# The numerical initial boundary value problem for the generalized conformal field equations

## TL;DR

This paper develops a numerical method for solving the initial boundary value problem of generalized conformal field equations, enabling long-term simulations of black hole spacetimes while maintaining stability and constraints.

## Contribution

It introduces a well-structured formulation using space-spinor formalism and boundary conditions that preserve constraints, suitable for long-time evolution of black hole perturbations.

## Key findings

- Successfully implemented a numerically well-posed system
- Demonstrated stability with Minkowski perturbations
- Extended applicability to Schwarzschild black hole perturbations

## Abstract

In this paper we study a numerical implementation for the initial boundary value formulation for the generalized conformal field equations. We propose a formulation which is well suited for the study of the long-time behaviour of perturbed exact solutions such as a Schwarzschild or even a Kerr black hole. We describe the derivation of the implemented equations which we give in terms of the space-spinor formalism. We discuss the conformal Gauss gauge, and a slight generalization thereof which seems to be particularly useful in the presence of boundaries. We discuss the structure of the equations at the boundary and propose a method for imposing boundary conditions which allow the correct number of degrees of freedom to be freely specified while still preserving the constraints. We show that this implementation yields a numerically well-posed system by testing it on a simple case of gravitational perturbations of Minkowski space-time and subsequently with gravitational perturbations of Schwarzschild space-time.