Discontinuous Galerkin methods for general-relativistic hydrodynamics: Formulation and application to spherically symmetric spacetimes

We have developed the formalism necessary to employ the discontinuous-Galerkin approach in general-relativistic hydrodynamics. The formalism is first presented in a general four-dimensional setting and then specialized to the case of spherical symmetry within a 3+1 splitting of spacetime. As a direct application, we have constructed a one-dimensional code, EDGES, which has been used to assess the viability of these methods via a series of tests involving highly relativistic flows in strong gravity. Our results show that discontinuous-Galerkin methods are able not only to handle strong relativistic shock waves but, at the same time, to attain very high orders of accuracy and exponential convergence rates in smooth regions of the flow. Given these promising prospects and their affinity with a pseudospectral solution of the Einstein equations, discontinuous-Galerkin methods could represent a new paradigm for the accurate numerical modelling in relativistic astrophysics.