Improving the convergence properties of the moving-mesh code AREPO
  Accurate numerical solutions of the equations of hydrodynamics play an ever
more important role in many fields of astrophysics. In this work, we
reinvestigate the accuracy of the moving-mesh code \textsc{Arepo} and show how
its convergence order can be improved for general problems. In particular, we
clarify that for certain problems \textsc{Arepo} only reaches first-order
convergence for its original formulation. This can be rectified by simple
modifications we propose to the time integration scheme and the spatial
gradient estimates of the code, both improving the accuracy of the code. We
demonstrate that the new implementation is indeed second-order accurate under
the $L^1$ norm, and in particular substantially improves conservation of
angular momentum. Interestingly, whereas these improvements can significantly
change the results of smooth test problems, we also find that cosmological
simulations of galaxy formation are unaffected, demonstrating that the
numerical errors eliminated by the new formulation do not impact these
simulations. In contrast, simulations of binary stars followed over a large
number of orbital times are strongly affected, as here it is particularly
crucial to avoid a long-term build up of errors in angular momentum
conservation.
