The Barnes–Hut simulation is an approximation algorithm for performing an n-body simulation. It is notable for having order O compared to a direct-sum algorithm which would be O. The simulation volume is usually divided up into cubic cells via an octree, so that only particles from nearby cells need to be treated individually, and particles in distant cells can be treated as a single large particle centered at the cell's center of mass. This can dramatically reduce the number of particle pair interactions that must be computed. Some of the most demanding high-performance computing projects do computational astrophysics using the Barnes–Hut treecode algorithm, such as DEGIMA.
Algorithm
The Barnes–Hut tree
In a three-dimensional n-body simulation, the Barnes–Hut algorithm recursively divides the n bodies into groups by storing them in an octree. Each node in this tree represents a region of the three-dimensional space. The topmost node represents the whole space, and its eight children represent the eightoctants of the space. The space is recursively subdivided into octants until each subdivision contains 0 or 1 bodies. There are two types of nodes in the octree: internal and external nodes. An external node has no children and is either empty or represents a single body. Each internal node represents the group of bodies beneath it, and stores the center of mass and the total mass of all its children bodies.
To calculate the net force on a particular body, the nodes of the tree are traversed, starting from the root. If the center of mass of an internal node is sufficiently far from the body, the bodies contained in that part of the tree are treated as a single particle whose position and mass is respectively the center of mass and total mass of the internal node. If the internal node is sufficiently close to the body, the process is repeated for each of its children. Whether a node is or isn't sufficiently far away from a body, depends on the quotient, where s is the width of the region represented by the internal node, and d is the distance between the body and the node's center of mass. The node is sufficiently far away when this ratio is smaller than a threshold valueθ. The parameter θ determines the accuracy of the simulation; larger values of θ increase the speed of the simulation but decreases its accuracy. If θ = 0, no internal node is treated as a single body and the algorithm degenerates to a direct-sum algorithm.
