Bragdon, Sophia Potoczak, authorPinaud, Olivier, advisorBangerth, Wolfgang, committee memberCheney, Margaret, committee memberGelfand, Martin, committee member2021-06-072021-06-072021https://hdl.handle.net/10217/232573This dissertation consists of two independent projects, where wave propagation is the common theme. The first project considers modeling the propagation of laser light through the atmosphere using an approximation procedure we call the variational scaling law (VSL). We begin by introducing the Helmholtz equation and the paraxial approximation to the Helmholtz equation, which is the starting point of the VSL. The approximation method is derived by pairing the variational formulation of the paraxial Helmholtz equation with a generalized Gaussian ansatz which depends on the laser beam parameters. The VSL is a system of stochastic ODEs that describe the evolution of the Gaussian beam parameters. We will conclude with a numerical comparison between the variational scaling law and the paraxial Helmholtz equation. Through exploring numerical examples for increasing strengths of atmospheric turbulence, we show the VSL provides, at least, an order-one approximation to the paraxial Helmholtz equation. The second project focuses on quantum transport by numerically studying the quantum Liouville equation (QLE) equipped with the BGK-collision operator. The collision operator is a relaxation-type operator which locally relaxes the solution towards a local quantum equilibrium. This equilibrium operator is nonlinear and is obtained by solving a moment problem under a local density constraint using the quantum entropy minimization principle introduced by Degond and Ringhofer in \cite{degondringhofer}. A Strang splitting scheme is defined for the QLE in which the collision and transport of particles is treated separately. It is proved that the numerical scheme is well-defined and convergent in-time. The splitting scheme for the QLE is applied in a numerical study of electron transport in different collision regimes by comparing the QLE with the ballistic Liouville equation and the quantum drift-diffusion model. The quantum drift-diffusion model is an example of a quantum diffusion model which is derived from the QLE through a diffusive limit. Finally, it is numerically verified that the QLE converges to the solution to the quantum drift-diffusion equation in the long-time limit.born digitaldoctoral dissertationsengCopyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright.Text