Context. Modified Newtonian dynamics (MOND) is a promising alternative to dark matter. To further test the theory, there is a need for fluid- and particle-dynamics simulations. The force in MOND is not a direct particle-particle interaction, but derives from a potential for which a nonlinear partial differential equation (PDE) needs to be solved. Normally, this makes the problem of simulating dynamical evolution computationally expensive. Aims. We intend to develop a fast particle-mesh (PM) code for MOND (the AQUAL formalism). Methods. We transformed the nonlinear equation for MOND into a system of linear PDEs plus one algebraic equation. An iterative scheme with the fast Fourier transform (FFT) produces successively better numerical approximations. Results. The algorithm was tested for dynamical systems in MOND where analytical solutions are known: the two-body problem, a body with a circular ring, and a spherical distribution of particles in thermal equilibrium in the self-consistent potential. Conclusions. The PM code can accurately calculate the forces at subpixel scale and reproduces the analytical solutions. Four iterations are required for the potential, but when the spatial steps are small compared to the kernel width, one iteration is suffices. The use of a smoothing kernel for the accelerations is inevitable in order to eliminate the self-gravity of the point particles. Our PDE solver is 15 to 42 times as slow as a standard Poisson solver. However, the smoothing and particle propagation takes up most of the time above one particle per 10 ³ pixels. The FFTs, the smoothing, and the propagation part in the code can all be parallelized.

Context. In a Keplerian system, a large number of bodies orbit a central mass. Accretion disks, protoplanetary disks, asteroid belts, and planetary rings are examples. Simulations of these systems require algorithms that are computationally efficient. The inclusion of collisions in the simulations is challenging but important. Aims. We intend to calculate the time of collision of two astronomical bodies in intersecting Kepler orbits as a function of the orbital elements. The aim is to use the solution in an analytic propagator (N-body simulation) that jumps from one collision event to the next. Methods. We outline an algorithm that maintains a list of possible collision pairs ordered chronologically. At each step (the soonest event on the list), only the particles created in the collision can cause new collision possibilities. We estimate the collision rate, the length of the list, and the average change in this length at an event, and study the efficiency of the method used. Results. We find that the collision-time problem is equivalent to finding the grid point between two parallel lines that is closest to the origin. The solution is based on the continued fraction of the ratio of orbital periods. Conclusions. Due to the large jumps in time, the algorithm can beat tree codes (octree and k-d tree codes can efficiently detect collisions) for specific systems such as the Solar System with N < 108. However, the gravitational interactions between particles can only be treated as gravitational scattering or as a secular perturbation, at the cost of reducing the time-step or at the cost of accuracy. While simulations of this size with high-fidelity propagators can already span vast timescales, the high efficiency of the collision detection allows many runs from one initial state or a large sample set, so that one can study statistics.