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In this paper, we study the stability and convergence of a fully discretefinite difference scheme for the initial value problem associated with theKorteweg-De Vries (KdV) equation. We employ the Crank-Nicolson method fortemporal discretization and establish that the scheme is $L^2$-conservative.The convergence analysis reveals that utilizing inherent Kato's local smoothingeffect, the proposed scheme converges to a classical solution for sufficientlyregular 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 $u_0 \inL^2(\mathbb{R})$. Optimal convergence rates in both time and space for thedevised scheme are derived. The theoretical results are justified throughseveral numerical illustrations.

Convergence of a conservative Crank-Nicolson finite difference scheme for the KdV equation with smooth and non-smooth initial data