Periodic Boundary Conditions and the Error-Controlled Fast Multipole Method

The simulation of pairwise interactions in huge particle ensembles is a vital issue in scientific research. Especially the calculation of long-range interactions poses limitations to the system size, since these interactions scale quadratically with the number of particles. Fast summation techniques like the Fast Multipole Method (FMM) can help to reduce the complexity to $\mathcal{O}$(N). This work extends the possible range of applications of the FMM to periodic systems in one, two and three dimensions with one unique approach. Together with a tight error control, this contribution enables the simulation of periodic particle systems for different applications without the need to know and tune the FMM specific parameters. The implemented error control scheme automatically optimizes the parameters to obtain an approximation for the minimal runtime for a given energy error bound