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

TL;DR

This paper develops a formalism for applying discontinuous Galerkin methods to general-relativistic hydrodynamics, demonstrating high accuracy and robustness in spherically symmetric spacetime simulations, with potential for advanced astrophysical modeling.

Contribution

It introduces a novel formalism for discontinuous Galerkin methods in relativistic hydrodynamics and implements a one-dimensional code to test their effectiveness in strong gravity scenarios.

Findings

Handles strong relativistic shock waves effectively

Achieves high order accuracy and exponential convergence in smooth regions

Demonstrates the potential of DG methods for relativistic astrophysics simulations

Abstract

We have developed the formalism necessary to employ the discontinuous-Galerkin approach in general-relativistic hydrodynamics. The formalism is firstly presented in a general 4-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 asses 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.