A high-order fully discrete scheme for the Korteweg–de Vries equation with a time-stepping procedure of Runge–Kutta-composition type

Abstract

We consider the periodic initial-value problem for the Korteweg–de Vries equation that we discretize in space by a spectral Fourier–Galerkin method and in time by an implicit, high-order, Runge–Kutta scheme of composition type based on the implicit midpoint rule. We prove L2 error estimates for the resulting semidiscrete and the fully discrete approximations. Some numerical experiments illustrate the results.