A treecode for potentials of the form r^−λ

Abstract

It is known that the classical multilevel fast multipole method is a good technique for accelerating the iterative solution of integral equations with the Green's function kernel 1/r. However, few methods have been proposed for problems in molecular dynamics and computational chemistry where the Green's function is of the form r ^−λ, λ≥1. In this paper, we describe an efficient algorithm for computing potentials in which the integral equations have r ^−λ kernels. We propose a treecode algorithm which uses spherical harmonics to compute multipole coefficients that are used to evaluate these potentials. The key idea in this algorithm is the use of ultraspherical polynomials to represent r ^−λ in a manner analogous to the use of Legendre polynomials for the expansion of the Coulomb potential r ⁻¹. We exploit the relationship between ultraspherical and Legendre polynomials to develop a natural generalization of the multipole expansion theorem which was used in the multilevel fast multipole method. This theorem is used along with a hierarchical scheme to compute potentials of the form r ^−λ. The complexity of the algorithm is O(p ³ Nlog N) where N is the number of interacting particles and p is a small constant. We discuss the advantages of our algorithm over existing single level methods based on Cartesian coordinate expansion schemes. The efficiency of the algorithm is illustrated with numerical experiments.

Keywords:

• Fast multipole method
• Gegenbauer
• Ultraspherical
• Van der Waals forces

Acknowledgements

This work has been supported in part by NSF under the grant NSF-CCF0431068.