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

作者：Li, Haochen; Mu, Zhenguo; Wang, Yushun^*

来源：Journal of Computational and Applied Mathematics, 2018, 344: 245-258.

DOI：10.1016/j.cam.2018.05.025

• Hamiltonian equations
• Energy-preserving schemes
• Nonlinear Schrodinger equation in two dimensions
• Galerkin spectral element discretization
• Crank-Nicolson method
• Error estimate

A Crank-Nicolson Galerkin spectral element method for solving the nonlinear Schrodinger (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 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.

• 出版日期2018-12-15
• 单位南京师范大学