Smoothed Particle Magnetohydrodynamics IV - Using the Vector Potential

TL;DR

This paper explores the use of the vector potential in Smoothed Particle Magnetohydrodynamics to maintain divergence constraints, deriving a variational formulation that conserves key physical quantities but ultimately finds the approach unstable for practical use.

Contribution

The paper develops a variational formulation of MHD with the vector potential in SPH, revealing stability issues and concluding the approach is not viable.

Findings

The vector potential formulation is unstable to tensile instability in SPH.

Hybrid vector potential approach works in 1D and 2D but not in 3D.

Using the vector potential in SPH is not a practical method due to instability.

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.