A Tetrad-First Approach to Robust Numerical Algorithms in General Relativity

TL;DR

This paper introduces a novel tetrad-based high-resolution shock-capturing algorithm for general relativity that simplifies complex Riemann solvers by transforming variables into a locally flat spacetime basis, improving stability and convergence.

Contribution

The paper presents a new tetrad-first approach enabling the use of special relativistic Riemann solvers in curved spacetimes, enhancing robustness in general relativistic simulations.

Findings

Demonstrated superior convergence and stability in black hole magnetosphere simulations.

Validated the algorithm's effectiveness in ultra-relativistic black hole accretion problems.

Showcased applicability to realistic astrophysical scenarios with high black hole spin.

Abstract

General relativistic Riemann solvers are typically complex, fragile and unwieldy, at least in comparison to their special relativistic counterparts. In this paper, we present a new high-resolution shock-capturing algorithm on curved spacetimes that employs a local coordinate transformation at each inter-cell boundary, transforming all primitive and conservative variables into a locally flat spacetime coordinate basis (i.e., the tetrad basis), generalizing previous approaches developed for relativistic hydrodynamics. This algorithm enables one to employ a purely special relativistic Riemann solver, combined with an appropriate post-hoc flux correction step, irrespective of the geometry of the underlying Lorentzian manifold. We perform a systematic validation of the algorithm using the Gkeyll simulation framework for both general relativistic electromagnetism and general relativistic hydrodynamics, highlighting the algorithm's superior convergence and stability properties in each case when compared against standard analytical solutions for black hole magnetosphere and ultra-relativistic black hole accretion problems. However, as an illustration of the generality and practicality of the algorithm, we also apply it to more astrophysically realistic magnetosphere and fluid accretion problems in the limit of high black hole spin, for which standard general relativistic Riemann solvers are often too unstable to produce useful solutions.