The gravitating N-body-problem is one of the grand challenges of theoretical physics and astrophysics. Its accurate solution for very large astrophysical particle numbers cannot generally be obtained by mathematical considerations (series evaluations) as it was possible for the historical treatment of the classical two- and three-body problems. Only computer modelling on the fastest available hardware using specialized mathematical-numerical algorithms can be used as an appropriate tool here.
In addition to the fundamental theoretical interest of large gravitating Nbody systems such models are essential for our understanding of the structure and evolution of many astrophysically relevant objects, as there are our planetary system, our own and other galaxies and the entire universe seen as an object forming structure via gravitational interaction between particles. Such numerical modelling is also important for the interpretation of a wealth of new observational data from space based instruments, as e.g. the Hubble Space Telescope (HST) or the ISO infrared observatory.
A particular class of ``high-accuracy'' numerical models following the orbit of each particle due to the ``exact'' gravitational forces of all the other particles in a many-body system is considered. It is necessary, because for models of star clusters and galactic nuclei two-body (thermal) relaxation is important and its relevant time scales extend over hundreds or thousands of dynamical times. So, very high accuracy for the force calculation is required at the individual time steps, which usually are of the order of a small fraction of a dynamical time. In such situation codes and algorithms using approximate potential calculation are not well-suited, such as TREE-codes or fast multipole codes (however, see Kawai, this proceedings for a fresh multipole approach). We use a fourth order Hermite scheme with hierarchically blocked individual time steps, Ahmad-Cohen (AC) neighbour scheme, and regularisation of close encounters (for a contemporary overview see Mikkola, 1997) and hierarchical subsystems (Mikkola & Aarseth 1996, 1998). Such scheme is widely known as Aarseth scheme, coming in flavours NBODY4 (without AC scheme) and NBODY6 (with AC scheme), and NBODY6++ (massively parallel AC scheme), see Aarseth (1985, 1993), Spurzem & Baumgardt (1999, preprint).
Such algorithms require most of their computational time to compute and accumulate the mutual pairwise gravitational forces between the particles (N**2-problem!). In successful projects special purpose computers named GRAPE for that purpose were developed in Japan (Makino et al. 1997) and are also used in Germany as well as in many other countries in the world. They have successfully been used, for example to prove the existence of gravothermal oscillations (Makino 1996), to study dynamical friction of a binary black hole in galactic nuclei (Makino & Ebisuzaki 1996, Makino 1997), to model star formation with a particle based method for gas dynamics (``smoothed particle hydrodynamics'', GRAPE-SPH, Steinmetz 1996, Klessen 1997), or to follow cold collapse and radialorbit instability with high central resolution (Theis & Spurzem 1999).
However, such special purpose machines reach their highest efficiency only for problems, which can be tackled with pure and clean N-body algorithms such as NBODY4 or KIRA. Already for SPH or standard N-body simulations using an AC neighbour scheme or a very large number of close (so-called primordial) binaries, or for molecular dynamics simulations with potentials other than the Coulomb potential (e.g. van der Waals) they are not the optimal choice.
We propose two ways out of this problem. One is to use general purpose massively parallel machines, on which competitive implementation of NBODY6++ exist if compared with one of the GRAPE-4 boards, but not if compared with a larger scale GRAPE machine or the coming GRAPE-6. But there is still work in progress to make better use of the general purpose parallel computers. The other solution is to build a hybrid machine, which uses for the intermediate range forces a new reconfigurable special purpose device (``field programmable gate array'', FPGA), which is programmable, but follows directly the increase in hardware performance with time, as in the case of special purpose computers like GRAPE. The cost for that is a performance loss compared to GRAPE, and a still fairly complicated programming method. We propose to combine such FPGA boards as they are for example being designed and manufactured at the University of Mannheim with GRAPE and a host computer with PCI bus, to make up the AHA-GRAPE (``Adaptive Hydrodynamics Architecture GRAPE''). For more details see Kugel (this proceedings) . There is also an FPGA based clone of GRAPE (PRO-GRAPE, ``PROgrammable GRAPE'') under construction in the GRAPE group in Tokyo (see Makino, this proceedings) .
What are the challenges for even further increasing computational speed for collisional N-body simulations? We know very little about core collapse and post-collapse behaviour of rotating star clusters (no survey of N-body models exist, only Fokker-Planck models, Einsel & Spurzem 1999), in the case of multi-mass models with mass segregation, stellar evolution effects and tidal fields the scaling to real large N is an unsolved problem (see Aarseth & Heggie 1998, and Heggie, this proceedings) , and deep in the centres of galactic nuclei there is a very dense star cluster, rotating, axisymmetric if not triaxial, loaded with a moving, massive, star-accreting black hole, maybe even two of them. Despite recent attempts to tackle this problem the physical interplay between relaxation, star accretion and black hole growth in such situations is an unsolved and challenging theoretical and numericalproblem.
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