# Numerical Hydrodynamics and Magnetohydrodynamics

in General Relativity

**José A. Font
**

Departamento de Astronomía y Astrofísica

Edificio de Investigación “Jeroni Muñoz”

Universidad de Valencia

Dr. Moliner 50

E-46100 Burjassot (Valencia), Spain

http://www.uv.es/jofontro/

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Abstract

This article presents a comprehensive overview of numerical hydrodynamics and
magnetohydrodynamics (MHD) in general relativity. Some significant additions have been
incorporated with respect to the previous two versions of this review (2000, 2003), most notably
the coverage of general-relativistic MHD, a field in which remarkable activity and progress has
occurred in the last few years. Correspondingly, the discussion of astrophysical simulations in
general-relativistic hydrodynamics is enlarged to account for recent relevant advances, while
those dealing with general-relativistic MHD are amply covered in this review for the first time.
The basic outline of this article is nevertheless similar to its earlier versions, save for the addition
of MHD-related issues throughout. Hence, different formulations of both the hydrodynamics and
MHD equations are presented, with special mention of conservative and hyperbolic formulations
well adapted to advanced numerical methods. A large sample of numerical approaches for
solving such hyperbolic systems of equations is discussed, paying particular attention to solution
procedures based on schemes exploiting the characteristic structure of the equations through
linearized Riemann solvers. As previously stated, a comprehensive summary of astrophysical
simulations in strong gravitational fields is also presented. These are detailed in three basic
sections, namely gravitational collapse, black-hole accretion, and neutron-star evolutions; despite
the boundaries, these sections may (and in fact do) overlap throughout the discussion. The
material contained in these sections highlights the numerical challenges of various representative
simulations. It also follows, to some extent, the chronological development of the field, concerning
advances in the formulation of the gravitational field, hydrodynamics and MHD equations and
the numerical methodology designed to solve them. To keep the length of this article reasonable,
an effort has been made to focus on multidimensional studies, directing the interested reader to
earlier versions of the review for discussions on one-dimensional works.