Smoothed Particle Magnetohydrodynamics IV - Using the Vector Potential

Abstract

In this paper we investigate the use of the vector potential as a means of maintaining the divergence constraint in the numerical solution of the equations of Magnetohydrodynamics (MHD) using the Smoothed Particle Hydrodynamics (SPH) method. We derive a self-consistent formulation of the equations of motion using a variational principle that is constrained by the numerical formulation of both the induction equation and the curl operator used to obtain the magnetic field, which guarantees exact and simultaneous conservation of momentum, energy and entropy in the numerical scheme. This leads to a novel formulation of the MHD force term, unique to the vector potential, which differs from previous formulations. We also demonstrate how dissipative terms can be correctly formulated for the vector potential such that the contribution to the entropy is positive definite and the total energy is conserved. On a standard suite of numerical tests in one, two and three dimensions we find firstly that the consistent formulation of the vector potential equations is unstable to the well-known SPH tensile instability, even more so than in the standard Smoothed Particle Magnetohydrodynamics (SPMHD) formulation where the magnetic field is evolved directly. Furthermore we find that, whilst a hybrid approach based on the vector potential evolution equation coupled with a standard force term gives good results for one and two dimensional problems (where dAz/dt = 0), such an approach suffers from numerical instability in three dimensions related to the unconstrained evolution of vector potential components. We conclude that use of the vector potential is not a viable approach for Smoothed Particle Magnetohydrodynamics.

accepted to MNRAS. 25 pages of pain, 8 figures. Don't try this at home. Some movies showing how bad it really is at http://users.monash.edu.au/~dprice/pubs/spmhd/index4.html

References in corpus (9)

• SPLASH: An interactive visualisation tool for Smoothed Particle Hydrodynamics simulations
• Producing ultra-strong magnetic fields in neutron star mergers
• Modelling discontinuities and Kelvin-Helmholtz instabilities in SPH
• The impact of magnetic fields on single and binary star formation
• The effect of magnetic fields on star cluster formation
• MAGMA: a 3D, Lagrangian magnetohydrodynamics code for merger applications
• Magnetic field structure due to the global velocity field in spiral galaxies
• Magnetic fields and the dynamics of spiral galaxies
• On some applications of Galilean electrodynamics of moving bodies

Cited by in corpus (34)

• Smoothed Particle Hydrodynamics and Magnetohydrodynamics
• Phantom: A smoothed particle hydrodynamics and magnetohydrodynamics code for astrophysics
• Modeling Collapse and Accretion in Turbulent Gas Clouds: Implementation and Comparison of Sink Particles in AMR and SPH
• Magnetohydrodynamics on an unstructured moving grid
• The origin of accreted stellar halo populations in the Milky Way using APOGEE, $\text{Gaia}$, and the EAGLE simulations
• Accurate, Meshless Methods for Magneto-Hydrodynamics
• The quenching and morphological evolution of central galaxies is facilitated by the feedback-driven expulsion of circumgalactic gas
• Can non-ideal magnetohydrodynamics solve the magnetic braking catastrophe?
• A comparison between grid and particle methods on the statistics of driven, supersonic, isothermal turbulence
• Constrained Hyperbolic Divergence Cleaning for Smoothed Particle Magnetohydrodynamics
• The origin of diverse $\alpha$-element abundances in galaxy discs
• Astrophysical Weighted Particle Magnetohydrodynamics
• The gas fractions of dark matter haloes hosting simulated $\sim L^{\star }$ galaxies are governed by the feedback history of their black holes
• There is no magnetic braking catastrophe: Low-mass star cluster and protostellar disc formation with non-ideal magnetohydrodynamics
• Constrained hyperbolic divergence cleaning in smoothed particle magnetohydrodynamics with variable cleaning speeds
• Dusty gas with one fluid in smoothed particle hydrodynamics
• Ambipolar diffusion in smoothed particle magnetohydrodynamics
• A switch to reduce resistivity in smoothed particle magnetohydrodynamics
• A quantification of the non-spherical geometry and accretion of collapsing cores
• Protostellar collapse and fragmentation using an MHD GADGET
• Divergence-free Interpolation of Vector Fields From Point Values - Exact divB=0 in Numerical Simulations
• Quenching and morphological evolution due to circumgalactic gas expulsion in a simulated galaxy with a controlled assembly history
• Magnetic helicity transport in the advective gauge family
• Magnetic field evolution and reversals in spiral galaxies
• A Vector Potential implementation for Smoothed Particle Magnetohydrodynamics
• The Kelvin-Helmholtz instability and smoothed particle hydrodynamics
• Phurbas: An Adaptive, Lagrangian, Meshless, Magnetohydrodynamics Code. I. Algorithm
• Stable anisotropic heat conduction in smoothed particle hydrodynamics
• Application and Benchmark of SPH for Modeling the Impact in Thermal Spraying
• Smoothed particle magnetohydrodynamics with the geometric density average force expression
• Implementing and comparing sink particles in AMR and SPH
• Meshless Methods for Magnetohydrodynamics with Vector Potential
• Diagnosing Turbulence in the Neutral and Molecular Interstellar Medium of Galaxies
• Magnetic fields and Turbulence in Star Formation using Smoothed Particle Hydrodynamics