Semi-implicit time integration for [formula omitted] thermal radiative transfer

Abstract

Implicit time integration involving the solution of large systems of equations is the current paradigm for time-dependent radiative transfer. In this paper we present a semi-implicit, linear discontinuous Galerkin method for the spherical harmonics ( P N ) equations for thermal radiative transfer in planar geometry. Our method is novel in that the material coupling terms are treated implicitly (via linearizing the emission source) and the streaming operator is treated explicitly using a second-order accurate Runge–Kutta method. The benefit of this approach is that each time step only involves the solution of equations that are local to each cell. This benefit comes at the cost of having the time step limited by a CFL condition based on the speed of light. To guarantee positivity and avoid artificial oscillations, we use a slope-limiting technique. We present analysis and numerical results that show the method is robust in the diffusion limit when the photon mean-free path is not resolved by the spatial mesh. Also, in the diffusion limit the time step restriction relaxes to a less restrictive explicit diffusion CFL condition. We demonstrate with numerical results that away from the diffusion limit our method demonstrates second-order error convergence as the spatial mesh is refined with a fixed CFL number.