Single and multidomain Chebyshev collocation methods for the Korteweg-de Vries equation

We propose spectral Chebyshev collocation algorithms for the approximation of the initial and boundary value problem for the Korteweg-de Vries equation. Both single and multidomain approaches are discussed. Different methods for the treatment of the boundary conditions are considered. The numerical analysis of the eigenvalues' behaviour of the spectral differentiation operators involved in the approximation suggests appropriate finite difference methods for time-marching. Several numerical experiments have been performed, which prove spectral convergence and stability of the proposed schemes.