Analysis of a Chebyshev-type spectral method for the Korteweg-de Vries equation in the real line

Abstract

Many problems in science and engineering involve nonlinear PDEs which are posed on the real line (${\mathbb {R}}$). Particular examples include a number of important nonlinear wave equations. A rigorous numerical analysis of such problems is especially challenging due to the spatial discretization of the nonlinear operators involved in the models. Classically, such problems have been treated using Fourier pseudo-spectral methods by truncating the spatial domain and imposing periodic boundary conditions. Unfortunately, in many cases this approach may not be appropriate to adequately capture the dynamics of the problem on the spatial domain. In this paper, we consider a Chebyshev-type pseudo-spectral method based on the algebraically mapped Chebyshev basis defined in ${\mathbb {R}}$. The approximation properties of this basis are naturally described on the scale of weighted Bessel potential spaces and lead to optimal error estimates for the method. Based on these estimates an efficient spectral scheme for solving the nonlinear Korteweg-de Vries equation globally in ${\mathbb {R}}$ is proposed. The numerical simulations presented in the paper confirm our theoretical results.