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Black hole-neutron star (BHNS) binary mergers areimportant gravitational wave sources and (possibly) gamma ray burst progenitors. Fully relativisticsimulations have only recently begun to try to capture neutron star physics beyond the polytrope approximation. In this talk we discuss the numerical challenges of replacing polytropes with realistic neutron stars--particularly those relating to sharper neutron star surface features and to the effects of neutrino radiation--and we present the current status of simulations by the (Caltech-Cornell-CITA-WSU) SXS collaboration and how they are dealing with these issues.

The Einstein Toolkit Consortium is developing and supporting open software for relativistic astrophysics. Our aim is to provide the core computational tools that can enable new science, broaden the community, facilitate interdisciplinary research, and take advantage of emerging petascale computers and advanced cyberinfrastructure. The Einstein Toolkit currently consists of an open set of over 100 components for computational relativity along with associated tools for simulation management and visualization. The toolkit includes a vacuum spacetime solver, a relativistic hydrodynamics solver, along with components for initial data, analysis, and computational infrastructure. These components have been developed and improved over many years by many different contributors.

New equations of state (EOSs) from extensive virial and relativistic mean field calculations will be presented. We construct thermodynamically consistent EOSs from slightly noisy free energy calculations, which satisfy the first law, and conserves entropy during adiabatic compression. However this requires a very careful procedure of numerically smoothing the entropy and then integrating the entropy to generate consistent free energies. We discuss various features of the EOS in different density, temperature, and proton fraction regimes. We compare the adiabatic index for our EOSs with those of the Lattimer Swesty and H. Shen et al. EOSs. We do not find a first order liquid vapor phase transition for the astrophysical EOS. Finally, we discuss neutrino interactions that are consistent with our EOSs. At low density, we present model independent neutrino responses that are based on the viral expansion .

Massive accretion disks may form from the merger of neutron star (NS)-NS or black hole-NS binaries, or following the accretion-induced collapse (AIC) of a white dwarf. These disks, termed `hyper-accreting' due to their accretion rates up to several solar masses per second, may power the relativistic jets responsible for short duration gamma-ray bursts. Using 1D time-dependent calculations of hyper-accreting disks, I show that a generic consequence of the disk's late-time evolution is the development of a powerful outflow, powered by viscous heating and the recombination of free nuclei into Helium. These outflows - in additional to any material dynamically-ejected during the merger - synthesize heavy radioactive elements as they expand into space. Nuclear heating from the r-process is not yet incorporated in merger simulations, yet has important consequences both for the dynamics of late `fall-back' accretion and in powering a supernova-like transient (`kilonova') 1 day following the merger or AIC.

Massive accretion disks may form from the merger of neutron star (NS)-NS or black hole-NS binaries, or following the accretion-induced collapse (AIC) of a white dwarf. These disks, termed `hyper-accreting' due to their accretion rates up to several solar masses per second, may power the relativistic jets responsible for short duration gamma-ray bursts. Using 1D time-dependent calculations of hyper-accreting disks, I show that a generic consequence of the disk's late-time evolution is the development of a powerful outflow, powered by viscous heating and the recombination of free nuclei into Helium. These outflows - in additional to any material dynamically-ejected during the merger - synthesize heavy radioactive elements as they expand into space. Nuclear heating from the r-process is not yet incorporated in merger simulations, yet has important consequences both for the dynamics of late `fall-back' accretion and in powering a supernova-like transient (`kilonova') 1 day following the merger or AIC.

The energy generated by the r-process can impact the dynamics of neutron star mergers. Solving a full r-process network coupled with the hydrodynamics becomes the necessary but it is computational very expensive. We have developed a simple model that can be implemented into hydrodynamic simulations and gives a very good estimate of the r-process heating.

A stalled core-collapse supernova shock is unstable to non-spherical perturbations, in what is known as the Standing Accretion Shock Instability (SASI). This instability is global and oscillatory, affecting the region between the protoneutron star surface and the shock. I'll discuss several insights into this instability obtained by combining linear stability analysis and time-dependent simulations, using simple prescriptions for the microphysics that capture the essential physics of the problem.

I would like to overview the progress of the EOS tables for core-collapse supernovae and their influence clarified by these EOS developments. Some topics I try to cover include the neutrino signal from the black hole formation as well as the composition in supernova cores. We would to like hear needs and comments from users of the series of Shen EOS tables for further developments. I would like to report also on recent development of our numerical code of the neutrino-transfer calculation in 3D.