A class of Galerkin schemes for time-dependent radiative transfer

TL;DR

This paper introduces a flexible Galerkin-based numerical framework for solving time-dependent radiative transfer problems, ensuring stability and decay properties, with error analysis and a specific discretization scheme demonstrated through computational results.

Contribution

It develops a general Galerkin discretization framework for radiative transfer, including error analysis and a specific mixed PN-finite element scheme with stability properties.

Findings

The proposed method preserves exponential stability and decay to equilibrium.

Error estimates are derived for a broad class of Galerkin schemes.

Computational results validate the effectiveness of the discretization scheme.

Abstract

The numerical solution of time-dependent radiative transfer problems is challenging, both, due to the high dimension as well as 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 to preserve basic properties like exponential stability and decay to equilibrium also 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 like 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 PN-finite element time stepping scheme is demonstrated by computational results.