A Very Fast and Momentum-conserving Tree Code

Abstract

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 and at the same level of accuracy, the new 5 N = 10 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.

Gasoline: a flexible, parallel implementation of TreeSPH

“…This leads to a force calculation step that is O(N) as each tree node interacts with a similar number of nodes independent of N and the number of nodes is proportional to the number of bodies. Building the tree is still O(N log N) but this is a small cost for simulations up to N ∼ 10 7 (Dehnen 2000). The Greengard & Rokhlin (1987) method used spherical harmonic expansions where the desired accuracy is achieved solely by changing the order of the expansions.…”

“…The Greengard & Rokhlin (1987) method used spherical harmonic expansions where the desired accuracy is achieved solely by changing the order of the expansions. For the majority of astrophysical applications the allowable force accuracies make it much more efficient to use fixed order Cartesian expansions and an opening angle criterion similar to standard tree codes (Salmon & Warren 1994, Dehnen 2000. This approach has the nice property of explicitly conserving momentum (as do PM and P 3 M codes).…”

On Universal Halos and the Radial Orbit Instability

“…Even if some of the progenitors of a given halo develop steep cusps through approximately spherical infall, one might well imagine these cusps becoming shallower through merging and subsequent relaxation. A recent paper by Dehnen (2005) suggests otherwise. On the basis of a theorem for phase mixing, Dehnen argues that in the merger of two (virialized) subhalos, the steepest cusp survives.…”

A natural symmetrization for the plummer potential

“…Elliott & Board (1996) successfully reduced the calculation cost in FMM from O(p 4 N) to O(p(log p) 1/2 N) by using a fast fourier transformation and optimal choice of the tree level. Dehnen (2000) proposed a Cartesian FMM, however, its calculation cost is estimated as O(p 6 N). Again this cost can be reduced to O(p 3/2 (log p) 1/2 N) by following Elliott & Board (1996).…”

“…Thus, it is widely used in simulations of self-gravitating systems and the GRAPE (GRAvity PipE) series also adopts this form of the gravitational potential (Sugimoto et al, 1990;Ito et al, 1991;Okumura et al, 1993;Makino et al, 1997;Kawai et al, 2000;Makino et al, 2003;Kawai & Fukushige, 2006;Makino, 2005. Due to the wide adoption of the Plummer potential, there have been several studies on the choice of the optimal gravitational softening length (Merritt, 1996;Athanassoula et al, 2000;Dehnen, 2001).…”