Cosmic explosions involving black holes and neutron stars generate some of the universe's most extreme conditions, where matter in strong gravity can exceed nuclear densities, or where neutrinos can mediate the production of heavy elements. Over the past decade, advances in gravitational wave observatories, transient surveys, neutrino detectors, and nuclear experiments have greatly improved our ability to probe these events. These advances also demand robust theoretical support to take full advantage of the data. Given the complexity of these phenomena, however, realistic observational predictions require intensive computational modeling through global simulations that include nuclear and neutrino physics as inputs. I will present an overview of how this computational approach deepens our understanding of these events, as illustrated by three examples: heavy element production in neutron star mergers, black hole formation in supernovae, and mass ejection in novel thermonuclear transients.