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

• Published in: International journal for numerical methods in engineering Vol. 79; no. 13; pp. 1577 - 1604
• Main Authors: Cruz, Felipe A., Barba, L. A.
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 24.09.2009; Wiley
• Subjects: Accuracy; Aids; Algorithms; Applied classical electromagnetism; Computational techniques; Computer simulation; Electromagnetic wave propagation, radiowave propagation; Electromagnetism; electron and ion optics; Errors; Exact sciences and technology; fast multipole method; Fluid dynamics; Fundamental areas of phenomenology (including applications); General theory; hierarchical algorithms; Mathematical analysis; Mathematical methods in physics; Mathematical models; Multipoles; order-N algorithms; particle methods; Physics; Solid dynamics (ballistics, collision, multibody system, stabilization...); Solid mechanics; vortex method; Fluid mechanics; hierarchical algorithms; particle methods; Swirling flow; Algorithmics; Error bound; fast multipole method; Particle models; Experimental study; Modeling; order-N algorithms; Multipole field; Particle method; Spatial distribution; Meshless method; vortex method; Fast algorithm; Vortex; Electrostatics
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.2611

The 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(N2) 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. Copyright © 2009 John Wiley & Sons, Ltd.