The ADM Einstein equations are reformulated into a conformal traceless system, an idea originally introduced by Shibata and Nakamura  (see also ), and further developed by Baumgarte and Shapiro . This ``BSSN'' formulation, which shows enhanced long-term stability compared to the original ADM system, makes use of a conformal decomposition of the 3-metric, and the trace-free part of the extrinsic curvature, , with the conformal factor chosen to satisfy . In this formulation, as shown by , in addition to the evolution equations for the conformal 3-metric and the conformal-traceless extrinsic curvature variables , there are evolution equations for the conformal factor , the trace of the extrinsic curvature K and the ``conformal connection functions'' . Further details can be found in [25, 258].
The code uses an approximate maximal slicing condition, which amounts to solving a parabolic equation for , and a minimal distortion gauge condition for the shift vector. It also admits -rotation symmetry around the z -axis, as well as plane symmetry with respect to the z =0 plane, allowing computations in a quadrant region. In addition, Shibata has also implemented in the code the ``cartoon'' method proposed by the AEI Numerical Relativity group , which makes possible axisymmetric computations with a Cartesian grid. ``Approximate'' outgoing boundary conditions are used at the outer boundaries; these do not completely eliminate numerical errors due to spurious back reflection of gravitational waves . A staggered leapfrog method is used for the time evolution of the metric quantities. Correspondingly, the hydrodynamic equations are updated using a second-order two-step Runge-Kutta scheme. In each time step, the staggered metric quantities needed for the hydrodynamics update are properly extrapolated to intermediate time levels to reach the desired order of accuracy.
The code developed by Shibata [258, 260] has been tested in a variety of problems, including spherical collapse of dust to a black hole, signalled by the appearance of the apparent horizon (comparing 1D and 3D simulations), stability of spherical stars and computation of the radial oscillation period, quadrupole oscillations of perturbed spherical stars and computation of the associated gravitational radiation, preservation of the rotational profile of (approximate) rapidly rotating stars, and the preservation of a co-rotating binary neutron star in a quasi-equilibrium state (assuming a conformally flat 3-metric) for more than one orbital period. Improvements of the hydrodynamical schemes have been considered very recently , in order to tackle problems in which shocks play an important role, e.g., stellar core collapse to a neutron star. Shibata's code has allowed important breakthroughs in the study of the morphology and dynamics of various general relativistic astrophysical problems, such as black hole formation resulting from both the gravitational collapse of rotating neutron stars and rotating supermassive stars, and, perhaps most importantly, the coalescence of binary neutron stars, a long-standing problem in numerical relativistic hydrodynamics. These applications are discussed in Section 4 . The most recent simulations of binary neutron star coalescence  have been performed on a FACOM VPP5000 computer with typical grid sizes of (505, 505, 253) in (x, y, z).
The hydrodynamics part of the code uses the conservative formulation discussed in Section 2.1.3 . A variety of state-of-the-art Riemann solvers are implemented, including a Roe-like solver  and Marquina's flux formula . The code uses slope-limiter methods to construct second-order TVD schemes by means of monotonic piecewise linear reconstructions of the cell-centered quantities for the computation of the numerical fluxes. The standard ``minmod'' limiter and the ``monotonized central-difference'' limiter  are implemented. Both schemes provide the desired second-order accuracy for smooth solutions, while still satisfying the TVD property. In addition, third-order piecewise parabolic (PPM) reconstruction has been recently implemented and used in .
The Einstein equations are solved using the following different approaches: (i) the standard ADM formalism, (ii) a hyperbolic formulation developed by , and (iii) the BSSN formulation of [197, 264, 25]. The time evolution of both the ADM and the BSSN systems can be performed using several numerical schemes [96, 4, 89]. Currently, a leapfrog (non-staggered in time), and an iterative Crank-Nicholson scheme have been coupled to the hydrodynamics solver. The code is designed to handle arbitrary shift and lapse conditions, which can be chosen as appropriate for a given spacetime simulation. The AEI Numerical Relativity group  is currently developing robust gauge conditions for (vacuum) black hole spacetimes (see, e.g.,  and references therein). Hence, all advances accomplished here can be incorporated in future versions of the code for non-vacuum spacetime evolutions. Similarly, since it is a general purpose code, a number of different boundary conditions can be imposed for either the spacetime variables or for the hydrodynamical variables. We refer the reader to [96, 4, 89] for additional details.
The code has been subjected to a series of convergence tests , with many different combinations of the spacetime and hydrodynamics finite differencing schemes, demonstrating the consistency of the discrete equations with the differential equations. The simulations performed in  include, among others, the evolution of equilibrium configurations of compact stars (solutions to the TOV equations) and the evolution of relativistically boosted TOV stars (v =0.87 c) transversing diagonally across the computational domain - a test for which an exact solution exists. In [4, 5] the improved stability properties of the BSSN formulation of the Einstein equations were compared to the ADM system. In particular, in  a number of strongly gravitating systems were simulated, including weak and strong gravitational waves, black holes, boson stars and relativistic stars. While the error growth-rate can be decreased by going to higher grid resolutions, the BSSN formulation requires grid resolutions higher than the ones needed in the ADM formulation to achieve the same accuracy. Furthermore, it was shown in  that the code successfully passed stringent long-term evolution tests, such as the evolution of both spherical and rapidly rotating, stationary stellar configurations, and the formation of an apparent horizon during the collapse of a relativistic star to a black hole. The high accuracy of the hydrodynamical schemes employed has allowed the detailed investigation of the frequencies of radial, quasi-radial and quadrupolar oscillations of relativistic stellar models, and use them as a strong assessment of the accuracy of the code. The frequencies obtained have been compared to the frequencies computed with perturbative methods and in axisymmetric nonlinear evolutions . In all of the cases considered, the code reproduces these results with excellent accuracy and is able to extract the gravitational waveforms that might be produced during non-radial stellar pulsations.
|Numerical Hydrodynamics in General Relativity
José A. Font
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