An implicit numerical algorithm for solving the general relativistic   hydrodynamical equations around accreting compact objects
  An implicit algorithm for solving the equations of general relativistic hydrodynamics in conservative form in three-dimensional axi-symmetry is presented. This algorithm is a direct extension of the pseudo-Newtonian implicit radiative magnetohydrodynamical solver -IRMHD- into the general relativistic regime.   We adopt the Boyer-Lindquist coordinates and formulate the hydrodynamical equations in the fixed background of a Kerr black hole. The set of equations are solved implicitly using the hierarchical solution scenario (HSS). The HSS is efficient, robust and enables the use of a variety of solution procedures that range from a purely explicit up to fully implicit schemes. The discretization of the HD-equations is based on the finite volume formulation and the defect-correction iteration strategy for recovering higher order spatial and temporal accuracies. Depending on the astrophysical problem, a variety of relaxation methods can be applied. In particular the vectorized black-white Line-Gauss-Seidel relaxation method is most suitable for modeling accretion flows with shocks, whereas the Approximate Factorization Method is for weakly compressible flows.   The results of several test calculations that verify the accuracy and robustness of the algorithm are shown. Extending the algorithm to enable solving the non-ideal MHD equations in the general relativistic regime is the subject of our ongoing research. 
Ahmad Hujeirat, Max Camenzind and Bernhard W. Keil (ZAH,   Landessternwarte K\"onigstuhl (LSW), Universit\"at Heidelberg)