Computation of the Schrödinger Equation in the Semiclassical Regime on an Unbounded Domain

Abstract

The study of this paper is twofold: On the one hand, we generalize the high-order local absorbing boundary conditions (LABCs) proposed in [J. Zhang et al., Comm. Comput. Phys., 10 (2011), pp. 742--766] to compute the Schrödinger equation in the semiclassical regime on an unbounded domain. We analyze the stability of the equation with LABCs and the convergence of the Crank--Nicolson scheme that discretizes it and we conclude that when the rescaled Planck constant $\varepsilon$ gets small, the accuracy deteriorates and the requirements on time step and mesh size get tough. This leads to the second part of our study. We propose an asymptotic method based on the frozen Gaussian approximation. The absorbing boundary condition is dealt with by a simple strategy that all the effects of the Gaussian functions which contribute to the outgoing waves will be eliminated by stopping the Hamiltonian flow of their centers when they get out of the domain of interest. We present numerical examples in both one and two dimensions to verify the performance of the proposed numerical methods.