Degree Type


Date of Award


Degree Name

Doctor of Philosophy




Applied Mathematics

First Advisor

James A. Rossmanith


The linear kinetic transport equations are ubiquitous in many application areas, including as a model for neutron transport in nuclear reactors and the propagation of electromagnetic radiation in astrophysics. The main computational challenge in solving the linear transport equations is that solutions live in a high-dimensional phase space that must be sufficiently resolved for accurate simulations. The three standard computational techniques for solving the linear transport equations are the (1) implicit Monte Carlo, (2) discrete ordinate(S$_N$), and (3) spherical harmonic(P$_N$) methods. Monte Carlo methods are stochastic methods for solving time-dependent nonlinear radiative transfer problems. In a traditional Monte Carlo method when photons are absorbed, they are reemitted in a distribution which is uniform over the entire spatial cell where the temperature is assumed constant, resulting in loss of information. In implicit Monte Carlo(IMC) methods, photons are reemitted from the place where they were actually absorbed, which improves the accuracy. Overall, IMC method improves stability, flexibility, and computational efficiency \cite{fleck}. The S$_N$ method solves the transport equation using a quadrature rule to reconstruct the energy density. This method suffers from so-called "ray effect", which are due to the approximation of the double integral over a unit sphere by a finite number of discrete angular directions \cite{chai}. The P$_N$ approximation is based on expanding the part of the solution that depends on velocity direction (i.e., two angular variables) into spherical harmonics. A big challenge with the P$_N$ approach is that the spherical harmonics expansion does not prevent the formation of negative particle concentrations. The idea behind my research is to develop on an alternative formulation of P$_N$ approximations that hybridizes aspects of both P$_N$ and S$_N$. Although the basic scheme does not guarantee positivity of the solution, the new formulation allows for the introduction of local limiters that can be used to enforce positivity.

Copyright Owner

Minwoo Shin



File Format


File Size

117 pages