Algorithmic comparisons of decaying, isothermal, supersonic turbulence

Context. Simulations of astrophysical turbulence have reached such a level of sophistication that quantitative results are now starting to emerge. However, contradicting results have been reported in the literature with respect to the performance of the numerical techniques employed for its study and their relevance to the physical systems modelled. Aims. We aim at characterising the performance of a variety of hydrodynamics codes including different particle-based and grid-based techniques on the modelling of decaying supersonic turbulence. This is the first such large-scale comparison ever conducted. Methods. We modelled driven, compressible, supersonic, isothermal turbulence with an rms Mach number of M-rms similar to 4, and then let it decay in the absence of gravity, using runs performed with four different grid codes (ENZO, FLASH, TVD, ZEUS) and three different SPH codes (GADGET, PHANTOM, VINE). We additionally analysed two calculations denoted as PHANTOM A and PHANTOM B using two different implementations of artificial viscosity in PHANTOM. We analysed the results of our numerical experiments using volume-averaged quantities like the rms Mach number, volume-and density-weighted velocity Fourier spectrum functions, and probability distribution functions of density, velocity, and velocity derivatives. Results. Our analysis indicates that grid codes tend to be less dissipative than SPH codes, though details of the techniques used can make large differences in both cases. For example, the Morris & Monaghan viscosity implementation for SPH results in less dissipation (PHANTOM B and VINE versus GADGET and PHANTOM A). For grid codes, using a smaller diffusion parameter leads to less dissipation, but results in a larger bottleneck effect (our ENZO versus FLASH runs). As a general result, we find that by using a similar number of resolution elements N for each spatial direction means that all codes (both grid-based and particle-based) show encouraging similarity of all statistical quantities for isotropic supersonic turbulence on spatial scales k less than or similar to N/32 (all scales resolved by more than 32 grid cells), while scales smaller than that are significantly affected by the specific implementation of the algorithm for solving the equations of hydrodynamics. At comparable numerical resolution (N-particles approximate to N-cells), the SPH runs were on average about ten times more computationally intensive than the grid runs, although with variations of up to a factor of ten between the different SPH runs and between the different grid runs. Conclusions. At the resolutions employed here, the ability to model supersonic to transonic flows is comparable across the various codes used in this study.