An Integral-based Approach for the Vector Potential in Smoothed Particle Magnetohydrodynamics

TL;DR

This paper introduces an integral-based method for evolving the magnetic vector potential in smoothed particle magnetohydrodynamics, but finds it to be numerically unstable and non-viable.

Contribution

It proposes a new integral discretisation for the vector potential in SPMHD and evaluates its stability and accuracy.

Findings

The new formulation is numerically unstable.

It does not solve the MHD equations in the continuum limit.

The approach is deemed non-viable due to instability.

Abstract

A new implementation for the time evolution of the magnetic vector potential is obtained for smoothed particle magnetohydrodynamics by considering the induction equation in integral form. Galilean invariance is achieved through proper gauge choice. This new discretisation is tested using the Orszag-Tang MHD vortex in a 3D configuration. The corresponding conservative equations of motion are derived, but are not found to solve the MHD equations in the continuum limit. Tests are performed using a hybrid approach instead, whereby the equations of motion based on the magnetic field instead of vector potential are used. Test results experience the same numerical instability as with the Price (2010) formulation. We conclude that this new formulation is non-viable.