The Convergence of ParticleinCell Schemes for Cosmological Dark Matter Simulations
Abstract
Particle methods are a ubiquitous tool for solving the VlasovPoisson equation in comoving coordinates, which is used to model the gravitational evolution of dark matter (DM) in an expanding universe. However, these methods are known to produce poor results on idealized test problems, particularly at late times, after the particle trajectories have crossed. To investigate this, we have performed a series of one and twodimensional “Zel’dovich pancake” calculations using the popular particleincell (PIC) method. We find that PIC can indeed converge on these problems provided that the following modifications are made. The first modification is to regularize the singular initial distribution function by introducing a small but finite artificial velocity dispersion. This process is analogous to artificial viscosity in compressible gas dynamics, and, as with artificial viscosity, the amount of regularization can be tailored so that its effect outside of a welldefined region—in this case, the highdensity caustics—is small. The second modification is the introduction of a particle remapping procedure that periodically reexpresses the DM distribution function using a new set of particles. We describe a remapping algorithm that is thirdorder accurate and adaptive in phase space. This procedure prevents the accumulation of numerical errors in integrating the particle trajectories from growing large enough to significantly degrade the solution. Once both of these changes are made, PIC converges at second order on the Zel’dovich pancake problem, even at late times, after many caustics have formed. Furthermore, the resulting scheme does not suffer from the unphysical, smallscale “clumping” phenomenon known to occur on the pancake problem when the perturbation wavevector is not aligned with one of the Cartesian coordinate axes.
 Publication:

The Astrophysical Journal
 Pub Date:
 January 2016
 DOI:
 10.3847/0004637X/816/2/56
 arXiv:
 arXiv:1503.05969
 Bibcode:
 2016ApJ...816...56M
 Keywords:

 cosmology: theory;
 dark matter;
 methods: numerical;
 Astrophysics  Cosmology and Nongalactic Astrophysics
 EPrint:
 29 pages, 29 figures. Accepted for publication in ApJ. The revised version includes a discussion of energy conservation in the remapping procedure, as well as some interpretive differences in the Conclusions made in response to the referee report. Results themselves are unchanged