On the measurements of numerical viscosity and resistivity in Eulerian   MHD codes
  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. 
Tomasz Rembiasz, Martin Obergaulinger, Pablo Cerd\'a-Dur\'an,   Miguel-\'Angel Aloy, and Ewald M\"uller
astro-ph.IM astro-ph.SR