A comparison between divergence-cleaning and staggered-mesh formulations for numerical magnetohydrodynamics

In recent years, several different strategies have emerged for evolving the magnetic field in numerical MHD. Some of these methods can be classified as divergence-cleaning schemes, in which one evolves the magnetic field components just like any other variable in a higher order Godunov scheme. The fact that the magnetic field is divergence-free is imposed post facto via a divergence-cleaning step. Other schemes for evolving the magnetic field rely on a staggered-mesh formulation that is inherently divergence-free. The claim has been made that the two approaches are equivalent. In this paper we compare three divergence-cleaning schemes based on scalar and vector divergence-cleaning and a popular divergence-free scheme. All schemes are applied to the same stringent test problem. Several deficiencies in all the divergence-cleaning schemes become clearly apparent, with the scalar divergence-cleaning schemes performing worse than the vector divergence-cleaning scheme. The vector divergence-cleaning scheme also shows some deficiencies relative to the staggered-mesh divergence-free scheme. The differences can be explained by realizing that all the divergence-cleaning schemes are based on a Poisson solver that introduces a nonlocality into the scheme, although other, subtler points of difference are also cataloged. By using several diagnostics that are routinely used in the study of turbulence, it is shown that the differences in the schemes produce measurable differences in physical quantities that are of interest in such studies.