A Comparison between Divergence‐Cleaning and Staggered‐Mesh Formulations for Numerical Magnetohydrodynamics

In recent years, several different strategies have emerged for evolving themagnetic field in numerical MHD. Some of these methods can be classified asdivergence-cleaning schemes, where one evolves the magnetic field componentsjust like any other variable in a higher order Godunov scheme. The fact thatthe magnetic field is divergence-free is imposed post-facto via adivergence-cleaning step. Other schemes for evolving the magnetic field rely ona staggered mesh formulation which is inherently divergence-free. The claim hasbeen made that the two approaches are equivalent. In this paper wecross-compare three divergence-cleaning schemes based on scalar and vectordivergence-cleaning and a popular divergence-free scheme. All schemes areapplied to the same stringent test problem. Several deficiencies in all thedivergence-cleaning schemes become clearly apparent with the scalardivergence-cleaning schemes performing worse than the vectordivergence-cleaning scheme. The vector divergence-cleaning scheme also showssome deficiencies relative to the staggered mesh divergence-free scheme. Thedifferences can be explained by realizing that all the divergence-cleaningschemes are based on a Poisson solver which introduces a non-locality into thescheme, though other subtler points of difference are also catalogued. By usingseveral diagnostics that are routinely used in the study of turbulence, it isshown that the differences in the schemes produce measurable differences inphysical quantities that are of interest in such studies