Formulation of discontinuous Galerkin methods for relativistic astrophysics

TL;DR

This paper develops and compares formulations of discontinuous Galerkin methods for solving PDEs in curved spacetimes relevant to relativistic astrophysics, including non-conservative equations like Einstein's field equations.

Contribution

It introduces new formulations of DG methods for curved spacetimes and non-conservative equations, including a simplified derivation of the ALE algorithm.

Findings

Two computationally distinct DG formulations are identified.

A new, simpler derivation of the ALE algorithm is provided.

Numerical experiments demonstrate the formulations' effectiveness and the impact of metric identities.

Abstract

The DG algorithm is a powerful method for solving pdes, especially for evolution equations in conservation form. Since the algorithm involves integration over volume elements, it is not immediately obvious that it will generalize easily to arbitrary time-dependent curved spacetimes. We show how to formulate the algorithm in such spacetimes for applications in relativistic astrophysics. We also show how to formulate the algorithm for equations in non-conservative form, such as Einstein's field equations themselves. We find two computationally distinct formulations in both cases, one of which has seldom been used before for flat space in curvilinear coordinates but which may be more efficient. We also give a new derivation of the ALE algorithm (Arbitrary Lagrangian-Eulerian) using 4-vector methods that is much simpler than the usual derivation and explains why the method preserves the conservation form of the equations. The various formulations are explored with some simple numerical experiments that also explore the effect of the metric identities on the results.