Symplectic and multi-symplectic wavelet collocation methods for two-dimensional Schrödinger equations

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• 作者：Huajun Zhu^a ; ^{girl-zhu@163.com} ; Yaming Chen^a ; ^{chenym-08@163.com} ; Songhe Song^a ; ^{shsong@nudt.edu.cn} ; Huayu Hu^b ; ^{yuyuideal@gmail.com}
• 关键词：Symplectic ; Multi-symplectic ; Wavelet collocation method ; Two-dimensional Schr&#xf6 ; dinger equation ; Autocorrelation function
• 刊名：Applied Numerical Mathematics
• 出版年：2011
• 出版时间：March 2011
• 年：2011
• 卷：61
• 期：3
• 页码：308-321
• 全文大小：878 K

文摘

In this paper, we develop symplectic and multi-symplectic wavelet collocation methods to solve the two-dimensional nonlinear Schrödinger equation in wave propagation problems and the two-dimensional time-dependent linear Schrödinger equation in quantum physics. The Hamiltonian and the multi-symplectic formulations of each equation are considered. For both formulations, wavelet collocation method based on the autocorrelation function of Daubechies scaling functions is applied for spatial discretization and symplectic method is used for time integration. The conservation of energy and total norm is investigated. Combined with splitting scheme, splitting symplectic and multi-symplectic wavelet collocation methods are also constructed. Numerical experiments show the effectiveness of the proposed methods.