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.

翻译：本文研究Korteweg-De Vries（KdV）方程初值问题全离散有限差分格式的稳定性与收敛性。采用Crank-Nicolson方法进行时间离散，并证明该格式具有$L^2$-守恒性。收敛性分析表明，利用Kato固有局部光滑效应，对于充分正则的初值$u_0 \in H^{3}(\mathbb{R})$，所提格式收敛至经典解；对于非光滑初值$u_0 \in L^2(\mathbb{R})$，则收敛至$L^2(0,T;L^2_{\text{loc}}(\mathbb{R}))$中的弱解。推导给出了该格式在时间和空间上的最优收敛速率。通过多个数值算例验证了理论结果。