An energy-preserving Crank–Nicolson Galerkin spectral element method for the two dimensional nonlinear Schrödinger equation

Abstract

A Crank–Nicolson Galerkin spectral element method for solving the nonlinear Schrödinger (NLS) equation in two dimensions is proposed in this paper. Our key idea is twofolds. First, the 2D NLS equation is rewritten as an infinite-dimensional Hamiltonian PDE and the Hamiltonian PDE is discreted by using the Galerkin spectral element (GSE) method in space. Second, we cast the resulted ODEs into a finite-dimensional canonical Hamiltonian system and discrete the system by using the Crank–Nicolson (CN) method. The relay leads to a fully discretized and energy-preserved scheme. Without grid ratio restrictions, the order of convergence of our new method is $\mathcal{O}({\tau }^2+h^2)$ if the discrete $L^2$-norm is employed. The Fast Fourier Transform and the matrix diagonalization method are applied to the new method to increase computing efficiency. Numerical examples are given to further illustrate the conservation properties and convergence of the energy-preserving scheme constructed.