A very fast and momentum-conserving tree code

The tree code for the approximate evaluation of gravitational forces is extended and substantially accelerated by including mutual cell-cell interactions. These are computed by a Taylor series in Cartesian coordinates and in a completely symmetric fashion, such that Newton's third law is satisfied by construction and that therefore momentum is exactly conserved. The computational effort is further reduced by exploiting the mutual symmetry of the interactions. For typical astrophysical problems with N = 10(5) and at the same level of accuracy, the new code is about 4 times faster than the tree code. For large N, the computational costs are found to scale almost linearly with N, which can also be supported by a theoretical argument, and the advantage over the tree code increases with ever larger N.