A linearized implicit pseudo-spectral method for certain non-linear water wave equations

An efficient numerical method is developed for the numerical solution of non-linear wave equations typified by the third- and fifth-order Korteweg-de Vries equations and their generalizations. The method developed uses a pseudo-spectral (Fourier transform) treatment of the space dependence together with a linearized implicit scheme in time. An important advantage to be gained from the use of this method over the pseudo-spectral scheme proposed by Fornberg and Whitham (a Fourier transform treatment of the space variable together with a leap-frog scheme in time) which is conditionally stable, is the ability to vary the mesh length, thereby reducing the computational time. Using a linearized stability analysis, it is shown that the proposed method is unconditionally stable. The method presented here is for the Korteweg-de Vries equations and their generalized forms, but it can be implemented to a broad class of non-linear wave equations (equation (1)), with obvious changes in the various formulae. To illustrate the application of this method, numerical results portraying a single soliton solution and the collision of two solitons are reported for the third- and fifth-order Korteweg-de Vries equations.