  We carry out the convergence analysis of the Scalar Auxiliary Variable (SAV)
method applied to the nonlinear Schr\"odinger equation which preserves a
modified Hamiltonian on the discrete level. We derive a weak and strong
convergence result, establish second-order global error bounds and present long
time error estimates on the modified Hamiltonian. In addition, we illustrate
the favorable energy conservation of the SAV method compared to classical
splitting schemes in certain applications.

Numerical Analysis;Numerical Analysis