  We provide a systematic comparison of two numerical methods to solve the
widely used nonlinear Schr\"odinger equation. The first one is the standard
second order split-step (SS2) method based on operator splitting approach. The
second one is the Hamiltonian integration method (HIM). It allows the exact
conservation of the Hamiltonian at the cost of requiring the implicit time
stepping. We found that numerical error for HIM method is systematically
smaller than the SS2 solution for the same time step. At the same time, one can
take orders of magnitude larger time steps in HIM compared with SS2 still
ensuring numerical stability. In contrast, SS2 time step is limited by the
numerical stability threshold.
