Departmental Papers (MEAM)

Document Type

Journal Article

Date of this Version

July 2003

Comments

Postprint version. Published in Journal of Computational Physics Volume 196, Issue 2 , 20 May 2004, Pages 591-626
Publisher URL: doi:10.1016/j.jcp.2003.11.021

NOTE: At the time of publication, author George Biros was affiliated with New York University. Currently (March 2005), he is a faculty member in the Department of Mechanical Engineering and Applied Mechanics at the University of Pennsylvania.

Abstract

We present a new fast multipole method for particle simulations. The main feature of our algorithm is that it does not require the implementation of multipole expansions of the underlying kernel, and it is based only on kernel evaluations. Instead of using analytic expansions to represent the potential generated by sources inside a box of the hierarchical FMM tree, we use a continuous distribution of an equivalent density on a surface enclosing the box. To find this equivalent density we match its potential to the potential of the original sources at a surface, in the far field, by solving local Dirichlet-type boundary value problems. The far field evaluations are sparsified with singular value decomposition in 2D or fast Fourier transforms in 3D. We have tested the new method on the single and double layer operators for the Laplacian, the modified Laplacian, the Stokes, the modified Stokes, the Navier, and the modified Navier operators in two and three dimensions. Our numerical results indicate that our method compares very well with the best known implementations of the analytic FMM method for both the Laplacian and modified Laplacian kernels. Its advantage is the (relative) simplicity of the implementation and its immediate extension to more general kernels.

Keywords

fast multipole methods, fast solvers, integral equations, single-layer potential, double-layer potential, particle methods, N-body problems

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Date Posted: 20 July 2004

This document has been peer reviewed.