Dear Colleagues,

The interaction between physics and mathematics has been very fruitful and necessary at all times. This is still true today. In the area of numerical relativity, the development of accurate and stable numerical methods in order to solve complex partial differential equations allows us to extract the most precise and valuable information from state-of-the-art models, and to compare them afterwards with real data from the different observatories and experiments.

General relativity was proposed in 1915 by Albert Einstein, and it is still a completely valid theory in almost all regimes, with the exception of its consistency at very small scales in which quantum theories are expected to reign. General relativity has been tested in many scenarios and it has survived all the attempts to find experimental results or observations which would lead to its falsification. Although there are analytical solutions of its equations, their complexity in realistic cases requires the use of numerical methods and supercomputers. In particular, the numerical solution of the merger of two black holes in general relativity was a primordial objective first achieved in 2005. Since then, mergers of binary systems have been successfully solved and even effective models of their most significant properties have been derived.

Not only space-time is affected by this relativistic approach, but relativistic fluids should also be taken into account in the modeling of compact objects in non-vacuum space-times, like neutron stars or black hole mimicking exotic objects (e.g., boson stars). These fluids need a matter description through equations of state which account for speed propagations bounded by the speed of light, non-negligible Lorentz factors which strongly couple the evolution of all the physical variables, or the significant presence of magnetic and electric fields described by the (relativistic) Maxwell equations, for example.

Recently, the data registered by the global network of the LIGO-Virgo Gravitational Wave observatories have been revolutionizing our understanding of several astrophysical processes. The improvements in the detectors demand more accurate numerical simulations to properly analyze the already-registered data, as well as new data coming in the next observation runs, in which the KAGRA observatory will join this global network. These observations have already ruled out some alternatives to general relativity, and we expect that they will also provide new and amazing challenges to our modeling in the fields of astrophysics and cosmology.

The development of explicit and implicit numerical methods, the combination of more traditional and novel approaches, and their implementation in highly relativistic regimes are currently hot topics. The aim of this Special Issue is to highlight contributions in the field of numerical relativity, connecting the areas of astrophysics and cosmology with the area of applied mathematics, and sharing new proposals with researchers in both sectors, hopefully giving the chance to establish further collaborations in the future.

It is a great pleasure for me to welcome manuscripts to this Special Issue, and I look forward to reading your contributions in this fascinating field.

Prof. Dr. Isabel Cordero-Carrión
Guest Editor

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• astrophysics and cosmology
• applied mathematics
• numerical relativity
• numerical methods for partial differential equations
• general relativity
• alternative theories of gravity