The analytic propagator Kepler-Collisions calculates collisions for Keplerian systems. The algorithm maintains a list of collision possibilities and jumps from one collision to the next; since collisions are rare in astronomical scales, jumping from collision to collision and calculating each one is more efficient than calculating all the time steps that are between collisions.

MONDPMesh provides a particle-mesh method to calculate the time evolution of an system of point masses under modified gravity, namely the AQUAL formalism. This is done by transforming the Poisson equation for the potential into a system of four linear PDEs, and solving these using fast Fourier transforms. The accelerations on the point masses are calculated from this potential, and the system is propagated using Leapfrog integration. The time complexity of the code is O(N⋅p⋅log(p)) for p pixels and N particles, which is the same as for a Newtonian particle-mesh code.