Comparing Numerical Methods for Isothermal Magnetized Supersonic
  Turbulence
  We employ simulations of supersonic super-Alfvenic turbulence decay as a
benchmark test problem to assess and compare the performance of nine
astrophysical MHD methods actively used to model star formation. The set of
nine codes includes: ENZO, FLASH, KT-MHD, LL-MHD, PLUTO, PPML, RAMSES, STAGGER,
and ZEUS. We present a comprehensive set of statistical measures designed to
quantify the effects of numerical dissipation in these MHD solvers. We compare
power spectra for basic fields to determine the effective spectral bandwidth of
the methods and rank them based on their relative effective Reynolds numbers.
We also compare numerical dissipation for solenoidal and dilatational velocity
components to check for possible impacts of the numerics on small-scale density
statistics. Finally, we discuss convergence of various characteristics for the
turbulence decay test and impacts of various components of numerical schemes on
the accuracy of solutions. We show that the best performing codes employ a
consistently high order of accuracy for spatial reconstruction of the evolved
fields, transverse gradient interpolation, conservation law update step, and
Lorentz force computation. The best results are achieved with divergence-free
evolution of the magnetic field using the constrained transport method, and
using little to no explicit artificial viscosity. Codes which fall short in one
or more of these areas are still useful, but they must compensate higher
numerical dissipation with higher numerical resolution. This paper is the
largest, most comprehensive MHD code comparison on an application-like test
problem to date. We hope this work will help developers improve their numerical
algorithms while helping users to make informed choices in picking optimal
applications for their specific astrophysical problems.
