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Numer. Math. Theor. Meth. Appl., 2 (2009), pp. 43-64. |
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Spectral Method for Three-Dimensional Nonlinear Klein-Gordon Equation by Using Generalized Laguerre and Spherical Harmonic Functions Xiao-Yong Zhang 1, Ben-Yu Guo 2*, Yu-Jian Jiao 2 1 Department of Mathematics, Shanghai Maritime University, Pudong Road, 1550, Shanghai, 200135, China.2 Department of Mathematics, Shanghai Normal University, Scientific Computing Key Laboratory of Shanghai Universities, Division of Computational Science of E-institute of Shanghai Universities, Guilin Road 100, Shanghai, 200234, China. Received 21 August 2007; Accepted (in revised version) 13 October 2008 Abstract In this paper, a generalized Laguerre-spherical harmonic spectral method is proposed for the Cauchy problem of three-dimensional nonlinear Klein-Gordon equation. The goal is to make the numerical solutions to preserve the same conservation as that for the exact solution. The stability and convergence of the proposed scheme are proved. Numerical results demonstrate the efficiency of this approach. We also establish some basic results on the generalized Laguerre-spherical harmonic orthogonal approximation, which play an important role in spectral methods for various problems defined on the whole space and unbounded domains with spherical geometry. AMS subject classifications: 65M70, 41A30, 81Q05Key words: Generalized Laguerre-spherical harmonic spectral method, Cauchy problem of nonlinear Klein-Gordon equation. *Corresponding author. Email: zhang-yong-mail@163.com (X.-Y. Zhang), byguo@shnu.edu.cn (B.-Y. Guo), yj-jiao@shnu.edu.cn (Y.-J. Jiao) |