Stability and Convergence analysis of a Crank-Nicolson Galerkin scheme for the fractional Korteweg-de Vries equation

In this paper we study the convergence of a fully discrete Crank-Nicolson Galerkin scheme for the initial value problem associated with the fractional Korteweg-de Vries (KdV) equation, which involves the fractional Laplacian and non-linear convection terms. Our proof relies on the Kato type local smoothing effect to estimate the localized $H^{{\alpha/2}$-norm} of the approximated solution, where $\alpha \in [1,2)$. We demonstrate that the scheme converges strongly in $L^{2(0,T;L2_{loc}(\mathbb{R}))$} to a weak solution of the fractional KdV equation provided the initial data in $L^{2(\mathbb{R})$.} Assuming the initial data is sufficiently regular, we obtain the rate of convergence for the numerical scheme. Finally, the theoretical convergence rates are justified numerically through various numerical illustrations.