  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 $L^{2}$ error estimates for the
resulting semidiscrete and the fully discrete approximations.
