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 - <p style="text-align: justify;">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 Schr<span style="text-align: justify;">ö</span>dinger 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.</p>