A comparison between grid and particle methods on the statistics of driven, supersonic, isothermal turbulence

By Daniel J. Price, Christoph Federrath. 2010

199 citations recorded by the source corpus.

We compare the statistics of driven, supersonic turbulence at a high Mach number using FLASH, a widely used Eulerian grid-based code, and PHANTOM, a Lagrangian smoothed particle hydrodynamics (SPH) code at resolutions of up to 512 3 in both grid cells and SPH particles. We find excellent agreement between codes on the basic statistical properties: a slope of k −1.95 in the velocity power spectrum for hydrodynamic, Mach 10 turbulence, evidence in both codes for a Kolmogorov-like slope of k −5/3 in the variable ρ 1/3 v as suggested by Kritsuk et al. and a lognormal probability distribution function (PDF) with a width that scales with the Mach number and proportionality constant b = 0.33–0.5 in the density variance–Mach number relation. The measured structure function slopes are not converged in either code at 512 3 elements. We find that for measuring volumetric statistics such as the power spectrum slope and structure function scaling, SPH and grid codes give roughly comparable results when the number of SPH particles is approximately equal to the number of grid cells. In particular, to accurately measure the power spectrum slope in the inertial range, in the absence of sub-grid turbulence models, requires at least 512 3 computational elements in either code. On the other hand the SPH code was found to be better at resolving dense structures, giving maximum densities at a resolution of 128 3 particles that were similar to the maximum densities resolved in the grid code at 512 3 cells, reflected also in the high density tail of the PDF. We find SPH to be more dissipative at comparable numbers of computational elements in statistics of the velocity field, but correspondingly less dissipative than the grid code in the statistics of density-weighted quantities such as ρ 1/3 v. For SPH simulations of high Mach number turbulence, we find it important to use sufficient non-linear β-viscosity in order to prevent particle interpenetration in shocks (we require βvisc = 4 instead of the widely used default value, βvisc = 2).

Subfields: Computational Mechanics, Ocean Engineering

View Paper: https://www.semanticscholar.org/paper/255098057d4dd43c50e52f5692327d2ee2c831c4

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Citations in the corpus

Gasoline2: a modern smoothed particle hydrodynamics code (2017, background): “SPH and Eulerian codes generally agree well in simulations that involve supersonic flow (e.g. comparison projects by Kitsionas et al. 2009; Price & Federrath 2010; Bauer & Springel 2012), highMach number shocks (Tasker et al. 2008; Sijacki et al. 2012) and gravitational structure formation (O’Shea…”
A NEW JEANS RESOLUTION CRITERION FOR (M)HD SIMULATIONS OF SELF-GRAVITATING GAS: APPLICATION TO MAGNETIC FIELD AMPLIFICATION BY GRAVITY-DRIVEN TURBULENCE (2011, methodology): “Measuring the power-law exponent of the turbulence accurately would require at least 512 cells inside the Jeans volume as expected from pure turbulence simulations in a periodic box (Federrath et al. 2010b; Price & Federrath 2010).”
Galaxies on FIRE (Feedback In Realistic Environments): stellar feedback explains cosmologically inefficient star formation (2013, background): “This includes Sod shock tubes; Sedov blastwaves; wind tunnel tests (radiative and adiabatic, up to Mach ∼ 104); linear sound wave propagation; oscillating polytropes; hydrostatic equilibrium “deformation”/surface tension tests (Saitoh & Makino 2013); Kelvin-Helmholtz and Rayleigh-Taylor instabilitie”
A new class of accurate, mesh-free hydrodynamic simulation methods (2014, methodology): “…in the literature to difficulties of SPH methods in dealing with these instabilities (see e.g. Morris 1996; Dilts 1999; Ritchie & Thomas 2001; Marri & White 2003; Okamoto et al. 2003; Agertz et al. 2007; Kitsionas et al. 2009; Price & Federrath 2010; Bauer & Springel 2012; Sijacki et al. 2012).”
FIRE-2 simulations: physics versus numerics in galaxy formation (2017, background): “…effect on predictions, since the turbulence is primarily super-sonic, cooling is fast compared to dynamical times, and phase structure is primarily driven by gravitational collapse and feedback – all limits where SPH performs well (see Kitsionas et al. 2009; Price & Federrath 2010; Hopkins 2015).”
Shocking results without shocks: Subsonic turbulence in smoothed particle hydrodynamics and moving-mesh simulations (2011, methodology): “In contrast, Price & Federrath (2010) claimed excellent agreement for driven supersonic Mach M = 10 turbulence between the SPH-code PHANTOM and the Eulerian mesh-code FLASH.”
Disc formation from tidal disruptions of stars on eccentric orbits by Schwarzschild black holes (2015, methodology): “2 Self-gravity is now implemented into PHANTOM.”
Star formation in galaxy mergers with realistic models of stellar feedback and the interstellar medium (2012, background): “The discrepancies above are also minimized when the flows of interest are supersonic (as opposed to subsonic), which is very much the case here (Kitsionas et al. 2009; Price & Federrath 2010; Bauer & Springel 2012).”
Phantom: A Smoothed Particle Hydrodynamics and Magnetohydrodynamics Code for Astrophysics (2017, methodology): “This is the switch used for numerous turbulence applications with Phantom (e.g. Price & Federrath, 2010; Price et al., 2011; Tricco et al., 2016b) where it is important to minimise numerical dissipation in order to maximise the Reynolds number (e.g. Valdarnini, 2011; Price, 2012b).”
THE STAR FORMATION RATE OF TURBULENT MAGNETIZED CLOUDS: COMPARING THEORY, SIMULATIONS, AND OBSERVATIONS (2012, background): “…we evolve the MHD equations for two turbulent crossing times, 2T = L/(Mcs) without self-gravity, in order to establish fully developed, compressible turbulence (e.g., Klessen et al. 2000; Klessen 2001; Heitsch et al. 2001; Federrath et al. 2009, 2010b; Price & Federrath 2010; Micic et al. 2012).”
Tearing up the disc: misaligned accretion on to a binary (2013, methodology): “We use the smoothed particle hydrodynamics (SPH; e.g. Price 2012) code PHANTOM (see e.g. Lodato & Price 2010; Price & Federrath 2010; Nixon 2012; Nixon et al. 2012a) to simulate the evolution of circumbinary discs.”
On the universality of supersonic turbulence (2013, background): “…cells, using solenoidal or weakly compressive driving, indicate a ρ1/3v scaling consistent with P(ρ1/3v) ∝ k−5/3 (Kritsuk et al. 2007; Federrath et al. 2010; Price & Federrath 2010; Kritsuk, Wagner & Norman 2013), even if a magnetic field is included (Kowal & Lazarian 2007; Kritsuk et al. 2009).”
A model for cosmological simulations of galaxy formation physics: multi-epoch validation (2013, background): “For example, Price & Federrath (2010) studied turbulence with the velocity field tracer implementation of the FLASH code and found that tracer particles tend to clump on sub-resolution scales in turbulence simulations.”
The structure of the interstellar medium of star‐forming galaxies (2011, methodology): “Such turbulence is reasonably well modelled numerically by SPH (based on comparisons to grid-based methods; e.g. Kitsionas et al. 2009; Price & Federrath 2010; Bauer & Springel 2011).”
Theoretical Challenges in Galaxy Formation (2016, background): “…ongoing efforts to better understand the strengths and weaknesses of different numerical schemes (e.g. Hayward et al. 2014; Heitsch, Naab & Walch 2011; Hubber, Falle & Goodwin 2013; Kim et al. 2014; Price & Federrath 2010) and to constantly improve on accuracy and performance of all major codes.”
The density variance–Mach number relation in supersonic turbulence – I. Isothermal, magnetized gas (2012, background): “This period of time is long enough to expect the turbulence to have reached a statistical steady state (Federrath, Klessen & Schmidt 2009; Federrath et al. 2010; Glover et al. 2010; Price & Federrath 2010).”