Abstract
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.
| Original language | English |
|---|---|
| Article number | e12774 |
| Number of pages | 22 |
| Journal | Studies in Applied Mathematics |
| Volume | 153 |
| Issue number | 4 |
| Early online date | 25 Oct 2024 |
| DOIs | |
| Publication status | Published - Nov 2024 |
Keywords
- Finite elements
- Nonconservative nonlinear Schrödinger equation
- Relaxation Crank–Nicolson scheme
Access to Document
- Athanassoulis_2024_SAM_Efficient-numerical-approximations_AAM_CCBY
Copyright © 2024 the Authors. This work has been made available online in accordance with the University of St Andrews Open Access policy. This accepted manuscript is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. The final published version of this work is available at https://doi.org/10.1111/sapm.12774.