Mass- and energy-conserving Gauss collocation methods for the nonlinear Schrödinger equation with a wave operator

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  • Shu Ma
  • Jilu Wang
  • Mingyan Zhang
  • Zhimin Zhang

Related Research Unit(s)


Original languageEnglish
Article number77
Journal / PublicationAdvances in Computational Mathematics
Issue number6
Online published23 Oct 2023
Publication statusPublished - Dec 2023


A fully discrete finite element method with a Gauss collocation in time is proposed for solving the nonlinear Schrödinger equation with a wave operator in the d-dimensional torus, ∈ { 1 , 2 , 3 } . Based on Gauss collocation method in time and the scalar auxiliary variable technique, the proposed method preserves both mass and energy conservations at the discrete level. Existence and uniqueness of the numerical solutions to the nonlinear algebraic system, as well as convergence to the exact solution with order O (hp+ τk+1) in the L(0 , T; H1) norm, are proved by using Schaefer’s fixed point theorem without requiring any grid-ratio conditions, where (p, k) is the degree of the space-time finite elements. The Newton iterative method is applied for solving the nonlinear algebraic system. The numerical results show that the proposed method preserves discrete mass and energy conservations up to machine precision, and requires only a few Newton iterations to achieve the desired accuracy, with optimal-order convergence in the L(0 , T; H1) norm.

© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023

Research Area(s)

  • Error estimate, Gauss collocation, High order, Mass and energy conservation, Nonlinear Schrödinger equation with wave operator, Scalar auxiliary variable