Notes on the Fast Multipole Method

TL;DR

This paper proposes modifications to the fast multipole method to eliminate potential energy discontinuities and improve boundary condition handling, enhancing its stability and applicability in simulations.

Contribution

It introduces a new approach that removes Legendre functions and fixes charges, leading to continuous potential energy and better periodic boundary condition implementation.

Findings

Elimination of potential energy discontinuity.

Implementation of continuous periodic boundary conditions.

Development of a version without shift process.

Abstract

Coulomb interactions of point charges can be calculated in $\mathcal{O}$(N) computation using the fast multipole method and direct calculations between charges nearby. It reduces computational cost dramatically, however, because of its method that combines direct and indirect calculations, there exists discontinuity of potential energy with respect to positions of charges. In this paper, we remove Legendre functions usually used in the fast multipole method and instead use charges fixed in positions. As an application of this method, we remove the discontinuity. It also leads us to a method of periodic boundary condition that is continuous even if a particle goes out from a wall of a simulation box and enters in opposite side of the box. Lastly, we show a version of the fast multipole method that do not use shift process.