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).
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