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Various formulations of smooth-particle hydrodynamics (SPH) have beenproposed, intended to resolve certain difficulties in the treatment of fluidmixing instabilities. Most have involved changes to the algorithm which eitherintroduce artificial correction terms or violate arguably the greatestadvantage of SPH over other methods: manifest conservation of energy, entropy,momentum, and angular momentum. Here, we show how a class of alternative SPHequations of motion (EOM) can be derived self-consistently from a discreteparticle Lagrangian (guaranteeing manifest conservation) in a manner whichtremendously improves treatment of instabilities and contact discontinuities.Saitoh & Makino recently noted that the volume element used to discretize theEOM does not need to explicitly invoke the mass density (as in the 'standard'approach); we show how this insight, and the resulting degree of freedom, canbe incorporated into the rigorous Lagrangian formulation that retains idealconservation properties and includes the 'Grad-h' terms that account forvariable smoothing lengths. We derive a general EOM for any choice of volumeelement (particle 'weights') and method of determining smoothing lengths. Wethen specify this to a 'pressure-entropy formulation' which resolves problemsin the traditional treatment of fluid interfaces. Implementing this in a newversion of the GADGET code, we show it leads to good performance in mixingexperiments (e.g. Kelvin-Helmholtz & blob tests). And conservation ismaintained even in strong shock/blastwave tests, where formulations withoutmanifest conservation produce large errors. This also improves the treatment ofsub-sonic turbulence, and lessens the need for large kernel particle numbers.The code changes are trivial and entail no additional numerical expense. Thisprovides a general framework for self-consistent derivation of different'flavors' of SPH.

A General Class of Lagrangian Smoothed Particle Hydrodynamics Methods and Implications for Fluid Mixing Problems