Energy-preserving methods for non-smooth nonlinear Schrödinger equations

Abstract

In this paper, we propose energy-preserving methods for nonlinear Schrödinger equations (NLSEs) with delta potentials. We reformulate the prototype model equation into an infinite-dimensional Hamiltonian system (IDHS), apply the average vector field (AVF) method to the time, and obtain the corresponding temporal semi-discrete system which possesses exact energy preservation properties. Then we apply the immersed interface method (IIM) to discretize the space of the semi-discrete system and get a scheme of full-discrete systems. The scheme preserves the discrete total energy precisely and admits, however, just first-order local accuracy. In order to improve the accuracy, we give some modifications for the scheme and get a second-order method as desired. The precise preservation of the discrete energy is rigorously proved for the modified scheme. Extensive numerical experimentation for cases of single- and double-delta potentials is examined and discussed, and different comparisons are made to validate the theoretical analyses. Together with the normalization conservation law and some convergence behaviours, it demonstrates that our methods are considerably good in the long-term calculations, especially the superiorities of preserving the energy conservation laws.