Abstract
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.
Highlights
Smoothed particle hydrodynamics (SPH) was introduced by Lucy (1977) and Gingold & Monaghan (1977) as an alternative to grid-based fluid solvers and has developed into a mature and popular simulation technique
To test various formulations of smoothed particle hydrodynamics (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
Based on the results presented above, we expect a significant difference in the fraction of gas that can cool in halos of a given size in full cosmological simulations when SPH is formulated either in terms of the thermal energy equation or the entropy equation
Summary
Smoothed particle hydrodynamics (SPH) was introduced by Lucy (1977) and Gingold & Monaghan (1977) as an alternative to grid-based fluid solvers and has developed into a mature and popular simulation technique. While cosmological SPH simulations have provided a successful instrument for exploring galaxy formation, it has recently become clear that subtle numerical properties of existing algorithms severely influence characteristics of model galaxies, such as their luminosities and colours These numerical complications follow once radiative cooling of the gas is included. SPH estimates of the density for hot particles passing close to cold particles will be biased high if they sample the dense cold phase, leading to accelerated cooling (Pearce et al, 1999; Thacker et al, 2000; Croft et al, 2001) Given these complications, and the highly nonlinear nature of radiative cooling, it is unclear whether or not cooling rates obtained by existing cosmological SPH codes are reliable, in particular with respect to the majority of halos which are resolved with relatively few particles.
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