  In this paper, we study the stability and convergence of a fully discrete
finite difference scheme for the initial value problem associated with the
Korteweg-De Vries (KdV) equation. We employ the Crank-Nicolson method for
temporal discretization and establish that the scheme is $L^2$-conservative.
The convergence analysis reveals that utilizing inherent Kato's local smoothing
effect, the proposed scheme converges to a classical solution for sufficiently
regular initial data $u_0 \in H^{3}(\mathbb{R})$ and to a weak solution in
$L^2(0,T;L^2_{\text{loc}}(\mathbb{R}))$ for non-smooth initial data $u_0 \in
L^2(\mathbb{R})$. Optimal convergence rates in both time and space for the
devised scheme are derived. The theoretical results are justified through
several numerical illustrations.
