Thornado-mini: Algorithm

The ExaStar project has been working to develop advanced transport techniques as the treatment of radiation transport is typically the most computational expensive component of current stellar explosion simulations. This is due to the need for high spatial resolution and unconstrained spatial dimensionality to follow important fluid instabilities as the explosion develops, as well as a sensitivity of neutrino heating rates to the neutrino energy distribution, which requires retention of the energy dimension of momentum space. To balance computational cost with the required physical fidelity, the number of unknowns can be reduced by truncating the moment expansion of the angular dimensions of momentum space; i.e., by solving for a finite number of angular moments.

Thornado-mini was developed in year one of the ExaStar project to solve the transport equation in the two-moment formalism. The two-moment model is obtained by truncating the moment expansion at the level of the so-called first order moments (akin to an expansion in spherical harmonics up to degree 1), so that the unknown moments are the spectral particle density, and particle flux—a two-moment model. To solve the neutrino transport equations the moment equations are discretized in energy-position space using high-order Discontinuous Galerkin (DG) methods. DG methods combine elements from both spectral and finite volume methods, and are considered a desirable option for solving hyperbolic partial differential equations (PDEs). Without modification, the DG methods recover the correct asymptotic behavior in the diffusion limit, characterized by frequent collisions.

Thornado-mini’s implementation of the radiation transport kernel was ported into both the FLASH and Castro source codes in the second year of the ExaStar project. The use of thornado-mini as a prototype or design exploration for the Clash toolkit demonstrates another increasingly common use case for proxy apps.

Stars more massive than about 8 solar masses end their lives as core-collapse supernovae. They are triggered from the collapse of the stellar core due to electron captures and the photodissociation of heavy nuclei. The collapse continues until normal nuclear matter density is reached. The nucleon pressure from the short-range repulsive nuclear interaction halts the collapse and the supersonically collapsing core bounces back. A sound wave turns into a shock wave which then propagates out of the core. The object which forms at core bounce is the protoneutron star (PNS), being hot and lepton rich in which sense it differs from the final supernova remnant, the neutron star. Deleptonization is the loss of leptons from the PNS.

Thornado-mini implements a simulation of deleptonization. The simulation is constructed by adopting analytic profiles for initial mass density, temperature, and electron fraction. The initial mass density is set such that the maximum density is consistent with nuclear matter, and neutrinos will be trapped. As the density decreases with radius, the neutrino mean free path increases, and the neutrinos created by electron capture will be able to escape the computational domain (deleptonization). The values set for the initial mass density, temperature and electron fraction profiles can be found here.
The radiation field is initialized by setting the distribution function equal to the local Fermi-Dirac distribution. The computational domain covers a radius of 0 to 100~km and energy of 0 to 300 MeV, and evolves until 100 ms.