Numerical Hydrodynamics in General Relativity

Abstract

The current status of numerical solutions for the equations of ideal general relativistic hydrodynamics is reviewed. With respect to an earlier version of the article, the present update provides additional information on numerical schemes, and extends the discussion of astrophysical simulations in general relativistic hydrodynamics. Different formulations of the equations are presented, with special mention of conservative and hyperbolic formulations well-adapted to advanced numerical methods. A large sample of available numerical schemes is discussed, paying particular attention to solution procedures based on schemes exploiting the characteristic structure of the equations through linearized Riemann solvers. A comprehensive summary of astrophysical simulations in strong gravitational fields is presented. These include gravitational collapse, accretion onto black holes, and hydrodynamical evolutions of neutron stars. 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 on the formulation of the gravitational field and hydrodynamic equations and the numerical methodology designed to solve them.

Electronic supplementary material: Supplementary material is available for this article at 10.12942/lrr-2003-4.

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 [54]. 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 [54] and the plot reproduces a similar one from Norman and Winkler [208].

Figure 2

Godunov’s scheme: local solutions of Riemann problems. At every interface, x_j−1/2, x_j+1/2 and x_j+3/2, 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 time step of the numerical scheme must satisfy the Courant-Friedrichs-Lewy condition, being small enough to prevent the waves from advancing more than Δx/2 in Δt.

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

mpg-Movie (2.28 MB)Stills from a movie showing the 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 [244]: 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

mpg-Movie (2.27 MB)Still from a movie showing the animation of the time evolution of the entropy in a core collapse supernova explosion [138]. The movie shows the evolution within the innermost 3000 km of the star and up to 220ms after core bounce. See text for explanation. Visualization by Konstantinos Kifonidis. (For video see appendix)

Figure 6

mpg-Movie (2.44 MB)Still from an animation showing the time evolution of a relativistic core collapse simulation (model A2B4G1 of [67]). Left: velocity field and isocontours of the density. Right: gravitational waveform (top) and central density evolution (bottom). The model exhibits a multiple bounce collapse (fizzler) with a type II signal. The camera follows the multiple bounces. Visualization by Harald Dimmelmeier. (For video see appendix)

Figure 7

Runaway instability of an unstable thick disk: contour levels of the rest-mass density ρ plotted at irregular times from t=0 to t=11.80 t_orb, once the disk has almost been entirely destroyed. See [87] for details.

Figure 8

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 [142] (used with permission).

Figure 9

Relativistic wind accretion onto a rapidly rotating Kerr black hole (a=0.999M, the hole spin is counter-clock wise) in Kerr-Schild coordinates (left panel). Isocontours of the logarithm of the density are plotted at the final stationary time t=500M. 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 12M units long. The simulation employed a (r, φ)-grid of 200×160 zones. Further details are given in [94].

Figure 10

mpg-Movie (2.01 MB)Stills from a movie showing the time evolution of the accretion/collapse of a quadrupolar shell onto a Schwarzshild black hole. The left panel shows isodensity contours and the right panel the associated gravitational waveform. The shell, initially centered at r_*=35M, 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 quasi-normal mode ringing. By the end of the simulation a spherical accretion solution is reached. Further details are given in [220]. (For video see appendix)

Figure 11

mpg-Movie (3.37 MB)Still from a movie showing the animation of a head-on collision simulation of two 1.4M_⊙ neutron stars obtained with a relativistic code [96, 183]. 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 [28] for download options of higher quality versions of this movie. (For video see appendix)

Figure 12

Snapshots of the density contours in the equatorial plane for a binary neutron star coalescence that leads to a rotating black hole (see [267] for the characteristics of the model). Vectors indicate the local velocity field (v^x, v^y). P_t=0 denotes the orbital period of the initial configuration. The thick solid circle in the final panel indicates the apparent horizon. The figure is taken from [267] (used with permission).