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

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 $u0 \in H^{{3}(\mathbb{R})$} and to a weak solution in $L^2(0,T;L2{\text{loc}}(\mathbb{R}))$ 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.