Simulating Radiation Transport in Curved Spacetimes

We are developing methods for simulation of radiation transport in systems governed by strong gravity (e.g., neutrino transport in core-collapse supernovae). By employing conservative formulations of the general relativistic Boltzmann equation\footnote{Cardall, C.Y., Endeve, E., \& Mezzacappa 2013, Phys. Rev. D {\bf 88}, 023011}, we aim to develop methods that are (i) high-order accurate for computational efficiency; (ii) robust in the sense that the phase space density $f$ preserves the maximum principle of the physical model ($f\in[0,1]$ for fermions); and (iii) applicable to curvilinear coordinate systems to accommodate curved spacetimes, which result in gravity-induced frequency shift and angular aberration. Our approach is based on the Runge-Kutta discontinuous Galerkin method\footnote{Cockburn \& Shu 2001, J. Sci. Comput. {\bf 16}, 173-261}, which has many attractive properties, including high-order accuracy on a compact stencil. We present the physical model, describe our numerical methods, and show results from implementations in spherical and axial symmetry. Our tests show that the method is high-order accurate and strictly preserves the maximum principle on $f$. We also demonstrate the ability of our method to accurately include effects of a strong gravitational field.