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We present a new general relativistic (GR) code for hydrodynamic supernovasimulations with neutrino transport in spherical and azimuthal symmetry(1D/2D). The code is a combination of the CoCoNuT hydro module, which is aRiemann-solver based, high-resolution shock-capturing method, and thethree-flavor, energy-dependent neutrino transport scheme VERTEX. VERTEXintegrates the neutrino moment equations with a variable Eddington factorclosure computed from a model Boltzmann equation and uses the ray-by-ray plusapproximation in 2D, assuming the neutrino distribution to be axially symmetricaround the radial direction, and thus the neutrino flux to be radial. Ourspacetime treatment employs the ADM 3+1 formalism with the conformal flatnesscondition for the spatial three-metric. This approach is exact in 1D and hasbeen shown to yield very accurate results also for rotational stellar collapse.We introduce new formulations of the energy equation to improve total energyconservation in relativistic and Newtonian hydro simulations with Eulerianfinite-volume codes. Moreover, a modified version of the VERTEX scheme isdeveloped that simultaneously conserves energy and lepton number with betteraccuracy and higher numerical stability. To verify our code, we conduct aseries of tests, including a detailed comparison with published 1D results forstellar core collapse. Long-time simulations of proto-neutron star cooling overseveral seconds both demonstrate the robustness of the new CoCoNuT-VERTEX codeand show the approximate treatment of GR effects by means of an effectivegravitational potential as in PROMETHEUS-VERTEX to be remarkably accurate in1D. (abridged)

A new multi-dimensional general relativistic neutrino hydrodynamics code for core-collapse supernovae. I. Method and code tests in spherical symmetry