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 _ { \mathrm { rms } } \sim 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 \lesssim 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 _ { \mathrm { particles } } \approx N _ { \mathrm { 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 .