Brown has recently introduced a covariant formulation of the BSSN equationswhich is well suited for curvilinear coordinate systems. This is particularlydesirable as many astrophysical phenomena are symmetric with respect to therotation axis or are such that curvilinear coordinates adapt better to theirgeometry. However, the singularities associated with such coordinate systemsare known to lead to numerical instabilities unless special care is taken(e.g., regularization at the origin). Cordero-Carrion will present a rigorousderivation of partially implicit Runge-Kutta methods in forthcoming papers,with the aim of treating numerically the stiff source terms in wave-likeequations that may appear as a result of the choice of the coordinate system.We have developed a numerical code solving the BSSN equations in sphericalsymmetry and the general relativistic hydrodynamic equations written influx-conservative form. A key feature of the code is that it uses asecond-order partially implicit Runge-Kutta method to integrate the evolutionequations. We perform and discuss a number of tests to assess the accuracy andexpected convergence of the code, namely a pure gauge wave, the evolution of asingle black hole, the evolution of a spherical relativistic star inequilibrium, and the gravitational collapse of a spherical relativistic starleading to the formation of a black hole. We obtain stable evolutions ofregular spacetimes without the need for any regularization algorithm at theorigin.
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