Comparative Analysis of Hall Effect Implementations in Hall Magnetohydrodynamics

Abstract There is no standard numerical implementation of the Hall effect, which is one of the nonideal magnetohydrodynamic (MHD) effects. Numerical instability arises when a simple implementation is used, in which the Hall electric field is added to the electric field to update magnetic fields without further modifications to the numerical scheme. In this paper, several implementations proposed in the literature are compared to identify an approach that provides stable and accurate results. We consider two types of implementations of the Hall effect. One is a modified version of the Harten–Lax–van Leer method (H all -HLL), in which the phase speeds of whistler waves are adopted as the signal speeds; the other involves adding a fourth-order hyper-resistivity to a Hall-MHD code. Based on an extensive series of test calculations, we find that hyper-resistivity yields more accurate results than H all -HLL, particularly in problems where the whistler wave timescale is shorter than the timescales of the physical processes of interest. Through both von Neumann stability analysis and numerical experiments, an appropriate coefficient for the hyper-resistivity is determined.