Cosmological smoothed particle hydrodynamics simulations: the entropy equation

We discuss differences in simulation results that arise between the use of either the thermal energy or the entropy as an independent variable in smoothed particle hydrodynamics (SPH). In this context, we derive a new version of SPH that, when appropriate, manifestly conserves both energy and entropy if smoothing lengths are allowed to adapt freely to the local mass resolution. To test various formulations of SPH, we consider point-like energy injection, as in certain models of supernova feedback, and find that powerful explosions are well represented by SPH even when the energy is deposited into a single particle, provided that the entropy equation is integrated. If the thermal energy is instead used as an independent variable, unphysical solutions can be obtained for this problem. We also examine the radiative cooling of gas spheres that collapse and virialize in isolation, and of haloes that form in cosmological simulations of structure formation. When applied to these problems, the thermal energy version of SPH leads to substantial overcooling in haloes that are resolved with up to a few thousand particles, while the entropy formulation is biased only moderately low for these haloes under the same circumstances. For objects resolved with much larger particle numbers, the two approaches yield consistent results. We trace the origin of the differences to systematic resolution effects in the outer parts of cooling flows. When the thermal energy equation is integrated and the resolution is low, the compressional heating of the gas in the inflow region is underestimated, violating entropy conservation and improperly accelerating cooling. The cumulative effect of this overcooling can be significant. In cosmological simulations of moderate size, we find that the fraction of baryons which cool and condense can be reduced by up to a factor ~2 if the entropy equation is employed rather than the thermal energy equation, partly explaining discrepancies with semi-analytic treatments of galaxy formation. We also demonstrate that the entropy method leads to a greatly reduced scatter in the density-temperature relation of the low-density Lyα forest relative to the thermal energy approach, in accord with theoretical expectations.