Relativistic simulations of rotational core collapse -: I.: Methods, initial models, and code tests

We describe an axisymmetric general relativistic code for rotational core collapse. The code evolves the coupled system of metric and fluid equations using the ADM 3+1 formalism and a conformally at metric approximation of the Einstein equations. Within this approximation the ADM 3+1 equations reduce to a set of five coupled non-linear elliptic equations for the metric components. The equations are discretized on a 2D grid in spherical polar coordinates and are solved by means of a Newton-Raphson iteration using a block elimination scheme to solve the diagonally dominant, sparse linear system arising within each iteration step. The relativistic hydrodynamics equations are formulated as a first-order flux-conservative hyperbolic system and are integrated using high-resolution shock-capturing schemes based on Riemann solvers. We assess the quality of the conformally at metric approximation for relativistic core collapse and present a comprehensive set of tests that the code successfully passed. The tests include relativistic shock tubes, the preservation of the rotation profile and of the equilibrium of rapidly and differentially rotating neutron stars (approximated as rotating polytropes), spherical relativistic core collapse, and the conservation of rest-mass and angular momentum in dynamic spacetimes. The application of the code to relativistic rotational core collapse, with emphasis on the gravitational waveform signature, is presented in an accompanying paper.