Nowadays the simulation of particle systems (N-body systems) is an important task in a broad range of research areas, eg physics, chemistry and electrical engineering. A straightforward approach to compute all pairwise interactions for such an N-body system has complexity O (N2). However, there exist methods, so-called Fast Summation Methods, which reduce the computational effort from O (N2) to O (N log N) or even to O (N)[1]. An additional obstacle occurs, if the interaction between particles is described by a long-range potential, eg gravitational or electrostatic potential. Then such simulations are even worse to tackle, since no short-range approximation can be applied. In the case of electrostatic interactions, the methods which can speed up the calculation of the Coulomb force in N-body systems are called fast Coulomb solvers. In fact the above mentioned gravitational interaction represents a special case of electrostatic interactions, since a mass is always positive, while a charge can be positive or negative. Since these kind of interactions are essential for simulations in a host of physical and chemical processes as well as in astrophysical simulations, there exist a growing demand for fast Coulomb solvers. The Fast Multipole Method (FMM), developed by Greengard and Rohklin and published in 1987 [2], is one famous method, which reduces the complexity of the electrostatic N-body problem to O (N).

In any case, there exist a wide range of methods able to compute the electrostatic interactions in N-body systems for both systems with open boundary conditions and systems with periodic boundary conditions, eg [3, 4, 2, 5]. However, this report is a first approach to evaluate the implementation of the Fast Multipole Method of our group at Forschungszentrum Jülich GmbH against other freely available methods.