# Hyperbolicity of Divergence Cleaning and Vector Potential Formulations of GRMHD

## TL;DR

This paper analyzes the hyperbolicity of divergence cleaning and vector potential formulations in general relativistic magnetohydrodynamics, demonstrating their strong hyperbolicity and suitability for well-posed initial value problems in numerical relativity.

## Contribution

It shows that divergence cleaning and vector potential formulations are strongly hyperbolic, making them preferable for numerical simulations in GRMHD.

## Key findings

- Divergence cleaning formulation is strongly hyperbolic.
- Vector potential formulations can be made strongly hyperbolic.
- Preferred formulations for numerical relativity are divergence cleaning and vector potential.

## Abstract

We examine hyperbolicity of general relativistic magnetohydrodynamics with divergence cleaning, a flux-balance law form of the model not covered by our earlier analysis. The calculations rely again on a dual-frame approach, which allows us to effectively exploit the structure present in the principal part. We find, in contrast to the standard flux-balance law form of the equations, that this formulation is strongly hyperbolic, and thus admits a well-posed initial value problem. Formulations involving the vector potential as an evolved quantity are then considered. Carefully reducing to first order, we find that such formulations can also be made strongly hyperbolic. Despite the unwieldy form of the characteristic variables we therefore conclude that of the free-evolution formulations of general relativistic magnetohydrodynamics presently used in numerical relativity, the divergence cleaning and vector potential formulations are preferred.