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# Notes on the Fast Multipole Method

Dec 2022

Coulomb interactions of point charges can be calculated in $\mathcal{O}$(N)computation using the fast multipole method and direct calculations betweencharges nearby. It reduces computational cost dramatically, however, because ofits method that combines direct and indirect calculations, there existsdiscontinuity of potential energy with respect to positions of charges. In thispaper, we remove Legendre functions usually used in the fast multipole methodand 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 boundarycondition that is continuous even if a particle goes out from a wall of asimulation box and enters in opposite side of the box. Lastly, we show aversion of the fast multipole method that do not use shift process.

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