A Fast O(N²) Fragmentation Algorithm

Collisional fragmentation is a ubiquitous phenomenon arising in a variety of astrophysical systems, from asteroid belts to debris and protoplanetary disks. Numerical studies of fragmentation typically rely on discretizing the size distribution of colliding objects into a large number N of bins in mass space, usually logarithmically spaced. A standard approach for redistributing the debris produced in collisions into the corresponding mass bins results in $O\left(N^3\right)$ calculation, which leads to significant computational overhead when N is large. Here, we formulate a more efficient explicit $O\left(N^2\right)$ fragmentation algorithm, which works when the size spectrum of fragments produced in an individual collision has a self-similar shape with only a single characteristic mass scale (which can have arbitrary dependence on the energy and masses of colliding objects). Fragment size spectra used in existing fragmentation codes typically possess this property. We also show that our $O\left(N^2\right)$ approach can be easily extended to work with non-self-similar fragment size distributions, for which we provide a worked example. This algorithm offers a substantial speedup of fragmentation calculations for large N ≳ 10², even over the implicit methods, making it an attractive tool for studying collisionally evolving systems.