We propose a class of numerical methods for the nonlinear Schr\"odinger (NLS) equation that conserves mass and energy, is of arbitrarily high-order accuracy in space and time, and requires only the solution of a scalar algebraic equation per time step. We show that some existing spatial discretizations, including the popular Fourier spectral method, are in fact energy-conserving if one considers the appropriate form of the energy density. We develop a new relaxation-type approach for conserving multiple nonlinear functionals that is more efficient and robust for the NLS equation compared to the existing multiple-relaxation approach. The accuracy and efficiency of the new schemes is demonstrated on test problems for both the focusing and defocusing NLS.
@article{arxiv.2510.14335,
title = {High-order mass- and energy-conserving methods for the nonlinear Schr\"odinger equation and its hyperbolization},
author = {Hendrik Ranocha and David I. Ketcheson},
journal= {arXiv preprint arXiv:2510.14335},
year = {2025}
}
Comments
The reproducibility repository is available at https://github.com/ranocha/2025_nls