High-order mass-, energy- and momentum-conserving methods for the nonlinear Schrödinger equation

Abstract

This paper introduces a novel formulation and an associated space-time finite element method for simulating solutions to the nonlinear Schrödinger equation. A major advantage of the proposed algorithm is its intrinsic ability to preserve the conservation of mass, energy, and momentum at the discrete level. This is proved for the numerical solutions determined by the fully discrete implicit scheme. An effective iterative scheme is proposed for solving the nonlinear system based on an equivalent formulation which suggests using Newton's iteration for the solution and no iteration for the Lagrange multipliers in the nonlinear system. Extensive numerical examples are provided to demonstrate the high-order convergence and effectiveness of the proposed algorithm in conserving mass, energy, and momentum in the simulation of one-dimensional Ma-solitons and bi-solitons, as well as of two-dimensional solitons governed by the nonlinear Schrödinger equation. The numerical results show that the mass-, energy- and momentum-conserving method designed in this paper also significantly reduces the errors of the numerical solutions in long-time simulations compared with methods which do not conserve these quantities.