A discontinuous Galerkin method for solving the fluid and MHD equations in astrophysical simulations

摘要

A discontinuous Galerkin (DG) method suitable for large-scale astrophysicalsimulations on Cartesian meshes as well as arbitrary static and moving Voronoimeshes is presented. Most major astrophysical fluid dynamics codes use a finitevolume (FV) approach. We demonstrate that the DG technique offers distinctadvantages over FV formulations on both static and moving meshes. The DG methodis also easily generalized to higher than second-order accuracy withoutrequiring the use of extended stencils to estimate derivatives (thereby makingthe scheme highly parallelizable). We implement the technique in the AREPO codefor solving the fluid and the magnetohydrodynamic (MHD) equations. By examiningvarious test problems, we show that our new formulation provides improvedaccuracy over FV approaches of the same order, and reduces post-shockoscillations and artificial diffusion of angular momentum. In addition, the DGmethod makes it possible to represent magnetic fields in a locallydivergence-free way, improving the stability of MHD simulations and moderatingglobal divergence errors, and is a viable alternative for solving the MHDequations on meshes where Constrained-Transport (CT) cannot be applied. We findthat the DG procedure on a moving mesh is more sensitive to the choice of slopelimiter than is its FV method counterpart. Therefore, future work to improvethe performance of the DG scheme even further will likely involve the design ofoptimal slope limiters. As presently constructed, our technique offers thepotential of improved accuracy in astrophysical simulations using the movingmesh AREPO code as well as those employing adaptive mesh refinement (AMR).