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arXiv · 2510.14335

High-order mass- and energy-conserving methods for the nonlinear Schr\"odinger equation and its hyperbolization

Hendrik Ranocha·David I. Ketcheson
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Abstract

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.

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Hendrik Ranocha, David I. Ketcheson. 2025-10-16. High-order mass- and energy-conserving methods for the nonlinear Schr\"odinger equation and its hyperbolization. https://arxiv.org/abs/2510.14335

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