@TechReport{ it:2013-019,
author = {Emil Kieri and Gunilla Kreiss and Olof Runborg},
title = {Coupling of {G}aussian Beam and Finite Difference Solvers
for Semiclassical {S}chr{\"o}dinger Equations},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2013},
number = {2013-019},
month = sep,
abstract = {In the semiclassical regime, solutions to the
time-dependent Schr{\"o}dinger equation are highly
oscillatory. The number of grid points required for
resolving the oscillations may become very large even for
simple model problems, making solution on a grid, e.g.,
using a finite difference method, intractable. Asymptotic
methods like Gaussian beams can resolve the oscillations
with little effort and yield good approximations when the
molecules are heavy and the potential is smooth. However,
when the potential has variations on a small length-scale,
quantum phenomena become important. Then asymptotic methods
are less accurate. The two classes of methods perform well
in different parameter regimes. This opens for hybrid
methods, using Gaussian beams where we can and finite
differences where we have to. We propose a new method for
treating the coupling between the finite difference method
and Gaussian beams. The new method reduces the needed
amount of overlap regions considerably compared to previous
methods, which improves the efficiency. We apply the method
to scattering problems in one and two dimensions. }
}