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 $L^2(0,T;L^2_{loc}(\mathbb{R}))$ 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.

翻译：本文研究分数阶Korteweg-de Vries (KdV)方程（涉及分数阶拉普拉斯算子与非线性对流项）初值问题的全离散Crank-Nicolson Galerkin格式的收敛性。我们的证明依赖于Kato型局部光滑效应来估计近似解的局部化$H^{\alpha/2}$范数（其中$\alpha \in [1,2)$）。我们证明，当初值位于$L^2(\mathbb{R})$时，该格式在$L^2(0,T;L^2_{loc}(\mathbb{R}))$中强收敛于分数阶KdV方程的弱解。假设初值具有充分正则性，我们得到了数值格式的收敛阶。最后，通过多种数值实验验证了理论收敛阶。