# A novel scheme for simulating the force-free equations: boundary conditions and the evolution of solutions towards stationarity

## TL;DR

This paper introduces a new 3D numerical method for simulating force-free electrodynamics around Kerr black holes, focusing on boundary conditions and the evolution towards stationary solutions in astrophysical contexts.

## Contribution

It presents a novel multi-block finite-difference scheme with covariant hyperbolization and stable boundary conditions for simulating FFE near black holes.

## Key findings

- Successfully reproduces known astrophysical jet solutions.
- Achieves equilibrium solutions with boundaries close to the black hole.
- Demonstrates stability and accuracy of the new numerical approach.

## Abstract

Force-Free Electrodynamics (FFE) describes a particular regime of magnetically dominated relativistic plasmas, which arises on several astrophysical scenarios of interest such as pulsars or active galactic nuclei. In this article, we present a full 3D numerical implementation of the FFE evolution around a Kerr black hole. The novelty of our approach is three-folded: i) We use the "multi-block" technique to represent a domain with $S^2 \times \mathbb{R}^{+}$ topology within a stable finite-differences scheme. ii) We employ as evolution equations those arising from a covariant hyperbolization of the FFE system. iii) We implement stable and constraint-preserving boundary conditions to represent an outer region given by a uniform magnetic field aligned or misaligned respect to the symmetry axis. We find stationary jet solutions which reach equilibrium --through boundary conditions-- with the outer numerical surface. This is so, even when the outer boundary is located very close to the central region (i.e. $r_{out}\sim 10M $). These numerical solutions reproduce most of the known results for analogue astrophysical settings.