@Article{CiCP-24-989,
author = {Lihui Chai, Shi Jin and Peter A. Markowich},
title = {A Hybrid Method for Computing the Schrödinger Equations with Periodic Potential with Band-Crossings in the Momentum Space},
journal = {Communications in Computational Physics},
year = {2018},
volume = {24},
number = {4},
pages = {989--1020},
abstract = {<p style="text-align: justify;">We propose a hybrid method which combines the Bloch decomposition-based
time splitting (BDTS) method and the Gaussian beam method to simulate the
Schrödinger equation with periodic potentials in the case of band-crossings. With the
help of the Bloch transformation, we develop a Bloch decomposition-based Gaussian
beam (BDGB) approximation in the momentum space to solve the Schrödinger equation.
Around the band-crossing a BDTS method is used to capture the inter-band transitions,
and away from the crossing, a BDGB method is applied in order to improve the
efficiency. Numerical results show that this method can capture the inter-band transitions
accurately with a computational cost much lower than the direct solver. We
also compare the Schrödinger equation with its Dirac approximation, and numerically
show that, as the rescaled Planck number ε→0, the Schrödinger equation converges to
the Dirac equations when the external potential is zero or small, but for general external
potentials there is an O(1) difference between the solutions of the Schrödinger
equation and its Dirac approximation.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.2018.hh80.01},
url = {http://global-sci.org/intro/article_detail/cicp/12315.html}
}