Skeletons from the Treecode Closet

We consider treecodes (N-body programs which use a tree data structure) from the standpoint of their worst-case behavior. That is, we derive upper bounds on the largest possible errors that are introduced into a calculation by use of various multipole acceptability criteria (MAC). We find that the conventional Barnes-Hut MAC can introduce potentially unbounded errors unless < 1=p3, and that this behavior while rare, is demonstrable in astrophysically reasonable examples. We consider two other MACs closely related to the BH MAC. While they don’t admit the same unbounded errors, they nevertheless require extraordinary amounts of CPU time to guarantee modest levels of accuracy. We derive new error bounds based on some additional, easily computed moments of the mass distribution. These error bounds form the basis for four new MACs which can be used to limit the absolute or relative error introduced by each multipole evaluation, or, with the introduction of some additional data structures, the absolute or rms error in the final acceleration of each particle. Using the Sum Squares MAC to analytically place a 1% bound the rms error in a series of test models, we find that it significantly outperforms the = 0:65 BH MAC in terms of both accuracy (mean, rms and maximum error) and performance (floating point operation count). Subject headings: treecodes — fast multipole methods — N-body methods Submitted to: Journal of Computational Physics, July 31, 1992; revised, March 10, 1993 1Department of Physics, University of California, Santa Barbara