Efficient numerical approximations for a nonconservative nonlinear Schrödinger equation appearing in wind‐forced ocean waves

We consider a nonconservative nonlinear Schrödinger 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 pa...

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Bibliographic Details
Published in:Studies in applied mathematics (Cambridge) 2024-11, Vol.153 (4), p.n/a
Main Authors: Athanassoulis, Agissilaos, Katsaounis, Theodoros, Kyza, Irene
Format: Article
Language:English
Subjects:
Energy conservation
finite elements
nonconservative nonlinear Schrödinger equation
relaxation Crank–Nicolson scheme
Schrodinger equation
Time dependence
Water conservation
Water waves
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Summary:We consider a nonconservative nonlinear Schrödinger 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 nonlinear Schrödinger equation. Both schemes are second‐order 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 nonconservative problems. We finally compare the two numerical schemes and discuss their performance.
ISSN:0022-2526
1467-9590
DOI:10.1111/sapm.12774