Fast Huygens Sweeping Methods for a Class of Nonlocal Schrödinger Equations

Abstract

We present efficient numerical methods for solving a class of nonlinear Schrödinger equations involving a nonlocal potential. Such a nonlocal potential is governed by Gaussian convolution of the intensity modeling nonlocal mutual interactions among particles. The method extends the Fast Huygens Sweeping Method (FHSM) that we developed in Leung et al. (Methods Appl Anal 21(1):31–66, 2014) for the linear Schrödinger equation in the semi-classical regime to the nonlinear case with nonlocal potentials. To apply the methodology of FHSM effectively, we propose two schemes by using the Lie’s and the Strang’s operator splitting, respectively, so that one can handle the nonlinear nonlocal interaction term using the fast Fourier transform. The resulting algorithm can then enjoy the same computational complexity as in the linear case. Numerical examples demonstrate that the two operator splitting schemes achieve the expected first-order and second-order accuracy, respectively. We will also give one-, two- and three-dimensional examples to demonstrate the efficiency of the proposed algorithm.