  A discontinuous Galerkin (DG) method suitable for large-scale astrophysical
simulations on Cartesian meshes as well as arbitrary static and moving Voronoi
meshes is presented. Most major astrophysical fluid dynamics codes use a finite
volume (FV) approach. We demonstrate that the DG technique offers distinct
advantages over FV formulations on both static and moving meshes. The DG method
is also easily generalized to higher than second-order accuracy without
requiring the use of extended stencils to estimate derivatives (thereby making
the scheme highly parallelizable). We implement the technique in the AREPO code
for solving the fluid and the magnetohydrodynamic (MHD) equations. By examining
various test problems, we show that our new formulation provides improved
accuracy over FV approaches of the same order, and reduces post-shock
oscillations and artificial diffusion of angular momentum. In addition, the DG
method makes it possible to represent magnetic fields in a locally
divergence-free way, improving the stability of MHD simulations and moderating
global divergence errors, and is a viable alternative for solving the MHD
equations on meshes where Constrained-Transport (CT) cannot be applied. We find
that the DG procedure on a moving mesh is more sensitive to the choice of slope
limiter than is its FV method counterpart. Therefore, future work to improve
the performance of the DG scheme even further will likely involve the design of
optimal slope limiters. As presently constructed, our technique offers the
potential of improved accuracy in astrophysical simulations using the moving
mesh AREPO code as well as those employing adaptive mesh refinement (AMR).
