Finite Element, Discontinuous Galerkin, and Finite Difference Evolution Schemes in Spacetime

TL;DR

This paper develops and compares finite difference, finite element, and discontinuous Galerkin schemes for solving Einstein's vacuum equations in spacetime, introducing new time-stepping methods and evaluating their performance on test problems.

Contribution

It introduces novel discretization schemes for Einstein's equations in spacetime, including new time-stepping methods and a comparison of different numerical approaches.

Findings

Discontinuous Galerkin methods related to Regge calculus are effective.

New time-stepping schemes improve stability for wave equations.

Schemes perform well on linear and non-linear test problems.

Abstract

Numerical schemes for Einstein's vacuum equation are developed. Einstein's equation in harmonic gauge is second order symmetric hyperbolic. It is discretized in four-dimensional spacetime by Finite Differences, Finite Elements, and Interior Penalty Discontinuous Galerkin methods, the latter related to Regge calculus. The schemes are split into space and time and new time-stepping schemes for wave equations are derived. The methods are evaluated for linear and non-linear test problems of the Apples-with-Apples collection.