three dimensions must be extended by imposing periodic boundary conditions on box boundaries. Even though the finite sum can be efficiently computed via fast summation algorithms, such as the fast multipole method (FMM), the periodized extension is usually treated via a different algorithm, Ewald summation, accelerated via the fast Fourier transform (FFT). A different approach to compute this periodized sum just using a blackbox finite fast summation algorithm is presented in this paper. The method splits the periodized sum in to two parts. The first, comprising the contribution of all points outside a large sphere enclosing the box, and some of its neighbors, is approximated inside the box by a collection of kernel functions ("sources") placed on the surface of the sphere or using an expansion in terms of spectrally convergent local basis functions. The second part, comprising the part inside the sphere, and including the box and its immediate neighborhood, is treated via available summation algorithms. The coefficients of the sources are determined by least squares collocation of the periodicity condition of the total potential, imposed on a circumspherical surface for the box. While the method is presented in general, details are worked out for the case of evaluating electrostatic potentials and forces. Results show that when used with the FMM, the periodized sum can be computed to any specified accuracy, at an additional cost of the order of the free-space FMM. Several technical details and efficient algorithms for auxiliary computations are provided, as are numerical comparisons.

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“…Two such methods are currently available, namely [10] and [11]. Issues related to the problem of velocity evaluation in periodic channel configurations were also addressed in [12].…”

Section: Conclusion and Further Perspectivesmentioning

confidence: 99%

Flow in a 2D duct with moving boundaries - vortex blob approach

Protas

Styczek

1999

ESAIM: Proc.

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In the paper we develop Random Vortex Blob method for the case of flows in periodic domains. It is shown how the formulation of the problem must be modified to account for the periodicity constraint. Properties of periodic vorticity particles are discussed. We also derive a rigorous method to satisfy the boundary conditions. Example computations concern the flow in an Xperiodic duct with moving boundaries and varying in time pressure gradient.

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On the Rokhlin-Greengard Method with Vortex Blobs for Problems Posed in All Space or Periodic in One Direction

Cited by 14 publications

References 0 publications

Characterization of the accuracy of the fast multipole method in particle simulations

Characterization of the accuracy of the fast multipole method in particle simulations

A method to compute periodic sums

Flow in a 2D duct with moving boundaries - vortex blob approach

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