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Supernova models with a full spectral treatment of the neutrino transport arepresented, employing the Prometheus/Vertex neutrino-hydrodynamics code with a``ray-by-ray plus'' approximation for treating two- (or three-) dimensionalproblems. The method is described in detail and critically assessed withrespect to its capabilities, limitations, and inaccuracies in the context ofsupernova simulations. In this first paper of a series, 1D and 2D core-collapsecalculations for a (nonrotating) 15 M_sun star are discussed, uncertainties inthe treatment of the equation of state -- numerical and physical -- are tested,Newtonian results are compared with simulations using a general relativisticpotential, bremsstrahlung and interactions of neutrinos of different flavorsare investigated, and the standard approximation in neutrino-nucleoninteractions with zero energy transfer is replaced by rates that includecorrections due to nucleon recoil, thermal motions, weak magnetism, and nucleoncorrelations. Models with the full implementation of the ``ray-by-ray plus''spectral transport were found not to explode, neither in spherical symmetry norin 2D with a 90 degree lateral wedge. The success of previous 2D simulationswith grey, flux-limited neutrino diffusion can therefore not be confirmed.Omitting the radial velocity terms in the neutrino momentum equation leads to``artificial'' explosions by increasing the neutrino energy density in theconvective gain layer by about 20--30% and thus the integral neutrino energydeposition in this region by about a factor of two. (abbreviated)Comment: 46 pages plus 13 pages online material; 49 figures; referee's  comments included, version accepted by Astronomy & Astrophysic

Two-dimensional hydrodynamic core-collapse supernova simulations with spectral neutrino transport