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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Communicated by WANG Biao
Project supported by the National Natural Science Foundation of China (No.10272118), the Hong Kong Polytechnic University Research Grant (No.A-PE28) and the Research Fund for the Doctoral Program of Ministry of Education of China (No.20020558013)
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Zhan, Jm., Li, Ys. Generalized finite spectral method for 1D Burgers and KdV equations. Appl Math Mech 27, 1635–1643 (2006). https://doi.org/10.1007/s10483-006-1206-z
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DOI: https://doi.org/10.1007/s10483-006-1206-z