A very fast and momentum-conserving tree code

The tree code for the approximate evaluation of gravitational forces is ext ended and substantially accelerated by including mutual cell-cell interacti ons. These are computed by a Taylor series in Cartesian coordinates and in a completely symmetric fashion, such that Newton's third law is satisfied b y construction and that therefore momentum is exactly conserved. The comput ational 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 linear ly with N, which can also be supported by a theoretical argument, and the a dvantage over the tree code increases with ever larger N.