Numerically solving the relativistic Grad-Shafranov equation in Kerr spacetimes: Numerical techniques

TL;DR

This paper develops and evaluates numerical algorithms for solving the relativistic Grad-Shafranov equation in Kerr spacetimes, crucial for understanding force-free magnetospheres around rotating black holes.

Contribution

It generalizes existing numerical methods, provides detailed implementation insights, and assesses their stability and convergence across key astrophysical configurations.

Findings

Algorithms are numerically stable for key setups

Convergence rates are quantified for various configurations

Method improvements enhance solution efficiency and accuracy

Abstract

The study of the electrodynamics of static, axisymmetric and force-free Kerr magnetospheres relies vastly on solutions of the so called relativistic Grad-Shafranov equation (GSE). Different numerical approaches to the solution of the GSE have been introduced in the literature, but none of them has been fully assessed from the numerical point of view in terms of efficiency and quality of the solutions found. We present a generalization of these algorithms and give detailed background on the algorithmic implementation. We assess the numerical stability of the implemented algorithms and quantify the convergence of the presented methodology for the most established setups (split-monopole, paraboloidal, BH-disk, uniform).