  We study the convergence of higher order schemes for the Cauchy problem
associated to the KdV equation. More precisely, we design a Galerkin type
implicit scheme which has higher order accuracy in space and first order
accuracy in time. The convergence is established for initial data in L^2, and
we show that the scheme converges strongly in L^2(0,T; L^2_loc(\R)) to a weak
solution. Finally, the convergence is illustrated by several examples.
