Date of Award

Summer 2011

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Computational Analysis and Modeling

First Advisor

Weizhong Dai

Abstract

This dissertation improves the accuracy of the Generalized Finite Difference Time Domain (FDTD) scheme by determining a differential operator that is capable of achieving reasonable accuracy when used to obtain even-order derivatives up to order fourteen. The Generalized FDTD scheme is an explicit, scheme used to solve the time-dependent Schrödinger equation, and being an explicit scheme, it must utilize a carefully devised ratio of the temporal step to the spatial step to maintain numerical stability. This ratio is called the mesh ratio, and the Generalized FDTD scheme allows this ratio to be significantly relaxed. As the mesh ratio increases, the generalized scheme requires the evaluation of increasingly high-order spatial derivatives.

In Chapter 3, two classes of differential operators are considered, the first being the repeated application of a central difference approximation of the Laplace operator using various orders of accuracy, and the second class being the differentiated Lagrange interpolating polynomials. This approach intentionally avoids attempting to approximate such derivatives using increasingly high-order finite differences, as the number of uncomputable points becomes very large as the order of the derivative increases.

Based on the conclusions from Chapter 3, a sixth-order accurate central difference operator is chosen to approximate the Laplace operator, and in Chapter 4 the order of accuracy is determined. The numerical stability is analyzed using the Von Neumann analysis and a stability condition is shown.

The validity of the analysis performed in Chapter 4 is verified by solving a Schrödinger equation with exact solution, and observing the numerical error and stability. The order of accuracy of the scheme is also verified through experimentation, it is shown both theoretically and empirically that the chosen differential operator is both stable and accurate when used to solve the time-dependent Schrödinger equation using the Generalized FDTD method.

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