Abstract
Various formulations of smooth-particle hydrodynamics (SPH) have been
proposed, intended to resolve certain difficulties in the treatment of fluid
mixing instabilities. Most have involved changes to the algorithm which either
introduce 'artificial' correction terms or violate what is arguably the
greatest advantage of SPH over other methods: manifest conservation of energy,
entropy, momentum, and angular momentum. Here, we show how a class of
alternative SPH equations of motion (EOM) can be derived self-consistently from
a discrete particle Lagrangian - guaranteeing manifest conservation - in a
manner which tremendously improves treatment of these instabilities and contact
discontinuities. Saitoh & Makino recently noted that the volume element used to
discretize the EOM does not need to explicitly invoke the mass density (as in
the 'standard' approach); we show how this insight can be incorporated into the
rigorous Lagrangian formulation that retains ideal conservation properties and
includes the 'Grad-h' terms that account for variable smoothing lengths. We
derive a general EOM for any choice of volume element (particle 'weights') and
method of determining smoothing lengths. We then specify this to a
'pressure-entropy formulation' which resolves problems in the traditional
treatment of fluid interfaces. Implementing this in a new version of the GADGET
code, we show it leads to good performance in mixing experiments (e.g.
Kelvin-Helmholtz & 'blob' tests). And conservation is maintained even in strong
shock/blastwave tests, where formulations without manifest conservation produce
large errors. This also improves the treatment of sub-sonic turbulence, and
lessens the need for large kernel particle numbers. The code changes are
trivial and entail no additional numerical expense. This provides a general
framework for self-consistent derivation of different 'flavors' of SPH.
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