An energy‐conserving formalism for adaptive gravitational force softening in smoothed particle hydrodynamics and N‐body codes

By Daniel J. Price, Joseph J Monaghan. 2006

270 citations recorded by the source corpus.

In this paper, we describe an adaptive softening length formalism for collisionless N-body and self-gravitating smoothed particle hydrodynamics (SPH) calculations which conserves momentum and energy exactly. This means that spatially variable softening lengths can be used in N-body calculations without secular increases in energy. The formalism requires the calculation of a small additional term to the gravitational force related to the gradient of the softening length. The extra term is similar in form to the usual SPH pressure force (although opposite in direction) and is therefore straightforward to implement in any SPH code at almost no extra cost. For N-body codes, some additional cost is involved as the formalism requires the computation of the density through a summation over neighbouring particles using the smoothing kernel. The results of numerical tests demonstrate that, for homogeneous mass distributions, the use of adaptive softening lengths gives a softening which is always close to the ‘optimal’ choice of fixed softening parameter, removing the need for fine-tuning. For a heterogeneous mass distribution (as may be found in any large-scale N-body simulation), we find that the errors on the least-dense component are lowered by an order of magnitude compared to the use of a fixed softening length tuned to the densest component. For SPH codes, our method presents a natural and an elegant choice of softening formalism which makes a small improvement to both the force resolution and the total energy conservation at almost zero additional cost.

Subfields: Astronomy and Astrophysics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/bd12f06d19fee4b4652411b99514b751fb60d338

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Citations in the corpus

A new class of accurate, mesh-free hydrodynamic simulation methods (2014, background): “Fortunately, Price & Monaghan (2007) show how the appropriate terms can be rigorously derived from the particle Lagrangian to maintain manifest conservation with variable softening lengths.”
Simulations of the grand design galaxy M51: a case study for analysing tidally induced spiral structure (2009, background): “All particles have variable smoothing lengths, the smoothing length and density solved iteratively according to h = ν ( m ρ )1/3 (1) (Price & Monaghan 2007).”
NEUTRINO SIGNATURES AND THE NEUTRINO-DRIVEN WIND IN BINARY NEUTRON STAR MERGERS (2008, methodology): “• A consistent implementation of adaptive gravitational softening lengths as described in Price & Monaghan (2007).”
Impacts of a flaring star-forming disc and stellar radial mixing on the vertical metallicity gradient (2016, methodology): “We apply the spline softening suggested by Price & Monaghan (2007).”
E pur si muove: Galilean-invariant cosmological hydrodynamical simulations on a moving mesh (2009, methodology): “It has to be included to make the system properly conservative when spatially adaptive gravitational softening lengths are used (Price & Monaghan, 2007) that are allowed to vary in time, which is our default approach to treat self-gravity in the moving-mesh scheme.”
Phantom: A Smoothed Particle Hydrodynamics and Magnetohydrodynamics Code for Astrophysics (2017, methodology): “This makes it straightforward to obtain the otherwise laborious expressions needed for each kernel (expressions for the cubic spline kernel are in Appendix A of Price & Monaghan 2007 and for the quintic spline in Appendix A of Hubber et al. 2011).”
Neutral hydrogen in galaxy haloes at the peak of the cosmic star formation history (2014, methodology): “The gravity solver in GIZMO is a modified version of the GADGET-3 gravity algorithm (Springel 2005) which implements the adaptive softening method of Price & Monaghan (2007) and a modified softening kernel that represents the exact solution for the potential of the SPH smoothing kernel following…”
On the diversity and statistical properties of protostellar discs (2018, methodology): “…(SPH) code, sphNG, based on the original version of Benz (1990; Benz et al. 1990), but substantially modified using the methods described in Bate et al. (1995), Price & Monaghan (2007), Whitehouse, Bate & Monaghan (2005), Whitehouse & Bate (2006) and parallelised using both OpenMP and MPI.”
SPLASH: An Interactive Visualisation Tool for Smoothed Particle Hydrodynamics Simulations (2007, methodology): “…is a spatially variable quantity set in such a way as either to fix (either exactly or approximately) the number of nearest neighbours (Hernquist & Katz 1989; Benz et al. 1990), or via an analytic relation to the (number) density (Springel & Hernquist 2002; Monaghan 2002; Price & Monaghan 2007).”
The Local Group on FIRE: dwarf galaxy populations across a suite of hydrodynamic simulations (2018, methodology): “Gravity is solved with an improved version of the TREEPM solver from GADGET-3 (Springel 2005), with fully adaptive (and fully conservative) gravitational force softenings for gas (so hydrodynamic and force softenings are always self-consistently matched), following Price & Monaghan (2007).”
Clumps in the outer disk by disk instability: Why they are initially gas giants and the legacy of disruption (2009, methodology): “howed that xed softening could lead either to either enhanced or suppressed clumping. Although there are ways of improving energy conservation that would permit the use of variable softening lengths (Price and Monaghan, 2007), this has yet to be implemented in GASOLINE and represents a future code d”
Remnant evolution after a carbon–oxygen white dwarf merger (2007, background): “Note that this could, in principle, be improved even further by taking into account the so-called ‘gradh’ terms (Monaghan 2002; Springel & Hernquist 2002; Price 2004) and extra terms arising from adapting gravitational smoothing terms (Price & Monaghan 2007).”
Numerical simulations of the dark universe: State of the art and the next decade (2012, background): “vates improvements in existing treatments, such as the introduction of adaptive gravitational softening [417, 418] or improved density estimators based on following the distortion of a tetrahedral tessellation of the initial density field [419, 420], as well as radical departures”
Galaxies on FIRE (Feedback In Realistic Environments): stellar feedback explains cosmologically inefficient star formation (2013, methodology): “…to allow for adaptive gravitational softenings35 and to more accurately symmetrize the force between interacting particles with different softening, following the fully Lagrangian method in Price & Monaghan (2007); this manifestly maintains conservation of energy, momentum, and angular momentum.”
FIRE in the field: Simulating the threshold of galaxy formation (2016, methodology): “…baryonic (dark matter) force resolution of hb = 2 pc ( DM = 35 pc); force softening for baryons uses the fully-conservative adaptive algorithm from Price & Monaghan (2007), meaning that the gravitational force assumes the identical mass distribution as the hydrodynamic equations (resulting in…”
The impact of baryonic physics on the structure of dark matter haloes: the view from the FIRE cosmological simulations (2015, methodology): “The gravity solver of the GIZMO code is an updated version of the PM+Tree c© 0000 RAS, MNRAS 000, 000–000 algorithm from Gadget-3 (Springel 2005) and uses fully conservative adaptive gravitational softening for gas (Price & Monaghan 2007).”
Smoothed Particle Hydrodynamics in Astrophysics (2010, methodology): “This term has to be included to make the system properly conservative when spatially adaptive gravitational softening lengths are used (Price & Monaghan, 2007).”