UC Santa Barbara

We will study high-frequency wave propagation in periodic media. A typical example is given by the Schr\"odinger equation in the semiclassical regime with a highly oscillatory periodic potential and external smooth potential. This problem presents a numerical challenge when in the semiclassical regime. For example, conventional methods such as finite differences and spectral methods leads to high numerical cost, especially in higher dimensions. For this reason, asymptotic methods like the frozen Gaussian approximation (FGA) was developed to provide an efficient computational tool. Prior to the development of the FGA, the geometric optics and Gaussian beam methods provided an alternative asymptotic approach to solving the Schr\"odinger equation efficiently. Unlike the geometric optics and Gaussian beam methods, the FGA does not lose accuracy due to caustics or beam spreading.

In this thesis, we will briefly review the geometric optics, Gaussian beam, and FGA methods. The mathematical techniques used by these methods will aid us in formulating the Bloch-decomposition based FGA. The Bloch-decomposition FGA generalizes the FGA to wave propagation in periodic media. We will establish the convergence of the Bloch-decomposition based FGA to the true solution for Schr\"odinger equation and develop a gauge-invariant algorithm for the Bloch-decomposition based FGA. This algorithm will avoid the numerical difficulty of computing the gauge-dependent Berry phase. We will show the numerical performance of our algorithm by several one-dimensional examples.

Lastly, we will propose a time-splitting FGA-based artificial boundary conditions for solving the one-dimensional nonlinear Schr\"odinger equation (NLS) on an unbounded domain. The NLS will be split into two parts, the linear and nonlinear parts. For the linear part we will use the following absorbing boundary strategy: eliminate Gaussian functions whose centers are too distant to a fixed domain.