A polytopal discrete de Rham scheme for the exterior calculus Einstein's equations

TL;DR

This paper develops a discrete exterior calculus scheme for Einstein's equations using a polytopal de Rham complex, enabling geometric discretization and conservation properties in numerical relativity.

Contribution

It introduces a novel polytopal discrete de Rham scheme for the exterior calculus formulation of Einstein's equations, linking geometric discretization with spacetime decomposition.

Findings

Discrete quantities are conserved in the proposed scheme

Two formulations are derived and tested

The scheme is based on the exterior calculus discrete de Rham complex

Abstract

In this work, based on the $3+1$ decomposition in [24, 33], we present a fully exterior calculus breakdown of spacetime and Einstein's equations. Links to the orthonormal frame approach [38] are drawn to help understand the variables in this context. Two formulations are derived, discretised and tested using the exterior calculus discrete de Rham complex [13], and some discrete quantities are shown to be conserved in one of the cases.