Year: 2023
Advances in Applied Mathematics and Mechanics, Vol. 15 (2023), Iss. 3 : pp. 583–601
Abstract
In this paper, we propose, analyze and numerically validate a conservative finite element method for the nonlinear Schrödinger equation. A scalar auxiliary variable (SAV) is introduced to reformulate the nonlinear Schrödinger equation into an equivalent system and to transform the energy into a quadratic form. We use the standard continuous finite element method for the spatial discretization, and the relaxation Runge-Kutta method for the time discretization. Both mass and energy conservation laws are shown for the semi-discrete finite element scheme, and also preserved for the full-discrete scheme with suitable relaxation coefficient in the relaxation Runge-Kutta method. Numerical examples are presented to demonstrate the accuracy of the proposed method, and the conservation of mass and energy in long time simulations.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/aamm.OA-2021-0255
Advances in Applied Mathematics and Mechanics, Vol. 15 (2023), Iss. 3 : pp. 583–601
Published online: 2023-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 19
Keywords: Schrödinger equation mass conservation energy conservation finite element method relaxation Runge-Kutta scalar auxiliary variable.
Author Details
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Recent advances in the numerical solution of the Nonlinear Schrödinger Equation
Barletti, Luigi
Brugnano, Luigi
Gurioli, Gianmarco
Iavernaro, Felice
Journal of Computational and Applied Mathematics, Vol. 445 (2024), Iss. P.115826
https://doi.org/10.1016/j.cam.2024.115826 [Citations: 1]