Young Researchers Workshop: Kinetic theory with applications in physical sciences

On Galerkin schemes for time-dependent radiative transfer

Matthias Schlottbom

University of Münster

The numerical solution of time dependent radiative transfer problems is challenging, both, due to the high dimension and the anisotropic structure of the underlying integro-partial differential equation. Starting from an appropriate variational formulations, we propose a general strategy for designing numerical methods based on a Galerkin discretization in space and angle combined with appropriate time stepping schemes. This allows us to systematically incorporate boundary conditions and to inherit basic properties like the conservation of mass and exponential stability from the continuous level. We also present the basic approximation error estimates. The starting point for our considerations is to rewrite the radiative transfer problem as a system of evolution equations which has a similar structure as more standard first order hyperbolic systems in acoustics or electrodynamics. This allows us to generalize the main arguments of the numerical analysis of such applications to the radiative transfer problems under investigation. We also discuss a particular discretization scheme based on a truncated spherical harmonic expansion in angle and a finite element discretization in space. The performance of the resulting mixed $P_N$-finite element scheme is demonstrated by computational results for various benchmark problems. Joint work with Herbert Egger (TU Darmstadt)