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Chebyshev finite spectral method with extended moving grids

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Abstract

A Chebyshev finite spectral method on non-uniform meshes is proposed. An equidistribution scheme for two types of extended moving grids is used to generate grids. One type is designed to provide better resolution for the wave surface, and the other type is for highly variable gradients. The method has high-order accuracy because of the use of the Chebyshev polynomial as the basis function. The polynomial is used to interpolate the values between the two non-uniform meshes from a previous time step to the current time step. To attain high accuracy in the time discretization, the fourth-order Adams-Bashforth-Moulton predictor and corrector scheme is used. To avoid numerical oscillations caused by the dispersion term in the Korteweg-de Vries (KdV) equation, a numerical technique on non-uniform meshes is introduced. The proposed numerical scheme is validated by the applications to the Burgers equation (nonlinear convectiondiffusion problems) and the KdV equation (single solitary and 2-solitary wave problems), where analytical solutions are available for comparisons. Numerical results agree very well with the corresponding analytical solutions in all cases.

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Authors and Affiliations

  1. Department of Applied Mechanics and Engineering, Sun Yat-sen University, Guangzhou, 510275, P. R. China

    Jie-min Zhan  (詹杰民) & Zhi Dong  (董 志)

  2. Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hong Kong, P. R. China

    Yok-sheung Li  (李毓湘)

Authors
  1. Jie-min Zhan  (詹杰民)
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  2. Yok-sheung Li  (李毓湘)
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  3. Zhi Dong  (董 志)
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Corresponding author

Correspondence to Jie-min Zhan  (詹杰民).

Additional information

Communicated by Biao WANG

Project supported by the Research Grants Council of Hong Kong (No. 522007) and the National Marine Public Welfare Research Projects of China (No. 201005002)

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Zhan, Jm., Li, Ys. & Dong, Z. Chebyshev finite spectral method with extended moving grids. Appl. Math. Mech.-Engl. Ed. 32, 383–392 (2011). https://doi.org/10.1007/s10483-011-1423-6

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  • DOI: https://doi.org/10.1007/s10483-011-1423-6

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2010 Mathematics Subject Classification

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