Abstract
We propose a hybrid method which combines the Bloch decomposition-based time splitting (BDTS) method and the Gaussian beam method to simulate the Schrodinger 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 Schrodinger 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 Schrodinger equation with its Dirac approximation, and numerically show that, as the rescaled Planck number epsilon -> 0, the Schrodinger 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 in between the solutions of the Schrodinger equation and its Dirac approximation.