  Modern astrophysical simulations aim to accurately model an ever-growing
array of physical processes, including the interaction of fluids with magnetic
fields, under increasingly stringent performance and scalability requirements
driven by present-day trends in computing architectures. Discontinuous Galerkin
methods have recently gained some traction in astrophysics, because of their
arbitrarily high order and controllable numerical diffusion, combined with
attractive characteristics for high performance computing. In this paper, we
describe and test our implementation of a discontinuous Galerkin (DG) scheme
for ideal magnetohydrodynamics in the AREPO-DG code. Our DG-MHD scheme relies
on a modal expansion of the solution on Legendre polynomials inside the cells
of an Eulerian octree-based AMR grid. The divergence-free constraint of the
magnetic field is enforced using one out of two distinct cell-centred schemes:
either a Powell-type scheme based on nonconservative source terms, or a
hyperbolic divergence cleaning method. The Powell scheme relies on a basis of
locally divergence-free vector polynomials inside each cell to represent the
magnetic field. Limiting prescriptions are implemented to ensure
non-oscillatory and positive solutions. We show that the resulting scheme is
accurate and robust: it can achieve high-order and low numerical diffusion, as
well as accurately capture strong MHD shocks. In addition, we show that our
scheme exhibits a number of attractive properties for astrophysical
simulations, such as lower advection errors and better Galilean invariance at
reduced resolution, together with more accurate capturing of barely resolved
flow features. We discuss the prospects of our implementation, and DG methods
in general, for scalable astrophysical simulations.
