Compact finite difference schemes with high accuracy for one-dimensional nonlinear Schrödinger equation

In this paper, two compact finite difference schemes are presented for the numerical solution of the one-dimensional nonlinear Schrödinger equation. The discrete L2L2-norm error estimates show that convergence rates of the present schemes are of order O(h4+τ2)O(h4+τ2). Numerical experiments on some model problems show that the present schemes preserve the conservation laws of charge and energy and are of high accuracy.