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

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.

24 pages, 19 figures. Small changes; matches version to appear in PRD

References in corpus (16)

• ECHO: an Eulerian Conservative High Order scheme for general relativistic magnetohydrodynamics and magnetodynamics
• Recoil velocities from equal-mass binary black-hole mergers: a systematic investigation of spin-orbit aligned configurations
• Accurate evolutions of inspiralling and magnetized neutron-stars: equal-mass binaries
• WhiskyMHD: a new numerical code for general relativistic magnetohydrodynamics
• Evolving black hole-neutron star binaries in general relativity using pseudospectral and finite difference methods
• Improved constrained scheme for the Einstein equations: An approach to the uniqueness issue
• Challenging the paradigm of singularity excision in gravitational collapse
• On the gravitational radiation from the collapse of neutron stars to rotating black holes
• Accurate evolutions of inspiralling neutron-star binaries: assessment of the truncation error
• Finite Element, Discontinuous Galerkin, and Finite Difference Evolution Schemes in Spacetime
• The trumpet solution from spherical gravitational collapse with puncture gauges
• Discontinuous Galerkin method for the spherically reduced BSSN system with second-order operators
• Nonlinear radial oscillations of neutron stars
• Critical Phenomena in Neutron Stars I: Linearly Unstable Nonrotating Models
• Type II critical phenomena of neutron star collapse
• A Numerical Study of Relativistic Fluid Collapse