Rational Chebyshev pseudospectral method for long-short wave equations

We consider the initial boundary value problem of the Long-Short wave equations on the whole line. Firstly, a three level linear fully discrete pseudospectral scheme are structured based on central difference in time and rational Chebyshev functions in space which are orthogonal in the L² space with weight 1. Secondly, the first-order differential matrix about rational Chebyshev functions is derived by the first-order differential matrix of Chebyshev polynomials, the entries of the matrix are just Chebyshev polynomials and Chebyshev Gauss collocation points. Thirdly, the numerical implementations are described and numerical results for the rational Chebyshev pseudospectral scheme are verified that a second-order accuracy in time and spectral accuracy in space.