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In order to find a way to have a better formulation for numerical evolution of the Einstein equations, we study the propagation equations of the constraints based on the Arnowitt-Deser-Misner formulation. By adjusting constraint terms in…

After a brief introduction to the black hole stability problem, we outline our recent proof of the linear stability of the non-extreme Kerr geometry.

In this paper, we propose a new numerical strategy for the stabilization of evolution systems. The method is based on the methodology given by Ammari, Nicaise andPignotti in ''Stabilization by switching time-delay, Asymptot. Anal., 83…

We study the stability properties of the standard ADM formulation of the 3+1 evolution equations of general relativity through linear perturbations of flat spacetime. We focus attention on modes with zero speed of propagation and conjecture…

We report new results which establish that the accurate 3-dimensional numerical simulation of generic single-black-hole spacetimes has been achieved by characteristic evolution with unlimited long term stability. Our results cover a…

Recent studies indicate that the configurational entropy is an useful tool to investigate the stability and (or) the relative dominance of states for diverse physical systems. Recent examples comprise the connection between the variation of…

Solving the 4-d Einstein equations as evolution in time requires solving equations of two types: the four elliptic initial data (constraint) equations, followed by the six second order evolution equations. Analytically the constraint…

We investigate the numerical stability of accelerated AdS black holes against linear scalar perturbations. In particular, we study the evolution of a probe non-minimally coupled scalar field on Schwarzschild and Reissner-Nordstr\"om AdS…

Can we conclude the stability of an unknown dynamical system from the knowledge of a finite number of snapshots of trajectories? We tackle this black-box problem for switched linear systems. We show that, for any given random set of…

We report on the potential occurrence of a numerical instability in the long-time simulation of black holes using the Baumgarte-Shapiro-Shibata-Nakamura formulation of numerical relativity, even in the simple set-up of a Schwarzschild black…

We have developed a numerical code to study the evolution of distorted, rotating black holes. This code is used to evolve a new family of black hole initial data sets corresponding to distorted ``Kerr'' holes with a wide range of rotation…

We present numerical evolutions of three equal-mass black holes using the moving puncture approach. We calculate puncture initial data for three black holes solving the constraint equations by means of a high-order multigrid elliptic…

We extend the notion of numerical stability of finite difference approximations to include hyperbolic systems that are first order in time and second order in space, such as those that appear in Numerical Relativity. By analyzing the symbol…

In recent years, many different numerical evolution schemes for Einstein's equations have been proposed to address stability and accuracy problems that have plagued the numerical relativity community for decades. Some of these approaches…

Algorithmic stability is a concept from learning theory that expresses the degree to which changes to the input data (e.g., removal of a single data point) may affect the outputs of a regression algorithm. Knowing an algorithm's stability…

We analyze the stability of generic spherically symmetric thin shells to linearized perturbations around static solutions. We include the momentum flux term in the conservation identity, deduced from the ''ADM'' constraint and the Lanczos…

We study quantum aspects of the accelerated black holes in some detail. Explicitly shown is the fact that a uniform acceleration stabilizes certain charged black holes against the well-known thermal evaporation. Furthermore, a close…

In this paper, we study existence and stability of static black holes in Lovelock theories with a particular focus on pure Lovelock black holes. We derive the equation of stability from action without using S-deformation approach. It turns…

In this work, we systematically investigate linear multi-step methods for differential equations with memory. In particular, we focus on the numerical stability for multi-step methods. According to this investigation, we give some…

We will address the problem of determining the existence and asymptotic stability of a non-trivial periodic orbit in dynamical systems described by polynomial vector fields. To this end, we will lean upon the celebrated results of Borg,…