Three-dimensional numerical general relativistic hydrodynamics: Formulations, methods, and code tests

This is the first in a series of papers on the construction and validation of a three-dimensional code for general relativistic hydrodynamics, and its application to general relativistic astrophysics. This paper studies the consistency and convergence of our general relativistic hydrodynamic treatment and its coupling to the spacetime evolutions described by the full set of Einstein equations with a perfect fluid source? complimenting a similar study of the (vacuum) spacetime part of the code. The numerical treatment of the general relativistic hydrodynamic equations is based on high resolution shock capturing schemes, specifically designed to solve non-linear hyperbolic systems of conservation laws. These schemes rely on the characteristic information of the system. A spectral decomposition for general relativistic hydrodynamics suitable far a general spacetime metric is presented. Evolutions based on different approximate Riemann solvers (flux-splitting, Roe, and Marquina) are studied and compared. The coupling between the hydrodynamics and the spacetime (the right and left hand side of the Einstein equations) is carried out in a treatment which is second order accurate in both space and time. The spacetime evolution allows for a choice of different formulations of the Einstein equations, and different numerical methods for each formulation. Together with the different hydrodynamical methods, there are twelve different combinations of spacetime and hydrodynamical evolutions. Convergence tests for all twelve combinations with a variety of test beds are studied, showing consistency with the differential equations and correct convergence properties. The test-beds examined include shack tubes, Friedmann-Robertson-Walker cosmology tests, evolutions of self-gravitating compact (TOV) stars, and evolutions of relativistically boosted TOV stars. Special attention is paid to the numerical evolution of strongly gravitating objects, e.g., neutron stars, in the full theory of general relativity, including a simple, yet effective treatment for the surface region of the star (where the rest mass density is abruptly dropping to;zero).