An Implicit Finite Volume Scheme to Solve the Time-dependent Radiation Transport Equation Based on Discrete Ordinates

We describe a new algorithm to solve the time dependent, frequency integrated radiation transport (RT) equation implicitly, which is coupled to an explicit solver for equations of magnetohydrodynamics (MHD) using {\sf Athena++}. The radiation filed is represented by specific intensities along discrete rays, which are evolved using a conservative finite volume approach for both cartesian and curvilinear coordinate systems. All the terms for spatial transport of photons and interactions between gas and radiation are calculated implicitly together. An efficient Jacobi-like iteration scheme is used to solve the implicit equations. This removes any time step constrain due to the speed of light in RT. We evolve the specific intensities in the lab frame to simplify the transport step. The lab-frame specific intensities are transformed to the co-moving frame via Lorentz transformation when the source term is calculated. Therefore, the scheme does not need any expansion in terms of $v/c$. The radiation energy and momentum source terms for the gas are calculated via direct quadrature in the angular space. The time step for the whole scheme is determined by the normal Courant -- Friedrichs -- Lewy condition in the MHD module. We provide a variety of test problems for this algorithm including both optically thick and thin regimes, and for both gas and radiation pressure dominated flows to demonstrate its accuracy and efficiency.