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Numerical hydrodynamics and magnetohydrodynamics in general relativity. (English) Zbl 1166.83003
Summary: 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.
Update to the author’s paper [Zbl 0944.83007], see also the update [Zbl 1068.83501]: The title has been changed to emphasize that numerical magnetohydrodynamics (MHD) is now also reviewed.
The abstract has been changed accordingly, regarding the MHD aspects of the new version.
The sectioning in Section 2 is unchanged, yet the contents may have seen slight updates wherever necessary.
Section 3 on MHD is entirely new.
Section 4 (which was Section 3 in the previous version) has seen changes throughout (new text and new descriptions to account for recent developments) along with new additions. The Section 3.2.3 in the previous version of the article has been removed and some of its contents have been moved into the ”going further” section (Section 4.2.3 in the updated version of the article). Section 4.3 on the magnetic field divergence-constraint is entirely new. Section 4.4. on state-of-the-art codes has undergone major changes with respect to the corresponding Section 3.3. of the previous version of the article. This section is splitted into two main subsections, each one of them devoted to describing hydrodynamical and MHD codes, respectively. Major updates have occurred in each of these subsections and two tables have been added.
Section 5 on astrophysical applications has been widely updated and enlarged to accommodate the spectacular level of activity witnessed in the field in recent years, both regarding hydrodynamics and MHD. While the basic subsectioning of this section still coincides (broadly speaking) with that of the previous version of the article, major modifications and additions have occurred throughout regarding the contents. In general, the description of one-dimensional studies has been reduced to limit the length of the article. The previous Section 4.1.2 on critical collapse has been removed. New figures and animations have been added, in particular those appearing in Figures 7, 8, 9, 11, 13, 16, and 18 (where the numbering here applies to the revised version of the article).
The reference list has been noticeably enlarged.

MSC:
83-02 Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory
83-03 History of relativity and gravitational theory
83C55 Macroscopic interaction of the gravitational field with matter (hydrodynamics, etc.)
85A30 Hydrodynamic and hydromagnetic problems in astronomy and astrophysics
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