High-Order Mass- and Energy-Conserving Methods for the Nonlinear Schrödinger Equation

SIAM Journal on Scientific Computing, Volume 46, Issue 2, Page A1026-A1046, April 2024.
Abstract. A class of high-order mass- and energy-conserving methods is proposed for the nonlinear Schrödinger equation based on Gauss collocation in time and finite element discretization in space, by introducing a mass- and energy-correction post-process at every time level. The existence, uniqueness, and high-order convergence of the numerical solutions are proved. In particular, the error of the numerical solution is [math] in the [math] norm after incorporating the accumulation errors arising from the post-processing correction procedure at all time levels, where [math] and [math] denote the degrees of finite elements in time and space, respectively, which can be arbitrarily large. Several numerical examples are provided to illustrate the performance of the proposed new method, including the conservation of mass and energy and the high-order convergence in simulating solitons and bi-solitons. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/jiashhu/ME-Conserved-NLS and in the supplementary materials (ME-Conserved-NLS-main-2-Reproducibility-badge.zip [14.4MB]).