  Brown has recently introduced a covariant formulation of the BSSN equations
which is well suited for curvilinear coordinate systems. This is particularly
desirable as many astrophysical phenomena are symmetric with respect to the
rotation axis or are such that curvilinear coordinates adapt better to their
geometry. However, the singularities associated with such coordinate systems
are known to lead to numerical instabilities unless special care is taken
(e.g., regularization at the origin). Cordero-Carrion will present a rigorous
derivation of partially implicit Runge-Kutta methods in forthcoming papers,
with the aim of treating numerically the stiff source terms in wave-like
equations that may appear as a result of the choice of the coordinate system.
We have developed a numerical code solving the BSSN equations in spherical
symmetry and the general relativistic hydrodynamic equations written in
flux-conservative form. A key feature of the code is that it uses a
second-order partially implicit Runge-Kutta method to integrate the evolution
equations. We perform and discuss a number of tests to assess the accuracy and
expected convergence of the code, namely a pure gauge wave, the evolution of a
single black hole, the evolution of a spherical relativistic star in
equilibrium, and the gravitational collapse of a spherical relativistic star
leading to the formation of a black hole. We obtain stable evolutions of
regular spacetimes without the need for any regularization algorithm at the
origin.
