Efficient numerical approximations for a non-conservative Nonlinear Schrodinger equation appearing in wind-forced ocean waves
Agissilaos Athanassoulis, Theodoros Katsaounis, Irene Kyza
TL;DR
The paper addresses the challenge of simulating a non-conservative nonlinear Schrödinger equation with time-dependent coefficients, modeling wind-forced ocean waves, while preserving discrete mass and energy balances. It adapts two NLS-conserving schemes—the linearly implicit relaxation scheme and the fully implicit Delfour-Fortin-Payre scheme—to the NCNLS setting and derives discrete balance laws that mirror the continuous ones, despite nonzero input. Numerical experiments in one dimension demonstrate that both schemes robustly track mass and energy under strong non-conservative forcing, with the relaxation scheme offering lower computational cost and the DFP scheme ensuring exact discrete energy conservation. The results suggest these schemes are effective for wind-driven wave modeling and provide a foundation for adaptive and higher-order extensions in non-conservative contexts.
Abstract
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.