Conservative discretization of the Einstein-Dirac equations in spherically symmetric spacetime

TL;DR

This paper introduces a conservative Galerkin numerical method for solving the Einstein-Dirac equations in spherically symmetric spacetime, ensuring charge conservation and robustness for studying black hole critical phenomena.

Contribution

The paper presents a novel spatial Galerkin discretization that conserves electric charge exactly and adapts the mesh based on physical arclength, improving numerical stability.

Findings

Method demonstrates excellent robustness and convergence.

Suitable for studying critical behavior near black hole thresholds.

Ensures exact conservation of total electric charge.

Abstract

In computational relativity, critical behaviour near the black hole threshold has been studied numerically for several models in the last decade. In this paper we present a spatial Galerkin method, suitable for finding numerical solutions of the Einstein-Dirac equations in spherically symmetric spacetime (in polar/areal coordinates). The method features exact conservation of the total electric charge and allows for a spatial mesh adaption based on physical arclength. Numerical experiments confirm excellent robustness and convergence properties of our approach. Hence, this new algorithm is well suited for studying critical behaviour.