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We consider the periodic initial-value problem for the Korteweg-de Vriesequation that we discretize in space by a spectral Fourier-Galerkin method andin time by an implicit, high order, Runge-Kutta scheme of composition typebased on the implicit midpoint rule. We prove $L^{2}$ error estimates for theresulting semidiscrete and the fully discrete approximations.

Notes on a high order fully discrete scheme for the Korteweg-de vries equation with a time-stepping procedure of Runge-Kutta-composition type