On the measurements of numerical viscosity and resistivity in Eulerian MHD codes

TL;DR

This paper introduces a simple method to estimate numerical viscosity and resistivity in Eulerian MHD codes, validated through simulations of wave propagation and tearing modes, and discusses how numerical schemes affect these parameters.

Contribution

It presents a novel ansatz for measuring numerical dissipation in Eulerian MHD codes and analyzes its dependence on simulation methods and resolution.

Findings

Numerical viscosity and resistivity depend on the numerical scheme used.

Ultra-high order methods significantly reduce numerical dissipation.

Proper resolution is crucial for accurately capturing magnetic field amplification.

Abstract

We propose a simple ansatz for estimating the value of the numerical resistivity and the numerical viscosity of any Eulerian MHD code. We test this ansatz with the help of simulations of the propagation of (magneto)sonic waves, Alfven waves, and the tearing mode instability using the MHD code Aenus. By comparing the simu- lation results with analytical solutions of the resistive-viscous MHD equations and an empirical ansatz for the growth rate of tearing modes we measure the numerical viscosity and resistivity of Aenus. The comparison shows that the fast-magnetosonic speed and wavelength are the characteristic velocity and length, respectively, of the aforementioned (relatively simple) systems. We also determine the dependance of the numerical viscosity and resistivity on the time integration method, the spatial reconstruction scheme and (to a lesser extent) the Riemann solver employed in the simulations. From the measured results we infer the numerical resolution (as a function of the spatial reconstruction method) required to properly resolve the growth and saturation level of the magnetic field amplified by the magnetorotational instability in the post-collapsed core of massive stars. Our results show that it is to the best advantage to resort to ultra-high order methods (e.g., 9th-order Monotonicity Preserving method) to tackle this problem properly, in particular in three dimensional simulations.