Korteweg-de Vries方程的时空谱配置方法

doi:10.12288/szjs.s2020-0692

数值计算与计算机应用 ›› 2021, Vol. 42 ›› Issue (4): 351-360.DOI: 10.12288/szjs.s2020-0692

Korteweg-de Vries方程的时空谱配置方法

马亚楠, 王天军, 李冰冰

河南科技大学数学与统计学院, 洛阳 471003

收稿日期:2020-08-21 出版日期:2021-12-15 发布日期:2021-12-07
基金资助:
国家自然科学基金（11371123，11171227）和河南省自然科学基金（202300410156）资助.

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马亚楠, 王天军, 李冰冰. Korteweg-de Vries方程的时空谱配置方法[J]. 数值计算与计算机应用, 2021, 42(4): 351-360.

Ma Yanan, Wang Tianjun, Li Bingbing. SPACE-TIME SPECTRAL COLLOCATION METHOD FOR KORTEWEG-DE VRIES EQUATION[J]. Journal on Numerica Methods and Computer Applications, 2021, 42(4): 351-360.

SPACE-TIME SPECTRAL COLLOCATION METHOD FOR KORTEWEG-DE VRIES EQUATION

Ma Yanan, Wang Tianjun, Li Bingbing

School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471003, China

Received:2020-08-21 Online:2021-12-15 Published:2021-12-07

针对全直线上的KdV方程构造了时空全离散Legendre-Hermite谱配置格式，也就是在时间方向上用Legendre-Gauss-Lobatto节点为配置点，在空间方向上用Hermite-Gauss节点为配置点，构造得到一个非线性矩阵方程，将其化为非线性方程组，利用通常的不动点迭代求解，数值实验表明这种方法的有效性.

关键词： Korteweg-de Vries方程, 时空谱配置方法, 全直线

A Legendre-Hermite spectral collocation scheme is constructed for Korteweg-de Vries (KdV) equation on the whole line by using the Legendre collocation method in time, and the Hermite spectral collocation method in space, which is a nonlinear matrix equation that is changed to a nonlinear systems and can be solved by the usual fixed point iteration Numerical results demonstrate the efficiency of the method.

Key words: Korteweg-de Vries equation, space-time spectral collocation method, the whole line

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[1] Zabusky N, Galvin C. Shallow water waves, the Korteveg-de Vries equation and solitons[J]. J. Fluid Mech., 1971, 47:811-824.
[2] Bernardi C, Maday Y. Spectral methods[M]. in Handbook of Numerical Analysis, 209-486, ed. by Ciarlet P G and Lions J L, Elsevier, Amsterdam, 1997.
[3] Canuto C, Hussaini M Y, Quarteroni A, Zang T A. Spectral Methods[M]. Springer-Verlag, Berlin, 2006.
[4] Guo B Y. Spectral Methods and Their Applications[M]. World Scientific, Singapore, 1998.
[5] Shen J, Tang T, Wang L L. Spectral Methods:Algorithms, Analysis and Applications[M]. Springer Series in Computational Mathematics, Vol. 41, Springer-Verlag, Berlin, Heidelberg, 2011.
[6] Bar-Yoseph P, Moses E, Zrahia U, Yarin A L. Space-time spectral element methods for onedimensional nonlinear advection-diffusion problems[J] J. Comput. Phys., 1995, 119:62-74
[7] Zrahia U, Bar-Yoseph P. Space-time spectral element method for solution of second-order hyperbolic equations[J] Comput. Methods Appl. Mech. Engrg., 1994, 116:135-146
[8] Tang J G, Ma H P Single and multi-interval Legendre spectral methods in time for parabolic equations[J] Numer. Methods Partial Differential Equations, 2005, 22:1007-1034
[9] Tang J G Ma H P A Legendre spectral method in time for first-order hyperbolic equations[J] Appl. Numer. Math. 2007, 57:1-11
[10] Liu L., Ma H. P. Space-time Spectral method for parabolic inverse problem with unknown control parameter[J]. J. Numer. Meth. Comput. Appl., 2020, 41(1):19-26.
[11] Cheng B. On the numerical simulation of Korteveg-de Vries equation with the Petrov-Galerkin finite element method[J]. J. Numer. Meth. & Comput. Appl., 1992, 1:73-80
[12] Jia H L, Wang Z Q. Chebyshev-Hermite spectral collocation method for KdV equations[J]. Commun. Appl. Math. Comput., 2013, 27(1):1-8.
[13] Bernard Bialecki, Andreas Karageorghis. Legendre Gauss spectral collocation for the Helmholtz equation on a rectangle[J]. Numer. Algori., 2004, 36:203-227.
[14] Wang Z Q Guo B Y Legendre-Gauss-Radau collocation method for solving initial value problems of first order ordinary differential equations[J] J. Sci. Comput. 201252226-255
[15] Wang T J, Yang S. Hermite spectral collocation method for nonlinear Fokker-Planck equation[J]. J. of Anhui University of Technology Natural Science), 2012, 29(4):381-384.
[16] Shen J, Tang T. Spectral and High-Order Methods with Applications[M]. Science Press, Beijing, 2006.
[17] Cui Y.F., Mao D. K. A Difference Scheme Satisfying Two Conservation Laws for KdV Equation[J]. Commun. Appl. Math. Comput., 2005, 19(2):15-22.

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Korteweg-de Vries方程的时空谱配置方法

SPACE-TIME SPECTRAL COLLOCATION METHOD FOR KORTEWEG-DE VRIES EQUATION

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