N-Body Simulations Using Message Passsing Parallel Computers

In this paper, we present new parallel formulations of the Barnes-Hut method for n-body simulations on message passing computers. These parallel formulations partition the domain eeciently incurring minimal communication overhead. This is in contrast to existing schemes that are based on sorting a large number of keys or on the use of global data structures. The new formulations are augmented by alternate communication strategies which serve to minimize communication overhead. The impact of these communication strategies is experimentally studied. We report on experimental results obtained from an astrophysical simulation on an nCUBE2 parallel computer. The n-body problem simulates the behavior of n particles, each of which is innuenced by every other particle during a time-step. An exact formulation of this problem therefore requires calculation of n 2 interactions between each pair of particles. Many approximate algorithms have been devised that reduce the complexity of this problem. Most of these algorithms are based on a hierarchical representation of the domain using a spatial tree data structure. The leaf nodes consist of aggregates of particles. Each node in the tree contains a series representation of the effect of the particles contained in the subtree rooted at the node. These representations are typically based on Taylor or Legendre polynomials. Interactions between nodes and particles are dictated by a multipole acceptance criteria (MAC). Diierent algorithms use diierent MAC. Selection of an appropriate MAC is critical to controlling the error in simulation. Methods in this class include those due to Appel, Barnes and Hut, and Greengard and Rokhlin (Fast Multi-pole Method). Even with the reduced complexity of these algorithms, the n-body problem takes a significant amount of time because of the large number of particles and time steps. EEcient parallel formulations therefore need to be developed. The Barnes-Hut method is one of the most popular methods due to its simplicity. Although its computational complexity of O(n log n) is more than that of the Fast Multipole Method (FMM), which is O(n))4], the associated constants are smaller for the Barnes-Hut method particularly for simulations in three dimensions. Furthermore, FMM uses a number of complicated data structures that make it diicult to program. However, FMM has proven error bounds unlike the Barnes-Hut method. Warren and Salmon 9] present a variant of the Barnes-Hut method that has a good worst-case error bound. In this paper, we describe parallel formulations of the Barnes-Hut method. The parallel formulations presented in …