TY - JOUR
T1 - Efficient numerical approximations for a nonconservative nonlinear Schrödinger equation appearing in wind-forced ocean waves
AU - Athanassoulis, Agis
AU - Katsaounis, Theodoros
AU - Kyza, Irene
N1 - © 2024 Wiley Periodicals LLC
PY - 2024/10/25
Y1 - 2024/10/25
N2 - 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.
AB - 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.
KW - finite elements
KW - nonconservative nonlinear Schrödinger equation
KW - relaxation Crank–Nicolson scheme
U2 - 10.1111/sapm.12774
DO - 10.1111/sapm.12774
M3 - Article
SN - 0022-2526
JO - Studies in Applied Mathematics
JF - Studies in Applied Mathematics
M1 - e12774
ER -