We discuss the numerical methods implemented in our HYDrodynamical RAdiation (HYDRA) code for rapidly expanding, line dominated, low-density envelopes commonly found in core collapse and thermonuclear supernovae. The code allows to calculate flux and polarization spectra including their evolution with time. Due to the low densities, non-thermal excitation by high
energy photons (e.g. from radioactive decays) and the time dependence of the problem, significant departures from LTE are common throughout the envelope even at large optical depths.
In principle, this physical system can be described by a large system of coupled integro-differential equations but a full solution of the general problem is computationally prohibitive, in particular, because past RT-methods of comoving are inadequate. The large velocity fields and the non-LTE problem result in a strong coupling of spatial, frequency and phase space (level populations). As a consequence, spatial and frequency discretization (and errors) are coupled.
For the numerical solution, we use variable separation, analytic solutions and approximations, and iterative schemes. The need for Adaptive Mesh Refinement (AMR) is demonstrated. For time-dependent and 3-D problems, we solve the radiation transport (RT)
via the moment equations. To construct the Eddington tensor elements, we use a
Monte Carlo scheme to determine the deviation (!) of the solution from isotropy in the radiation field (ALI of second kind) to avoid problems commonly inherent to MC at large optical depths. Line opacities are treated in the 'narrow line' limit to, formally, 'decouple' spatial frequency space for the RT. The method of Accelerated Lambda Iteration (ALI) is instrumental for both the coupling of the statistical equations and the hydrodynamical equations with the radiation transport.
We employ several concepts to improve the stability, and convergence
rate/ control including the {\sl concept of leading elements}, the use of net rates, level locking, reconstruction of global photon redistribution functions, equivalent-2-level approach, and predictive corrector methods. For appropriate conditions, the solution of the time-dependent rate equations can be reduced to the time-independent problem plus the (analytic)
solution of an ODE.
At the example of thermonuclear Supernovae, we demonstrate the status and power of our approach, and its limitations.
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