Numerical {3+1} general relativistic hydrodynamics: A local characteristic approach

We present a general procedure to solve numerically the three-dimensional general relativistic hydrodynamic system of equations within the framework of the {3 + 1} formalism. The equations are written in conservation form to exploit their hyperbolic character. We derive the theoretical ingredients that are necessary in order to build up a numerical scheme based on the solution of local Riemann problems. Hence the spectral decomposition of the Jacobian matrices of the system, i.e., the eigenvalues and eigenvectors, is explicitly shown. We have taken advantage of the analytic solution of the relativistic Riemann problem, recently derived in Minkowski spacetime, to extend the well-known battery of standard shock tube tests to general spacetimes as an important tool for calibrating any hydrocode. A selection of spherical and nonspherical accretion scenarios is presented and compared with the corresponding analytic or numerical solutions obtained by previous authors.