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 generalrelativistic hydrodynamics. The formalism is first presented in a general four-dimensional setting and then specialized to the case of spherical symmetry within a 3 þ 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.

A new spherical harmonics scheme for multi-dimensional radiation transport I. Static matter configurations

“…Filtering is a common procedure to reduce the effects of the Gibbs phenomenon in spectral methods for numerical solution of partial differential equations [14,43]. Filtered spherical harmonics expansions have been successfully used in, e.g., meteorology (see e.g.…”

Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement

“…Discontinuous Galerkin (DG) schemes Cockburn et al , 1990Cockburn & Shu 1998), on the contrary, do not need any spatial reconstruction and they allow for an arbitrary order of accuracy. DG schemes are still relatively unknown in high energy astrophysics, and only a few investigations have been performed so far in the relativistic regime (Zumbusch 2009;Radice & Rezzolla 2011;Zanotti & Dumbser 2011). Unfortunately, DG schemes suffer from a serious problem, which has negatively affected their popularity.…”

“…The procedures that have been adopted to overcome this difficulty can be roughly divided in two classes. From one side, it is possible to introduce additional numerical dissipation, either in the form of artificial viscosity (R. Hartmann & P.Houston 2002;Persson & Peraire 2006;Cesenek et al 2013), or by means of filtering (Radice & Rezzolla 2011). From another side, it is possible to isolate the so-called troubled cells, namely those affected by spurious oscillations, and adopt for them some sort of nonlinear finite-volume-type slope-limiting procedure (Cockburn & Shu 1998;J.Qiu & C-W.Shu 2004;Balsara et al 2007; J.…”

General relativistic monopole magnetosphere of neutron stars: a pseudo-spectral discontinuous Galerkin approach

“…Such scheme are also useful to solve Maxwell equations. Recently, another arbitrary high order method, the discontinuous Galerkin approach, has been tested in general relativity by Radice & Rezzolla (2011).…”