Date of Award

Spring 5-2020

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Computational Analysis and Modeling

First Advisor

Weizhong Dai

Abstract

The finite­difference time­domain (FDTD) method and its generalized variant (G­FDTD) are efficient numerical tools for solving the linear and nonlinear Schrödinger equations because not only are they explicit, allowing parallelization, but they also provide high­order accuracy with relatively inexpensive computational costs. In addition, the G­FDTD method has a relaxed stability condition when compared to the original FDTD method. It is important to note that the existing simulations of the G­FDTD scheme employed analytical solutions to obtain function values at the points along the boundary; however, in simulations for which the analytical solution is unknown, theoretical approximations for values at points along the boundary are desperately needed. Hence, the objective of this dissertation research is to develop absorbing boundary conditions (ABCs) so that the G­FDTD method can be used to solve the nonlinear Schrödinger equation when the analytical solution is unknown.

To create the ABCs for the nonlinear Schrödinger equation, we initially determine the associated Engquist­Majda one­way wave equations and then proceed to develop a finite difference scheme for them. These ABCs are made to be adaptive using a windowed Fourier transform to estimate a value of the wavenumber of the carrier wave. These ABCs were tested using the nonlinear Schrödinger equation for 1D and 2D soliton propagation as well as Gaussian packet collision and dipole radiation. Results show that these ABCs perform well, but they have three key limitations. First, there are inherent reflections at the interface of the interior and boundary domains due to the different schemes used the two regions; second, to use the ABCs, one needs to estimate a value for the carrier wavenumber and poor estimates can cause even more reflection at the interface; and finally, the ABCs require different schemes in different regions of the boundary, and this domain decomposition makes the ABCs tedious both to develop and to implement.

To address these limitations for the FDTD method, we employ the fractional­order derivative concept to unify the Schrödinger equation with its one­way wave equation over an interval where the fractional order is allowed to vary. Through careful construction of a variable­order fractional momentum operator, outgoing waves may enter the fractionalorder region with little to no reflection and, inside this region, any reflected portions of the wave will decay exponentially with time. The fractional momentum operator is then used to create a fractional­order FDTD scheme. Importantly, this single scheme can be used for the entire computational domain, and the scheme smooths the abrupt transition between the FDTD method and the ABCs. Furthermore, the fractional FDTD scheme relaxes the precision needed for the estimated carrier wavenumber. This fractional FDTD scheme is tested for both the linear and nonlinear Schrödinger equations. Example cases include a 1D Gaussian packet scattering off of a potential, a 1D soliton propagating to the right, as well as 2D soliton propagation, and the collision of Gaussian packets. Results show that the fractional FDTD method outperforms the FDTD method with ABCs.

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