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Abstract Motivated by their role as the direct or indirect source of many of the elements in the Universe, numerical modeling of core collapse supernovae began more than five decades ago. Progress toward ascertaining the explosion mechanism(s) has been realized through increasingly sophisticated models, as physics and dimensionality have been added, as physics and numerical modeling have improved, and as the leading computational resources available to modelers have become far more capable. The past five to ten years have witnessed the emergence of a consensus across the core collapse supernova modeling community that had not existed in the four decades prior. For the majority of progenitors – i.e., slowly rotating progenitors – the efficacy of the delayed shock mechanism, where the stalled supernova shock wave is revived by neutrino heating by neutrinos emanating from the proto-neutron star, has been demonstrated by all core collapse supernova modeling groups, across progenitor mass and metallicity. With this momentum, and now with a far deeper understanding of the dynamics of these events, the path forward is clear. While much progress has been made, much work remains to be done, but at this time we have every reason to be optimistic we are on track to answer one of the most important outstanding questions in astrophysics: How do massive stars end their lives? 

Abstract We compare the core-collapse evolution of a pair of 15.8M_☉stars with significantly different internal structures, a consequence of the bimodal variability exhibited by massive stars during their late evolutionary stages. The 15.78 and 15.79M_☉progenitors have core masses (masses interior to an entropy of 4k_Bbaryon⁻¹) of 1.47 and 1.78M_☉and compactness parametersξ_1.75of 0.302 and 0.604, respectively. The core-collapse simulations are carried out in 2D to nearly 3 s postbounce and show substantial differences in the times of shock revival and explosion energies. The 15.78M_☉model begins exploding promptly at 120 ms postbounce when a strong density decrement at the Si–Si/O shell interface, not present in the 15.79M_☉progenitor, encounters the stalled shock. The 15.79M_☉model takes 100 ms longer to explode but ultimately produces a more powerful explosion. Both the larger mass accretion rate and the more massive core of the 15.79M_☉model during the first 0.8 s postbounce time result in largerν_e/ ${\overline{\nu }}_e$luminosities and RMS energies along with a flatter and higher-density heating region. The more-energetic explosion of the 15.79M_☉model resulted in the ejection of twice as much⁵⁶Ni. Most of the ejecta in both models are moderately proton rich, though counterintuitively the highest electron fraction (Y_e= 0.61) ejecta in either model are in the less-energetic 15.78M_☉model, while the lowest electron fraction (Y_e= 0.45) ejecta in either model are in the 15.79M_☉model. 

Abstract The proposal that core collapse supernovae are neutrino driven is still the subject of active investigation more than 50 years after the seminal paper by Colgate and White. The modern version of this paradigm, which we owe to Wilson, proposes that the supernova shock wave is powered by neutrino heating, mediated by the absorption of electron-flavor neutrinos and antineutrinos emanating from the proto-neutron star surface, or neutrinosphere. Neutrino weak interactions with the stellar core fluid, the theory of which is still evolving, are flavor and energy dependent. The associated neutrino mean free paths extend over many orders of magnitude and are never always small relative to the stellar core radius. Thus, neutrinos are never always fluid like. Instead, a kinetic description of them in terms of distribution functions that determine the number density of neutrinos in the six-dimensional phase space of position, direction, and energy, for both neutrinos and antineutrinos of each flavor, or in terms of angular moments of these neutrino distributions that instead provide neutrino number densities in the four-dimensional phase-space subspace of position and energy, is needed. In turn, the computational challenge is twofold: (i) to map the kinetic equations governing the evolution of these distributions or moments onto discrete representations that are stable, accurate, and, perhaps most important, respect physical laws such as conservation of lepton number and energy and the Fermi–Dirac nature of neutrinos and (ii) to develop efficient, supercomputer-architecture-aware solution methods for the resultant nonlinear algebraic equations. In this review, we present the current state of the art in attempts to meet this challenge.