Pseudo-Spectral Methods for Linear Advection and Dispersive Problems

The most important feature of numerical methods based on a spectral decomposition is the best convergence rate (even infinite for infinitely regular functions) with respect to all other methods used in dealing with the solution to most of the differential equations. However this is true under the mandatory condition that at each time step of the evolving numerical solution no discontinuity occurs, either advected from the initial condition or self-generated (shock wave) by the non-linearity of the problem. In the first part of this paper we will point out that also by using any second or higher order pseudo-spectral method applied to the linear advection equation, one can experience the appearance in the numerical solution of the celebrated Gibbs phenomenon, located at the discontinuity points of the initial condition. As matter of fact in order to avoid such a drawback, the only applicable methods are the first order ones. In particular for stability reasons, we chose to consider the implicit Euler method and experienced that, even the Richardson’s extrapolation is not able in increasing the accuracy without falling into the same problem. As a partial remedy, we propose a novel way to improve the accuracy of the implicit Euler first order pseudo-spectral method by reducing the coefficient of its truncation error leading term via a time one-step extrapolation-like technique. The second part of the paper is instead devoted to show how for dispersive differential problems the use of pseudo-spectral methods represents a very powerful numerical approach in finding out the notorious solitons dynamics. In particular we will deal with the celebrated KdV equation in 1D and its generalization in 2D.