A class of Galerkin Schemes for Time-Dependent Radiative Transfer

The numerical solution of time-dependent radiative transfer problems is challenging, due to the high dimension and to the anisotropic structure of the underlying integro-partial differential equation. In this paper we propose a general framework for designing numerical methods for time-dependent radiative transfer based on a Galerkin discretization in space and angle combined with appropriate time stepping schemes. This allows us to systematically incorporate boundary conditions and also to preserve basic properties such as exponential stability and decay to equilibrium on the discrete level. We present the basic a priori error analysis and provide abstract error estimates that cover a wide class of methods. The starting point for our considerations is to rewrite the radiative transfer problem as a system of evolution equations which has a similar structure to first order hyperbolic systems in acoustics or electrodynamics. This analogy allows us to generalize the main arguments of the numerical analysis for such applications to the radiative transfer problem under investigation. We also discuss a particular discretization scheme based on a truncated spherical harmonic expansion in angle, a finite element discretization in space, and the implicit Euler method in time. The performance of the resulting mixed $P_N$-finite element time stepping scheme is demonstrated by computational results.