Date of Award

Summer 2013

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Engineering Physics

First Advisor

Weizhong Dai

Abstract

The nonlinear Schrödinger equation (NLSE) is one of the most widely applicable equations in physical science, and characterizes nonlinear dispersive waves, optics, water waves, and the dynamics of molecules. The NLSE satisfies many mathematical conservation laws. Moreover, due to the nonlinearity, the NLSE often requires a numerical solution, which also satisfies the conservation laws. Some of the more popular numerical methods for solving the NLSE include the finite difference, finite element, and spectral methods such as the pseudospectral, split-step with Fourier transform, and integrating factor coupled with a Fourier transform. With regard to the finite difference and finite element methods, higher-order accurate and stable schemes are often required to solve a large-scale linear system. Conversely, spectral methods via Fourier transforms for space discretization coupled with Runge-Kutta methods for time stepping become too complex when applied to multidimensional problems. One of the most prevalent challenges in developing these numerical schemes is that they satisfy the conservation laws.

The objective of this dissertation was to develop a higher-order accurate and simple finite difference scheme for solving the NLSE. First, the wave function was split into real and imaginary components and then substituted into the NLSE to obtain coupled equations. These components were then approximated using higher-order Taylor series expansions in time, where the derivatives in time were replaced by the derivatives in space via the coupled equations. Finally, the derivatives in space were approximated using higher-order accurate finite difference approximations. As such, an explicit and higher order accurate finite difference scheme for solving the NLSE was obtained. This scheme is called the explicit generalized finite-difference time-domain (explicit G-FDTD). For purposes of completeness, an implicit G-FDTD scheme for solving the NLSE was also developed.

In this dissertation, the discrete energy method is employed to prove that both the explicit and implicit G-FDTD scheme satisfy the discrete analogue form of the first conservation law. To verify the accuracy of the numerical solution and the applicability of the schemes, both schemes were tested by simulating bright and dark soliton propagation and collision in one and two dimensions. Compared with other popular existing methods (e.g., pseudospectral, split-step, integrating factor), numerical results showed that the G-FDTD method provides a more accurate solution, particularly when the time step is large. This solution is particularly important during the long-time period simulations. The explicit G-FDTD method proved to be advantageous in that it was simple and fast in computation. Furthermore, the G-FDTD showed that the solution propagates through the boundary with analytical solution continuation.

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