Introduction

Direct N-body simulations are of necessity expensive because of the need to evaluate all the N(N –1)/2 force interaction terms. We have seen that the Ahmad–Cohen [1973] neighbour scheme only alleviates the problem to some extent. However, once the particle number becomes sufficiently large, the dynamical behaviour begins to change because close encounters are less important. This behaviour has inspired methods for collisionless systems to be developed, such as tree codes or multipole expansions. In this chapter, we are concerned with tree codes since some relevant aspects of the latter have already been discussed. First we review the basic features of the pioneering Barnes & Hut [1986] scheme which is widely used in a variety of applications. Since the emphasis in this book is on collisional stellar dynamics, we devote a section to describing a tree code for point-mass interactions [McMillan & Aarseth, 1993] in the hope that it might be revived. The final section deals with an independent development for flattened systems [Richardson, 1993a,b] that has been used to study different stages of planetary formation as well as ring dynamics, where collisions play an important role.

Basic formulation

In view of the rapid growth in the computational requirements for increasing particle numbers when using direct summation, it is not surprising that several tree-based approaches have been made to speed up the expensive force calculation.