TreePM

TreePM is a class of algorithms for computing gravitational forces in N-body simulations, combining a Barnes-Hut tree method with a Particle-Mesh (PM) approach. It is widely used in astrophysics and cosmology to simulate the formation of large-scale structure because it balances accuracy and efficiency for systems with very large numbers of particles. The PM component handles long-range forces on a fixed spatial mesh, while the tree component resolves short-range forces within localized regions.

In a typical TreePM scheme, the gravitational force is split into long-range and short-range parts using a scale parameter, such as a cutoff radius. The long-range potential is computed on a grid by solving Poisson’s equation via a Fast Fourier Transform, producing a smooth, grid-based field. The short-range force is obtained with a hierarchical tree algorithm (for example, Barnes-Hut) using the actual particle distribution, with an appropriate real-space cutoff to avoid double counting with the PM part. The total force on each particle is the sum of the long-range PM force and the short-range tree force.

TreePM offers efficient scaling for large simulations, enabling runs with millions to billions of particles. It preserves large-scale accuracy from the PM method while improving small-scale fidelity with the tree, making it a common choice in cosmological codes such as Gadget and PKDGRAV. Parameters such as the mesh size, the splitting scale, and the tree opening criterion are tuned to balance speed and accuracy and may be adjusted to match the needs of a given simulation.

Limitations include sensitivity to the chosen split scale and mesh resolution, potential artifacts at the transition between long- and short-range forces, and memory requirements associated with the grid and tree data structures. Proper calibration is important to control force errors and convergence.