Convergence of a Conservative Crank–Nicolson Finite Difference Scheme for the KdV Equation

ABSTRACT In this paper, we study the stability and convergence of a conservative Crank–Nicolson finite difference scheme applied to the Korteweg–de Vries (KdV) equation endowed with initial data. We design a three‐point average scheme associated with the convective term, and the dispersion term is discretized by certain discrete operators along with the Crank–Nicolson scheme for the temporal discretization to establish that the proposed scheme is 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 and to a weak solution in for nonsmooth initial data . Optimal convergence rates in both space and time for the devised scheme are derived. The theoretical results are justified through several numerical illustrations.