Doctoral Dissertations

Date of Award

8-2025

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Cory D. Hauck

Committee Members

Cory Hauck, Steven Wise, Anthony Mezzacappa, Terry Haut

Abstract

This dissertation focuses on angular approximations of the radiation transport equation (RTE), which governs the evolution of particle density in space, angle, and time. We investigate the spherical harmonics and discrete ordinates methods, two widely used angular discretizations known for their spectral convergence. While spectral convergence offers high accuracy for smooth solutions, its performance deteriorates in the context of non-smooth solutions.

To address this limitation, we explore a regularization strategy known as filtering, which modifies angular discretizations by introducing a dissipative operator that damps high-frequency, angular modes.

We derive an error estimate of the filtered spherical harmonic solution using the theory of hypocoercivity, which characterizes decay to equilibrium, even in the absence of coercivity. This analysis highlights the trade-off between stability and consistency in filtered regularization.

We propose a novel approach for tuning the filter strength of the filtered spherical harmonic method locally, in space and time. The filter strength is modeled by a neural network, whose input features include local moment information and material properties. The network parameters are determined by solving a PDE-constrained minimization problem. This adaptive approach suppresses nonphysical oscillations, while preserving the accuracy of the spherical harmonic solution in smooth regions. Numerical experiments demonstrate that this method can substantially reduce the error of the original filtered spherical harmonic method with constant filter strength.

We study filtered approximations in the context of thermal radiative transfer (TRT), where the discrete ordinates method is employed. The variable Eddington factor (VEF) method couples the first two angular moments of the transport equation with the TRT equations. We develop a filtering strategy that maintains consistency between the discrete ordinates equations and the moment equations. Numerical experiments demonstrate that this filtering strategy significantly improves the quality of the TRT approximation, particularly when the solution lacks regularity.

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