Special high-accuracy direct force summation N -body algorithms and their relevance for the simulation of the dynamical evolution of star clusters and other gravitating N -body systems in astrophysics are presented, explained and compared with other methods. Other methods means here approximate physical models based on the Fokker-Planck equation as well as other, approximate algorithms to compute the gravitational potential in N -body systems. Questions regarding the parallel implementation of direct "brute force" N -body codes are discussed. The astrophysical application of the models to the theory of relaxing rotating and non-rotating collisional star clusters is presented, briefly mentioning the questions of the validity of the Fokker-Planck approximation, the existence of gravothermal oscillations and of rotation and primordial binaries. ⋆

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In a previous paper we have shown that direct gravitational N-body simulations in astrophysics scale very well for moderately parallel supercomputers (order 10-100 nodes). The best balance between computation and communication is reached if the nodes are accelerated by special purpose hardware; in this paper we describe the implemen-R. Spurzem (u) · P. tation of particle based astrophysical simulation codes on new types of accelerator hardware (field programmable gate arrays, FPGA, and graphical processing units, GPU). In addition to direct gravitational N-body simulations we also use the algorithmically similar "smoothed particle hydrodynamics" method as test application; the algorithms are used for astrophysical problems as e.g. evolution of galactic nuclei with central black holes and gravitational wave generation, and star formation in galaxies and galactic nuclei. We present the code performance on a single node using different kinds of special hardware (traditional GRAPE, FPGA, and GPU) and some implementation aspects (e.g. accuracy). The results show that GPU hardware for real application codes is as fast as GRAPE, but for an order of magnitude lower price, and that FPGA is useful for acceleration of complex sequences of operations (like SPH). We discuss future prospects and new cluster computers built with new generations of FPGA and GPU cards.

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Observations of Globular Cluster ellipticity distributions related to some fundamental parameters give strong evidence for a decay of rotational energy in these systems with time. In order to study the effectiveness of angular momentum transport (or loss, resp.) a code has been written which solves the Fokker-Planck equation in (E, Jz)-space and follows the evolution from some initial conditions through core collapse (and possibly gravothermal oscillations) up to the post-collapse phase. For the purpose of comparability with N-body simulations rotating initial model configurations according to the prescriptions of Lupton & Gunn (1987) have been constructed. These models are intended to continue previous work by Goodman (1983, Fokker-Planck) and Akiyama & Sugimoto (1989, N-Body). In this contribution the derivation of the flux coefficients is given.