Discontinuous Galerkin methods for the radiative transfer equation and its approximations

• Main Author: Eichholz, Joseph A.
• Other Authors: Han, Weimin
• Format: Others
• Language: English
• Published: University of Iowa 2011
• Subjects: Galerkin methods; radiative transfer; Applied Mathematics
• Online Access: https://ir.uiowa.edu/etd/1135; https://ir.uiowa.edu/cgi/viewcontent.cgi?article=2519&context=etd

Radiative transfer theory describes the interaction of radiation with scattering and absorbing media. It has applications in neutron transport, atmospheric physics, heat transfer, molecular imaging, and others. In steady state, the radiative transfer equation is an integro-differential equation of five independent variables. This high dimensionality and presence of integral term present a serious challenge when trying to solve the equation numerically. Over the past 50 years, several techniques for solving the radiative transfer equation have been introduced. These include, but are certainly not limited to, Monte Carlo methods, discrete-ordinate methods, spherical harmonics methods, spectral methods, finite difference methods, and finite element methods. Methods involving discrete ordinates have received particular attention in the literature due to their relatively high accuracy, flexibility, and relatively low computational cost. In this thesis we present a discrete-ordinate discontinuous Galerkin method for solving the radiative transfer equation. In addition, we present a generalized Fokker-Planck equation that may be used to approximate the radiative transfer equation in certain circumstances. We provide well posedness results for this approximation, and introduce a discrete-ordinate discontinuous Galerkin method to approximate a solution. Theoretical error estimates are derived, and numerical examples demonstrating the efficacy of the methods are given.