Smoothed Particle Magnetohydrodynamics – II. Variational principles and variable smoothing-length terms

By Daniel J. Price, Joseph J Monaghan. 2003

156 citations recorded by the source corpus.

In this paper we show how a Lagrangian variational principle can be used to derive the Smoothed Particle Magnetohydrodynamics (SPMHD) equations for ideal Magnetohydrodynamics (MHD). We also consider the effect of a variable smoothing length in the Smoothed Particle Hydrodynamics (SPH) kernels, after which we demonstrate by numerical tests that the consistent treatment of terms relating to the gradient of the smoothing length in the SPMHD equations significantly improves the accuracy of the algorithm. Our results complement those obtained in the companion paper for non-ideal MHD where artificial dissipative terms were included to handle shocks.

Subfields: Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/456f0e0cc88efd9ab9bb1a1b4602dc58de13737b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

Citations in the corpus

The impact of magnetic fields on single and binary star formation (2007, unknown): “…non-ideal MHD by incorporating shock-capturing dissipation terms, the form of which is given, using a simple generalisation of the terms derived in Price & Monaghan (2004b), by „ dα dt « diss = X j mj ᾱBijvsig ρ̄ij (αi − αj)|∇Wij |, (10) „ dβ dt « diss = X j mj ᾱBijvsig ρ̄ij (βi − βj)|∇Wij |,…”
An MHD gadget for cosmological simulations (2008, methodology): “Another possibility to regularize the magnetic field was presented by Price & Monaghan (2004a), who suggested including an artificial dissipation for the magnetic field, analogous to the artificial viscosity used in SPH.”
On the diffusive propagation of warps in thin accretion discs (2010, methodology): “PHANTOM implements the full variable smoothing length SPH formulation developed by Price & Monaghan (2004) and Price & Monaghan (2007), whereby the smoothing length,h, and density,ρ, are mutually dependent via the density sum (for particle a) ρa = ∑ b mbWab(ha), (20) which is an exact solution to…”
Modelling discontinuities and Kelvin-Helmholtz instabilities in SPH (2007, methodology): “This differs from previous formulations (e.g. Monaghan 1997; Price and Monaghan 2004a, 2005) which have assumed that the same signal velocity is used to treat jumps in all variables.”
Cosmic ray feedback in hydrodynamical simulations of galaxy formation (2006, background): “While recent advances in modelling MHD with SPH are promising (Dolag et al., 1999; Price & Monaghan, 2004), these techniques still face severe numerical challenges when applied to simulations with radiative cooling.”
A comparison between grid and particle methods on the statistics of driven, supersonic, isothermal turbulence (2010, methodology): “For hydrodynamics PHANTOM implements the full variable smoothing length SPH formulation developed by Price & Monaghan (2004) and Price & Monaghan (2007), whereby the smoothing length,, and density,ρ, are mutually dependent via the density sum (for particlea) ρa = ∑ b mbWab(ha), (6) which is an…”
An energy‐conserving formalism for adaptive gravitational force softening in smoothed particle hydrodynamics and N‐body codes (2006, methodology): “The method we use for setting the softening length is identical to the method used by Price (2004) (see Price & Monaghan 2004) for setting the smoothing length in SPH calculations.”
Astrophysical smooth particle hydrodynamics (2009, methodology): “The most obvious approach is a straight-forward SPH discretization of the ideal MHD equations (Gingold and Monaghan, 1977; Phillips and Monaghan, 1985; Dolag et al., 2002; Price and Monaghan, 2004a,b, 2005; Price, 2004).”
SPLASH: An Interactive Visualisation Tool for Smoothed Particle Hydrodynamics Simulations (2007, methodology): “Enforcing this relation rigourously (e.g. as in Springel & Hernquist 2002; Price & Monaghan 2004, 2007) thus corresponds to using constant weights (Equation 7) in the interpolation with the value related to the parameter η.”
Simulating cosmic rays in clusters of galaxies – II. A unified scheme for radio haloes and relics with predictions of the γ-ray emission (2007, background): “In principle, cosmological structure formation calculations with SPH are capable of following magneto-hydrodynamics (Dolag et al. 1999, 2005; Price & Monaghan 2004, 2005), although this is presently still fraught with numerical and physical difficulties.”
Phantom: A Smoothed Particle Hydrodynamics and Magnetohydrodynamics Code for Astrophysics (2017, methodology): “However, the error is usually small (see discussion in Bonet & Lok 1999, Section 5 of Price & Monaghan 2004b or Hu & Adams 2006).”
The effect of magnetic fields on star cluster formation (2008, methodology): “…by adopting the energy and entropy-conserving variable smoothing length algorithmsdeveloped by Springel & Hernquist (2002), Monaghan (2002) and Price & Monaghan (2004b) and by the introduction of additional physics in the form of magnetic fields (Price & Bate 2007) and radiative transfer…”
Smoothed Particle Hydrodynamics in Astrophysics (2010, background): “…research on developing improved SPH realizations of MHD has therefore concentrated on constructing formulations thateither eliminate the∇ ·B 6= 0 error, or at least keep it reasonably small (Dolag, Bartelmann & Lesch, 1999; Price & Monaghan, 2004a,b, 2005; Price, 2009; Dolag & Stasyszyn, 2009).”
Smoothed particle hydrodynamics with radiative transfer in the flux-limited diffusion approximation (2004, background): “Problems encountered include numerical tensional instabilities, non-conservation of momentum, and/or not enforcing the divergence of the magnetic field to be zero (see the introduction of Price & Monaghan 2004a).”