Volume 15, Issue 3
A Conservative SAV-RRK Finite Element Method for the Nonlinear Schrödinger Equation

Adv. Appl. Math. Mech., 15 (2023), pp. 583-601.

Published online: 2023-02

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• 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.

65M15, 65M60

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@Article{AAMM-15-583, author = {Yang , Jun and Yi , Nianyu}, title = {A Conservative SAV-RRK Finite Element Method for the Nonlinear Schrödinger Equation}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2023}, volume = {15}, number = {3}, pages = {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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2021-0255}, url = {http://global-sci.org/intro/article_detail/aamm/21442.html} }
TY - JOUR T1 - A Conservative SAV-RRK Finite Element Method for the Nonlinear Schrödinger Equation AU - Yang , Jun AU - Yi , Nianyu JO - Advances in Applied Mathematics and Mechanics VL - 3 SP - 583 EP - 601 PY - 2023 DA - 2023/02 SN - 15 DO - http://doi.org/10.4208/aamm.OA-2021-0255 UR - https://global-sci.org/intro/article_detail/aamm/21442.html KW - Schrödinger equation, mass conservation, energy conservation, finite element method, relaxation Runge-Kutta, scalar auxiliary variable. AB -

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.

Jun Yang & Nianyu Yi. (2023). A Conservative SAV-RRK Finite Element Method for the Nonlinear Schrödinger Equation. Advances in Applied Mathematics and Mechanics. 15 (3). 583-601. doi:10.4208/aamm.OA-2021-0255
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