A Conservative SAV-RRK Finite Element Method for the Nonlinear Schrödinger Equation

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.