Constrained hyperbolic divergence cleaning in smoothed particle magnetohydrodynamics with variable cleaning speeds

TL;DR

This paper introduces an improved divergence cleaning algorithm for smoothed particle magnetohydrodynamics that allows for variable cleaning speeds, ensuring stability and near machine-precision divergence reduction.

Contribution

The authors develop a conservative divergence cleaning method with spatially and temporally variable wave speeds by evolving a normalized quantity, simplifying implementation and preventing energy growth.

Findings

Achieves divergence reduction to machine precision.

Allows each particle to have an evolving cleaning speed.

Maintains conservation and stability with variable wave speeds.

Abstract

We present an updated constrained hyperbolic/parabolic divergence cleaning algorithm for smoothed particle magnetohydrodynamics (SPMHD) that remains conservative with wave cleaning speeds which vary in space and time. This is accomplished by evolving the quantity $\psi / c_h$ instead of $\psi$. Doing so allows each particle to carry an individual wave cleaning speed, $c_h$, that can evolve in time without needing an explicit prescription for how it should evolve, preventing circumstances which we demonstrate could lead to runaway energy growth related to variable wave cleaning speeds. This modification requires only a minor adjustment to the cleaning equations and is trivial to adopt in existing codes. Finally, we demonstrate that our constrained hyperbolic/parabolic divergence cleaning algorithm, run for a large number of iterations, can reduce the divergence of the field to an arbitrarily small value, achieving $\nabla \cdot B=0$ to machine precision.