New Preprint ‘High-order mass- and energy-conserving methods for the nonlinear Schrödinger equation and its hyperbolization’ on arXiv

David I. Ketcheson and I have published our new preprint High-order mass- and energy-conserving methods for the nonlinear Schrödinger equation and its hyperbolization on arXiv.

We propose a class of numerical methods for the nonlinear Schrödinger (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.

The reproducibility repository is available on GitHub.