Convergence of a fully discrete finite difference scheme for the Korteweg-de Vries equation

We prove convergence of a fully discrete finite difference scheme for the Korteweg–de Vries equation. Both the decaying case on the full line and the periodic case are considered. If the initial data $u{\mathrm{\mid }}_{t=0}=u_0$ is of high regularity, $u_0\in H^3(\mathbb{R})$, the scheme is shown to converge to a classical solution, and if the regularity of the initial data is smaller, $u_0\in L^2(\mathbb{R})$, then the scheme converges strongly in $L^2(0,T;L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2(\mathbb{R}))$ to a weak solution.