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.