Yasuhiro Kajima | Searcharxiv

arXiv2022

Notes on the Fast Multipole Method

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.

physics.comp-ph↗

arXiv2020

Summation of certain locally bilinear forms and its applications to the Fast Multipole Method

The Fast Multipole Method (FMM) reduces the computation of pairwise two-body interactions among $N$-particles to order $N$, whose computation cost should be of order $N^2$ by brute force. However, its implementation is somewhat complicated and requires a considerable amount of time to write the code. In this paper, I show a method that enables us to implement and write FMM algorithm code simply and briefly. FMM algorithm is composed of several steps. The main steps are Upward Pass and Downward Pass. Both the Upward Pass and Downward Pass include shift processes by which we move the centers of local expansions and multipole expansions. In this paper, I show a method that enables us to get rid of these processes.As a result of this simplification, the coding of FMM becomes much easier, and we can save considerable computation time. I compared the accuracy and time required to calculate potential fields with that of the existing FMM code.

physics.comp-ph↗