The Gaussian Wave Packet Transform for the Semi-Classical Schrödinger Equation with Vector Potentials (Communications in Computational Physics, 2019)
In this paper, we reformulate the semi-classical Schrödinger equation in thepresence of electromagnetic field by the Gaussian wave packet transform. With thisapproach, the highly oscillatory Schrödinger equation is equivalently transformed intoanother Schrödinger type wave equation, the w equation, which is essentially not oscillatory and thus requires much less computational effort. We propose two numericalmethods to solve the w equation, where the Hamiltonian is either divided into thekinetic, 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 accuratein space and second order accurate in time, and in principle they can be extended tohigher order schemes in time. Various one dimensional and multidimensional numerical tests are provided to justify the properties of the proposed methods.