Barreto’s group in Venezuela applied characteristic methods to study the self-similar collapse of spherical matter and charge distributions [26, 30, 27]. The assumption of self-similarity reduces the problem to a system of ODE’s, subject to boundary conditions determined by matching to an exterior Reissner–Nordström–Vaidya solution. Heat flow in the internal fluid is balanced at the surface by the Vaidya radiation. Their simulations illustrate how a nonzero total charge can halt gravitational collapse and produce a final stable equilibrium . It is interesting that the pressure vanishes in the final equilibrium state so that hydrostatic support is completely supplied by Coulomb repulsion. In subsequent work [24, 25], they applied their characteristic code to the evolution of a polytropic fluid sphere coupled to a scalar radiation field to study the central equation of state, conservation of the Newman–Penrose constant, the scattering of the scalar radiation off the polytrope and its late time decay. The work illustrates how characteristic evolution can be used to simulate radiation from a matter source in the simple context of spherical symmetry.
Font and Papadopoulos  have given a state-of-the-art treatment of relativistic fluids, which is applicable to either spacelike or null foliations. Their approach is based upon a high-resolution shock-capturing (HRSC) version of relativistic hydrodynamics in flux conservative form, which was developed by the Valencia group (for a review see ). In the HRSC scheme, the hydrodynamic equations are written in flux conservative, hyperbolic form. In each computational cell, the system of equations is diagonalized to determine the characteristic fields and velocities, and the local Riemann problem is solved to obtain a solution consistent with physical discontinuities. This allows a finite-differencing scheme along the characteristics of the fluid that preserves the conserved physical quantities and leads to a stable and accurate treatment of shocks. Because the general relativistic system of hydrodynamical equations is formulated in covariant form, it can equally well be applied to spacelike or null foliations of the spacetime. The null formulation gave remarkable performance in the standard Riemann shock tube test carried out in a Minkowski background. The code was successfully implemented first in the case of spherical symmetry, using a version of the Bondi–Sachs formalism adapted to describe gravity coupled to matter with a worldtube boundary . They verified second-order convergence in curved space tests based upon Tolman–Oppenheimer–Volkoff equilibrium solutions for spherical fluids. In the dynamic self-gravitating case, simulations of spherical accretion of a fluid onto a black hole were stable and free of numerical problems. Accretion was successfully carried out in the regime where the mass of the black hole doubled. Subsequently the code was used to study how accretion modulates both the decay rates and oscillation frequencies of the quasi-normal modes of the interior black hole .
The characteristic hydrodynamic approach of Font and Papadopoulos was first applied to spherically-symmetric problems of astrophysical interest. Linke, Font, Janka, Müller, and Papadopoulos  simulated the spherical collapse of supermassive stars, using an equation of state that included the effects due to radiation, electron-positron pair formation, and neutrino emission. They were able to follow the collapse from the onset of instability to black-hole formation. The simulations showed that collapse of a star with mass greater than solar masses does not produce enough radiation to account for the gamma ray bursts observed at cosmological redshifts.
Next, Siebel, Font, and Papadopoulos  studied the interaction of a massless scalar field with a neutron star by means of the coupled Klein–Gordon–Einstein-hydrodynamic equations. They analyzed the nonlinear scattering of a compact ingoing scalar pulse incident on a spherical neutron star in an initial equilibrium state obeying the null version of the Tolman–Oppenheimer–Volkoff equations. Depending upon the initial mass and radius of the star, the scalar field either excites radial pulsation modes or triggers collapse to a black hole. The transfer of scalar energy to the star was found to increase with the compactness of the star. The approach included a compactification of null infinity, where the scalar radiation was computed. The scalar waveform showed quasi-normal oscillations before settling down to a late time power law decay in good agreement with the dependence predicted by linear theory. Global energy balance between the star’s relativistic mass and the scalar energy radiated to infinity was confirmed.
Living Rev. Relativity 15, (2012), 2
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