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Review
, 11 (1), 7

Numerical Hydrodynamics and Magnetohydrodynamics in General Relativity

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Review

Numerical Hydrodynamics and Magnetohydrodynamics in General Relativity

José A Font. Living Rev Relativ.

Abstract

This article presents a comprehensive overview of numerical hydrodynamics and magneto-hydrodynamics (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.

Electronic supplementary material: Supplementary material is available for this article at 10.12942/lrr-2008-7.

Figures

Figure 1
Figure 1
Results for the shock heating test of a cold, relativistically inflowing gas against a wall using the explicit Eulerian techniques of Centrella and Wilson [73]. The plot shows the dependence of the relative errors of the density compression ratio versus the Lorentz factor W for two different values of the adiabatic index of the flow, Γ = 4/3 (triangles) and Γ = 5/3 (circles) gases. The relative error is measured with respect to the average value of the density over a region in the shocked material. The data are from Centrella and Wilson [73] and the plot reproduces a similar one from Norman and Winkler [291].
Figure 2
Figure 2
Godunov’s scheme: local solutions of Riemann problems. At every interface, formula image and formula image, a local Riemann problem is set up as a result of the discretization process (bottom panel), when approximating the numerical solution by piecewise constant data. At time tn these discontinuities decay into three elementary waves, which propagate the solution forward to the next time level tn+1 (top panel). The timestep of the numerical scheme must satisfy the Courant-Friedrichs-Lewy condition, being small enough to prevent the waves from advancing more than Ωx/2 in Ωt.
Figure 3
Figure 3
Schematic profiles of the velocity versus radius at three different times during core collapse: at the point of “last good homology”, at bounce and at the time when the shock wave has just detached from the inner core.
Figure 4
Figure 4
mpg-Movie (9.02 MB) Stills from movies showing Animations of a relativistic adiabatic core collapse using HRSC schemes (snapshots of the radial profiles of various variables are shown at different times). The simulations are taken from [339]: Velocity (top-left), logarithm of the rest-mass density (top-right), gravitational mass (bottom-left), and lapse function squared (bottom-right). See text for details of the initial model. Visualization by José V. Romero. (For video see appendix)
Figure 5
Figure 5
mpg-Movie (3.37 MB) Still from a movie showing The time evolution of the entropy in a core collapse supernova explosion [190]. The movie shows the evolution within the innermost 3000 km of the star and up to 220 ms after core bounce. See text for explanation. Visualization by Konstantinos Kifonidis. (For video see appendix)
Figure 6
Figure 6
mpeg-Movie (11.9 MB) Still from a movie showing The time evolution of a relativistic core collapse simulation (model A2B4G1 of [96]). Left: Velocity field and isocontours of the density. Right: gravitational waveform (top) and central density evolution (bottom). Multiple bounce collapse (fizzler), type II signal. The camera follows the multiple bounces. Visualization by Harald Dimmelmeier. (For video see appendix)
Figure 7
Figure 7
Top panel: time evolution of the normalized four lowest Fourier mode amplitudes for the three-dimensional relativistic core collapse model E20A of [305]. Lower panel: gravitational wave strains h+ and h× along the polar axis. Figure taken from [305] (used with permission).
Figure 8
Figure 8
mpeg-Movie (4.71 MB) Still from a movie showing An equatorial 2D slice from the Ott et al. [305] 3D simulation of their collapse model E20A that has a post-core-bounce β = T/|W| of ∼ 9%. The snapshot displayed corresponds to t = 71.02 ms after core bounce. Shown is a colormap encoding specific entropy per baryon s in units of the Boltzmann constant kB. The colormap is cut at s = 3 kB/baryon to emphasize the dynamics in the unshocked (hence low-entropy) inner proto-neutron star regions. Visualization by Christian Ott. (For video see appendix)
Figure 9
Figure 9
mpeg-Movie (88.4 MB) Still from a movie showing Gravitational radiation from the three-dimensional collapse of a neutron star to a rotating black hole [30]. The figure shows a snapshot in the evolution of the system once the black hole has been formed and the gravitational waves are being emitted in large amounts. See [30] for further details. Visualization made in AEI/ZIB. (For video see appendix)
Figure 10
Figure 10
Runaway instability of an unstable thick disk: contour levels of the rest-mass density ρ plotted at various times from t = 0 to t = 11.80 torb, once the disk has almost entirely been destroyed. See [129] for details.
Figure 11
Figure 11
Development of the MRI in a constant-angular-momentum magnetized torus around a Kerr black hole with a/M = 0.5 at t = 0 (left) and at t = 2000 M (right). The color is mapped from the logarithm of the density field; black is low and dark red is high. The resolution is 3002. The figure is taken from [149] (used with permission).
Figure 12
Figure 12
Jet formation: the twisting of magnetic field lines around a Kerr black hole (black sphere). The yellow surface is the ergosphere. The red tubes show the magnetic field lines that cross into the ergosphere. Figure taken from [197] (used with permission).
Figure 13
Figure 13
Ultrarelativistic outflow from the remnant of the merger of a neutron-star-binary system. The figure shows the gamma-ray burst model B01 from [8] corresponding to a time t = 50 ms after the start of the energy deposition, at a rate of 2 × 1051 erg/s. The (r, θ) grid resolution in this axisymmetric simulation is 1000 × 200. The outflow in the white-colored regions in the left panel (closer to the black hole), magnified in the right panel, achieves Lorentz factors of ∼ 100, with terminal values of about 1000. See [8] for further details. (Figure used with permission.)
Figure 14
Figure 14
Relativistic wind accretion onto a rapidly-rotating Kerr black hole (a = 0.999 M, the black-hole spin is counterclockwise) in Kerr-Schild coordinates (left panel). Isocontours of the logarithm of the density are plotted at the final stationary time t = 500 M. Brighter colors (yellow-white) indicate high density regions, while darker colors (blue) correspond to low density zones. The right panel shows how the flow solution looks when transformed to Boyer-Lindquist coordinates. The shock appears here totally wrapped around the horizon of the black hole. The box is 12 M units long. The simulation employed a (r, ϕ)-grid of 200 × 160 zones. Further details are given in [135].
Figure 15
Figure 15
mpeg-Movie (2.03 MB) Still from a movie showing The time evolution of the accretion/collapse of a quadrupolar shell onto a Schwarzschild black hole. The left panel shows isodensity contours and the right panel the associated gravitational waveform. The shell, initially centered at r* = 35 M, is gradually accreted by the black hole, a process that perturbs the black hole and triggers the emission of gravitational radiation. After the burst, the remaining evolution shows the decay of the black-hole quasinormal-mode ringing. By the end of the simulation a spherical accretion solution is reached. Further details are given in [310]. (For video see appendix)
Figure 16
Figure 16
mpeg-Movie (8.78 MB) Still from a movie showing The time evolution of the bar-mode instability of one of the differentially rotating neutron-star models of [29]. The animation depicts the distribution of the rest-mass density. The exponential growth of the instability, which is followed by its saturation, the development of spiral arms, and the attenuation of the bar deformation, is all clearly visible in the movie. The last part of the dynamics is dominated by an almost axisymmetric configuration. Visualization developed at SISSA. Used with permission. (For video see appendix)
Figure 17
Figure 17
mpeg-Movie (558 KB) Still from a movie showing An animation of a headon collision simulation of two 1.4 M neutron stars obtained with a relativistic code [137, 261]. The movie shows the evolution of the density and internal energy. The formation of the black hole in prompt timescales is demonstrated by the sudden appearance of the apparent horizon at t = 0.16 ms (t = 63.194 in code units), which is indicated by violet dotted circles representing the trapped photons. See [245] for download options of higher-quality versions of this movie. (For video see appendix)
Figure 18
Figure 18
Isodensity contours in the equatorial plane for the merger of a 1.3 M − 1.3 M neutron-star binary (left panel) and a 1.35 M − 1.35 M case. Vectors indicate the local velocity field (vx, vy). The image corresponds to the final snapshot of the evolution for the two models. The outcome of the merger for the more massive case is the delayed formation of a black hole, as signalled by the collapse of the lapse function shown in the lower panel of the image. In the less massive case, a long-lived hypermassive neutron star supported by rotation is formed. Simulations performed by [372]. (Used with permission.)

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