@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} }