Numerical Relativity: A review

TL;DR

Numerical Relativity provides a comprehensive framework for solving Einstein’s equations in strong-field regimes via multiple formulations (3+1, characteristic, and conformal approaches). It explains the core numerical strategies, including gauge choices, initial data construction, and boundary handling, and surveys the transition from vacuum spacetimes to matter-filled scenarios such as relativistic hydrodynamics and scalar field models. The article highlights key accomplishments, ongoing challenges, and synergistic approaches that couple full NR with perturbative methods, outlining a path toward realistic, predictive simulations for gravitational-wave sources and broader GR phenomena. The work underscores NR’s critical role in advancing gravitational-wave astronomy and our understanding of spacetime structure under extreme gravity.

Abstract

Computer simulations are enabling researchers to investigate systems which are extremely difficult to handle analytically. In the particular case of General Relativity, numerical models have proved extremely valuable for investigations of strong field scenarios and been crucial to reveal unexpected phenomena. Considerable efforts are being spent to simulate astrophysically relevant simulations, understand different aspects of the theory and even provide insights in the search for a quantum theory of gravity. In the present article I review the present status of the field of Numerical Relativity, describe the techniques most commonly used and discuss open problems and (some) future prospects.