TY - JOUR
T1 - The Gaussian Wave Packet Transform for the Semi-Classical Schrödinger Equation with Vector Potentials
JO - Communications in Computational Physics
VL - 2
SP - 469
EP - 505
PY - 2019
DA - 2019/04
SN - 26
DO - http://doi.org/10.4208/cicp.OA-2018-0131
UR - https://global-sci.org/intro/article_detail/cicp/13099.html
KW - Semi-classical Schrödinger equation, Gaussian wave packets, splitting methods,
Fourier-spectral methods.
AB - <p style="text-align: justify;">In this paper, we reformulate the semi-classical Schr<span style="text-align: justify;">ö</span>dinger equation in the
presence of electromagnetic field by the Gaussian wave packet transform. With this
approach, the highly oscillatory Schr<span style="text-align: justify;">ö</span>dinger equation is equivalently transformed into
another Schrödinger type wave equation, the w equation, which is essentially not oscillatory and thus requires much less computational effort. We propose two numerical
methods to solve the w equation, where the Hamiltonian is either divided into the
kinetic, the potential and the convection part, or into the kinetic and the potential-convection part. The convection, or the potential-convection part is treated by a semi-Lagrangian method, while the kinetic part is solved by the Fourier spectral method.
The numerical methods are proved to be unconditionally stable, spectrally accurate
in space and second order accurate in time, and in principle they can be extended to
higher order schemes in time. Various one dimensional and multidimensional numerical tests are provided to justify the properties of the proposed methods.</p>