  We prove strong convergence of a semi-discrete finite difference method for
the KdV and modified KdV equations. We extend existing results to non-smooth
data (namely, in $L^2$), without size restrictions. Our approach uses a fourth
order (in space) stabilization term and a special conservative discretization
of the nonlinear term. Convergence follows from a smoothing effect and energy
estimates. We illustrate our results with numerical experiments, including a
numerical investigation of an open problem related to uniqueness posed by Y.
Tsutsumi.
