  We consider a non-conservative nonlinear Schrodinger equation (NCNLS) with
time-dependent coefficients, inspired by a water waves problem. This problem
does not have mass or energy conservation, but instead mass and energy change
in time under explicit balance laws. In this paper we extend to the particular
NCNLS two numerical schemes which are known to conserve energy and mass in the
discrete level for the cubic NLS. Both schemes are second oder accurate in
time, and we prove that their extensions satisfy discrete versions of the mass
and energy balance laws for the NCNLS. The first scheme is a relaxation scheme
that is linearly implicit. The other scheme is a modified Delfour-Fortin-Payre
scheme and it is fully implicit. Numerical results show that both schemes
capture robustly the correct values of mass and energy, even in strongly
non-conservative problems. We finally compare the two numerical schemes and
discuss their performance.
