Chebyshev Finite-Spectral Method for 1D Boussinesq-Type Equations

Abstract

In this paper, an accurate Chebyshev finite-spectral method for one-dimensional (1D) Boussinesq-type equations is proposed. The method combines the advantages of both the finite-difference and spectral methods. The spatial derivatives in the governing equations can be calculated accurately in an efficient way, while some flexibility is allowed for treating irregular grids. The efficiency and accuracy of the proposed method are verified by successfully solving the problem of solitary wave propagation over a flat bottom where analytical solutions are available for comparison. A simple formula to calculate the wave celerity of the solitary wave propagation has also been derived. Finally, the applicability of the numerical method to periodic and random waves was validated by the simulation of nonlinear wave propagation over a bar where laboratory data are available for comparison. The method can be easily extended to treat 2D problems.