Journal Article

Smoothed Particle Magnetohydrodynamics – IV. Using the vector potential

Daniel J. Price

in Monthly Notices of the Royal Astronomical Society

Published on behalf of The Royal Astronomical Society

Volume 401, issue 3, pages 1475-1499
Published in print January 2010 | ISSN: 0035-8711
Published online January 2010 | e-ISSN: 1365-2966 | DOI: http://dx.doi.org/10.1111/j.1365-2966.2009.15763.x
Smoothed Particle Magnetohydrodynamics – IV. Using the vector potential

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In this paper, we investigate the use of the vector potential as a means of maintaining the divergence constraint in the numerical solution to 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 first find 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 SPMHD.

Keywords: magnetic fields; MHD; shock waves; methods: numerical; stars: formation

Journal Article.  13951 words.  Illustrated.

Subjects: Astronomy and Astrophysics

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