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.

翻译：本文研究了一类受水波问题启发、含时系数的非保守非线性薛定谔方程（NCNLS）。该问题不满足质量或能量守恒，其质量与能量随时间变化遵循显式平衡律。我们将两种已知在离散层面保持立方非线性薛定谔方程（NLS）能量与质量守恒的数值格式推广至特定的NCNLS方程。两种格式在时间上均为二阶精度，我们证明其推广形式满足了NCNLS质量与能量平衡律的离散版本。第一种格式为线性隐式松弛格式，另一种为改进的全隐式Delfour-Fortin-Payre格式。数值结果表明，即使在强非保守问题中，两种格式均能稳健地捕获质量与能量的正确值。最后，我们对两种数值格式进行了比较并讨论了其性能。