A general class of Lagrangian smoothed particle hydrodynamics methods and implications for fluid mixing problems

By P. Hopkins. 2012

352 citations recorded by the source corpus.

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 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 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, and the resulting degree of freedom, 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.

Subfields: Atmospheric Science, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/4b541e568719436c2e93731c3f873574b6bf7bb7

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Citations in the corpus

Neutral hydrogen in galaxy haloes at the peak of the cosmic star formation history (2014, methodology): “All simulations use the newly developed GIZMO5 simulation code (Hopkins 2014) in “P-SPH” mode (Hopkins 2013).”
Bimodality of low-redshift circumgalactic O VI in non-equilibrium EAGLE zoom simulations (2016, methodology): “…SPH implementation of the Gadget-3 code (Springel & Hernquist 2003a) comprise i) the pressure-entropy SPH formulation described by Hopkins (2013), ii) the artificial viscosity switch of Cullen & Dehnen (2010), iii) the artificial conduction switch of Price (2008), iv) the…”
Forged in FIRE: cusps, cores and baryons in low-mass dwarf galaxies (2015, methodology): “The P-SPH method adopts the Lagrangian “pressure-entropy” formulation of the SPH equations developed in (Hopkins 2013); this eliminates the major differences between SPH, moving mesh, and grid (adaptive mesh) codes, and resolves the well-known issues with fluid mixing instabilities in…”
Galactic r-process enrichment by neutron star mergers in cosmological simulations of a Milky Way-mass galaxy (2014, methodology): “Our simulations are also based on a formulation of SPH, called P-SPH, that resolves the historical problems of SPH in capturing hydrodynamic instabilities that are important for mixing, in particular the Kelvin–Helmholtz and Rayleigh–Taylor instabilities (Hopkins 2013).”
Galaxies on FIRE (Feedback In Realistic Environments): stellar feedback explains cosmologically inefficient star formation (2013, methodology): “As noted in the text, these runs adopt the P-SPH formulation of TreeSPH in GIZMO, which features many improvements to SPH and has been tested in a wide range of problems (listed in the main text).”
Colours and luminosities of z = 0.1 galaxies in the eagle simulation (2015, methodology): “We modified the treatment of hydrodynamics in GADGET-3 to use the conservative pressure-entropy SPH formulation of Hopkins (2013), the artificial viscosity switch introduced by Cullen & Dehnen (2010), an artificial conduction switch inspired by Price (2008), the C2 kernel from Wendland (1995) and…”
The APOSTLE simulations: solutions to the Local Group's cosmic puzzles (2015, methodology): “…using the standard Tree-PM scheme of P-GADGET-3, while hydrodynamic forces are computed in the smoothed particle hydrodynamics scheme of ANARCHY described in Dalla-Vecchia et al. (in preparation) and Schaller et al. (2015b), which uses the pressure– entropy formalism introduced by Hopkins (2013).”
(Star)bursts of FIRE: observational signatures of bursty star formation in galaxies (2015, methodology): “…for the simulations analysed in this work is a heavily modified version of the GADGET code (Springel et al. 2001; Springel 2005), GIZMO (Hopkins 2014, 2015).3 The hydrodynamical equations were solved with the pressure-based formulation of the smoothed particle hydrodynamics method (Hopkins 2013).”
Star formation in galaxy mergers with realistic models of stellar feedback and the interstellar medium (2012, result): “We have also compared a subset of our merger simulations run with an alternative formulation of SPH from Hopkins (2013), which produces results much more similar to grid codes; in these tests, we confirm all of the qualitative conclusions here.”
Gasoline2: a modern smoothed particle hydrodynamics code (2017, methodology): “Referring to SPH formalisms prior to ∼2008 as ‘traditional’ is fairly common in the literature (e.g. Gibson et al. 2009; Hopkins 2013).”
The cosmic baryon cycle and galaxy mass assembly in the FIRE simulations (2016, methodology): “All simulations are run using the “P-SPH" mode of the GIZMO simulation code2 (Hopkins 2015), employing a pressure-entropy formulation of SPH that minimizes some of the major problems previous formulations of SPH have with fluid mixing instabilities (Hopkins 2013; Saitoh & Makino 2013).”
BREATHING FIRE: HOW STELLAR FEEDBACK DRIVES RADIAL MIGRATION, RAPID SIZE FLUCTUATIONS, AND POPULATION GRADIENTS IN LOW-MASS GALAXIES (2015, methodology): “All 8 http://fire.northwestern.edu simulations were run using the GIZMO code (Hopkins 2015) with pressure-entropy based smoothed particle hydrodynamics (P-SPH; Hopkins 2013) and an updated version of the PM+TREE gravity solver from GADGET3 (Springel 2005).”
Gravitational torque-driven black hole growth and feedback in cosmological simulations (2016, neutral): “…0.16 105 0.5 103 20 0 Large momentum boost 600 3.3e-2 n256-nf 6.4e6 0.16 105 0.5 0 0 0 No feedback 0 0 n256-mfm 6.4e6 0.16 105 0.5 103 1 0 Meshless hydrodynamics 30 1.6e-3 n512 8.0e5 0.08 105 0.5 103 1 2 High resolution 30 1.6e-3 fluid mixing instabilities (Hopkins 2013; Saitoh & Makino 2013).”
Molecular hydrogen abundances of galaxies in the EAGLE simulations (2015, background): “Among the main features of Anarchy are (i) the pressure-entropy formulation of SPH described in Hopkins (2013), (ii) the artificial viscosity switch of Cullen & Dehnen (2010), (iii) an artificial conduction switch described in Price (2008), (iv) a C2 Wendland (1995) kernel with 58 neighbours to…”
Physical Models of Galaxy Formation in a Cosmological Framework (2014, methodology): “Hopkins (2013) reformulated DI-SPH in terms of entropy to incorporate the improved conservation properties of EC-SPH.”
Gusty, gaseous flows of FIRE: Galactic winds in cosmological simulations with explicit stellar feedback (2015, methodology): “Feedback from star formation contributes to the stability of this disc by stirring turbulence (Thompson, Quataert & Murray 2005; Ostriker & Shetty 2011; Faucher-Giguère, Quataert & Hopkins 2013; Hopkins, Kereš & Murray 2013).”
The EAGLE simulations of galaxy formation: the importance of the hydrodynamics scheme (2015, methodology): “(10) The use of the smoothed quantities P̄i and ρ̄i in the equations of motion smooths out the spurious pressure jumps appearing at contact discontinuities in older formulations of SPH (Saitoh & Makino 2013; Hopkins 2013).”