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

In this paper, two compact finite difference schemes are presented for the numerical solution of the one-dimensional nonlinear Schrödinger equation. The discrete L 2 -norm error estimates show that convergence rates of the present schemes are of order O ( h 4 + τ 2 ) . Numerical experiments on some model problems show that the present schemes preserve the conservation laws of charge and energy and are of high accuracy.