Journal
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
Volume 344, Issue -, Pages 245-258Publisher
ELSEVIER SCIENCE BV
DOI: 10.1016/j.cam.2018.05.025
Keywords
Hamiltonian equations; Energy-preserving schemes; Nonlinear Schrodinger equation in two dimensions; Galerkin spectral element discretization; Crank-Nicolson method; Error estimate
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Funding
- National Natural Science Foundation of China [11771213]
- National Key Research and Development Project of China [2016YFC0600310]
- Major Projects of Natural Sciences of University in Jiangsu Province of China [15KJA110002]
- Priority Academic Program Development of Jiangsu Higher Education Institutions
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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. (C) 2018 Elsevier B.V. All rights reserved.
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