A conservative Fourier pseudo-spectral method for the nonlinear Schrödinger equation

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• 作者：Yuezheng Gong^a ; Qi Wang^a ; ^b ; ^c ; Yushun Wang^d ; ^{wangyushun@njnu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Jiaxiang Cai^e
• 关键词：Nonlinear Schrö ; dinger equation ; Fourier pseudo-spectral method ; Unconditional convergence ; FFT
• 刊名：Journal of Computational Physics
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：328
• 期：Complete
• 页码：354-370
• 全文大小：2235 K

文摘

A Fourier pseudo-spectral method that conserves mass and energy is developed for a two-dimensional nonlinear Schrödinger equation. By establishing the equivalence between the semi-norm in the Fourier pseudo-spectral method and that in the finite difference method, we are able to extend the result in Ref. [56] to prove that the optimal rate of convergence of the new method is in the order of mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021999116305204&_mathId=si1.gif&_user=111111111&_pii=S0021999116305204&_rdoc=1&_issn=00219991&md5=5695a9546a04547722633e0ba065be98" title="Click to view the MathML source">O(N^−r+τ²)mathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll">O(N−r+τ2)math> in the discrete mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021999116305204&_mathId=si2.gif&_user=111111111&_pii=S0021999116305204&_rdoc=1&_issn=00219991&md5=78aa1244b4077567382cf51c80c0ff4d" title="Click to view the MathML source">L²mathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll">L2math> norm without any restrictions on the grid ratio, where N is the number of modes used in the spectral method and τ is the time step size. A fast solver is then applied to the discrete nonlinear equation system to speed up the numerical computation for the high order method. Numerical examples are presented to show the efficiency and accuracy of the new method.