A Comparison of Cosmological Hydrodynamic Codes

We present a detailed comparison of the simulation results of various cosmological hydrodynamic codes. Starting with identical initial conditions based on the Cold Dark Matter scenario for the growth of structure, we integrate from redshift $z=20$ to $z=0$ to determine the physical state within a representative volume of size $L^3$ where $L=64 h^{-1} {\rm Mpc}$. Five independent codes are compared: three of them Eulerian mesh based and two variants of the Smooth Particle Hydrodynamics "SPH" Lagrangian approach. The Eulerian codes were run at $N^3=(32^3,~64^3,~128^3,~{\rm and},~256^3)$ cells, the SPH codes at $N^3= 32^3$ and $64^3$ particles. Results were then rebinned to a $16^3$ grid with the expectation that the rebinned data should converge, by all techniques, to a common and correct result as $N \rightarrow \infty$. We find that global averages of various physical quantities do, as expected, tend to converge in the rebinned model, but that uncertainties in even primitive quantities such as $\langle T \rangle$, $\langle \rho^2\rangle^{1/2}$ persists at the 3\%-17\% level after completion of very large simulations. The two SPH codes and the two shock capturing Eulerian codes achieve comparable and satisfactory accuracy for comparable computer time in their treatment of the high density, high temperature regions as measured in the rebinned data; the variance among the five codes (at highest resolution) for the mean temperature (as weighted by $\rho^2$) is only 4.5\%. Overall the comparison allows us to better estimate errors, it points to ways of improving this current generation of hydrodynamic codes and of suiting their use to problems which exploit their individually best features.