Generalized Finite Spectral Method for 1D Burgers and KdV Equations
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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Adams-Bashforth 预报格式和Adams-Moulton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了比较,结果非常吻合.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.
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