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一维Burgers方程和KdV方程的广义有限谱方法
引用本文:詹杰民,李毓湘.一维Burgers方程和KdV方程的广义有限谱方法[J].应用数学和力学,2006,27(12):1431-1438.
作者姓名:詹杰民  李毓湘
作者单位:1. 中山大学,应用力学与工程系,广州,510275
2. 香港理工大学,土木及结构工程系
基金项目:国家自然科学基金;高等学校博士学科点专项科研项目
摘    要:给出了高精度的广义有限谱方法.为使方法在时间离散方面保持高精度,采用了Adars-Bashfoth预报格式和Adams-Mouhon校正格式,为了避免由Korteweg-de Vries(KdV)方程的弥散项引起的数值振荡,给出了两种数值稳定器.以Legendre多项式、Chebyshev多项式和Hermite多项式为基函数作为例子,给出的方法与具有分析解的Burgers方程的非线性对流扩散问题和KdV方程的单孤独波和双孤独波传播问题进行比较,结果非常吻合.

关 键 词:特殊函数  广义有限谱方法  非线性波
文章编号:1000-0887(2006)12-1431-08
收稿时间:2005-05-15
修稿时间:2006-06-30

Generalized Finite Spectral Method for 1D Burgers and KdV Equations
ZHAN Jie-min,LI Yok-sheung.Generalized Finite Spectral Method for 1D Burgers and KdV Equations[J].Applied Mathematics and Mechanics,2006,27(12):1431-1438.
Authors:ZHAN Jie-min  LI Yok-sheung
Abstract:A generalized finite spectral method is proposed. The method is of high_order accuracy. To attain high accuracy in time discretization, the fourth_order Adams_Bashforth_Moulton predictor and corrector scheme was used. To avoid numerical oscillations caused by the dispersion term in the KdV equation, two numerical techniques were introduced to improve the numerical stability. The Legendre, Chebyshev and Hermite polynomials were used as the basis functions. The proposed numerical scheme is validated by applications to the Burgers equation (nonlinear convection_diffusion problem) and KdV equation(single solitary and 2_solitary wave problems), where analytical solutions are available for comparison. Numerical results agree very well with the corresponding analytical solutions in all cases.
Keywords:special orthogonal functions  generalized finite spectral method  nonlinear wave
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