@Article{CiCP-9-3,
author = {},
title = {Semi-Eulerian and High Order Gaussian Beam Methods for the Schrödinger Equation in the Semiclassical Regime},
journal = {Communications in Computational Physics},
year = {2011},
volume = {9},
number = {3},
pages = {668--687},
abstract = {<p style="text-align: justify;">A novel Eulerian Gaussian beam method was developed in [8] to compute
the Schrödinger equation efficiently in the semiclassical regime. In this paper, we introduce
an efficient semi-Eulerian implementation of this method. The new algorithm
inherits the essence of the Eulerian Gaussian beam method where the Hessian is computed
through the derivatives of the complexified level set functions instead of solving
the dynamic ray tracing equation. The difference lies in that, we solve the ray tracing
equations to determine the centers of the beams and then compute quantities of interests
only around these centers. This yields effectively a local level set implementation,
and the beam summation can be carried out on the initial physical space instead of the
phase plane. As a consequence, it reduces the computational cost and also avoids the
delicate issue of beam summation around the caustics in the Eulerian Gaussian beam
method. Moreover, the semi-Eulerian Gaussian beam method can be easily generalized
to higher order Gaussian beam methods, which is the topic of the second part
of this paper. Several numerical examples are provided to verify the accuracy and
efficiency of both the first order and higher order semi-Eulerian methods.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.091009.160310s},
url = {https://global-sci.com/article/80863/semi-eulerian-and-high-order-gaussian-beam-methods-for-the-schrodinger-equation-in-the-semiclassical-regime}
}