Towards quantitative control on discreteness error in the non-linear regime of cosmologicalN-body simulations

Abstract

The effects of discreteness arising from the use of the N‐body method on the accuracy of simulations of cosmological structure formation are not currently well understood. In the first part of this paper, we discuss the essential question of how the relevant parameters introduced by this discretization should be extrapolated in convergence studies if the goal is to recover the Vlasov–Poisson limit. In the second part of the paper, we study numerically, and with analytical methods developed recently by us, the central issue of how finite particle density affects the precision of results above the force‐smoothing scale. In particular, we focus on the precision of results for the power spectrum at wavenumbers around and above the Nyquist wavenumber, in simulations in which the force resolution is taken to be smaller than the initial interparticle spacing. Using simulations of identical theoretical initial conditions sampled on four different ‘pre‐initial’ configurations (three different Bravais lattices and a glass), we obtain a lower bound on the real discreteness error. With the guidance of our analytical results, which match extremely well this measured dispersion into the weakly non‐linear regime, and of further controlled tests for dependences on the relevant discreteness parameters, we establish with confidence that the measured dispersion is not contaminated either by finite box size effects or by subtle numerical effects. Our results notably show that, at wavenumbers below the Nyquist wavenumber, the dispersion increases monotonically in time throughout the simulation, while the same is true above the Nyquist wavenumber once non‐linearity sets in. For normalizations typical of cosmological simulations, we find lower bounds on errors at the Nyquist wavenumber of the order of 1 per cent, and larger above this scale. Our main conclusion is that the only way this error may be reduced below these levels at these physical scales, and indeed convergence to the physical limit firmly established, is by extrapolation, at fixed values of the other relevant parameters, to the regime in which the mean comoving interparticle distance becomes less than the force‐smoothing scale.

Generation of vorticity and velocity dispersion by orbit crossing

“…This does not test the scalar mode of the stress tensor though, which ends up inducing the largest correction to the PPF approximation. In this respect, it would be interesting to test how robust the scalar part of the stress tensor is to details of the numerical simulations, as spurious effects due to discreteness may amplify velocity dispersion in simulations [56,57] (see also [58]). …”

Cold uniform spherical collapse revisited

“…The study of this problem for these initial conditions thus allow one to understand some of the subtleties which may be involved in practically testing for collisionality effects in N body systems. This is in principle an important issue in cosmological simulations (see [1] for a detailed discussion). This proceedings essentially summarizes synthetically results reported in much greater detail in [2].…”

A toy model to test the accuracy of cosmologicalN-body simulations

“…We note that most of the discussion in the literature has focused the attention on the role of the softening length (see e.g. Splinter et al 1988;Heitmann et al 2005;Romeo et al 2008;Joyce et al 2009a;Power et al 2003): here we have found that a crucial parameter is also represented by the time-stepping accuracy.…”

“…However, it is not a simple issue to control the numerical accuracy of the non-linear gravitational clustering in a cosmological simulation; for instance the question of determining the optimal spatial resolution, i.e. the best value for the gravitational force softening length, is still an open problem (see e.g., Splinter et al 1988;Power et al 2003;Romeo et al 2008;Joyce et al 2009a).…”