  We present Superbox, a particle-mesh code with high resolution sub-grids and
an NGP (nearest grid point) force-calculation scheme based on the second
derivatives of the potential. Superbox implements a fast low-storage
FFT-algorithm, giving the possibility to work with millions of particles on
desk-top computers. Test calculations show energy and angular momentum
conservation to one part in 10^5 per crossing-time. The effects of grid and
numerical relaxation remain negligible, even when these calculations cover a
Hubble-time of evolution. As the sub-grids follow the trajectories of
individual galaxies, the code allows a highly resolved treatment of
interactions in clusters of galaxies, such as high-velocity encounters between
elliptical galaxies and the tidal disruption of dwarf galaxies. Excellent
agreement is obtained in a comparison with a direct-summation N-body code
running on special-purpose Grape3 hardware. The orbital decay of satellite
galaxies due to dynamical friction obtained with Superbox agrees with
Chandrasekhar's treatment when the Coulomb logarithm is approximately 1.5.
