We propose a local discontinuous Galerkin (LDG) method for fractional Korteweg-de Vries equation involving the fractional Laplacian with exponent $\alpha\in (1,2)$ in one and two space dimensions. By decomposing the fractional Laplacian into a first order derivative and a fractional integral, we prove $L^2$-stability of the semi-discrete LDG scheme incorporating suitable interface and boundary fluxes. We analyze the error estimate by considering linear convection term and utilizing the estimate, we derive the error estimate for general nonlinear flux and demonstrate an order of convergence $\mathcal{O}(h^{k+1/2})$. Moreover, the stability and error analysis have been extended to multiple space dimensional case. Additionally, we devise a fully discrete LDG scheme using the four-stage fourth-order Runge-Kutta method. We prove that the scheme is strongly stable under an appropriate time step constraint by establishing a \emph{three-step strong stability} estimate. Numerical illustrations are shown to demonstrate the efficiency of the scheme by obtaining an optimal order of convergence.

翻译：我们针对含分数阶Laplacian算子（指数$\alpha\in (1,2)$）的一维和二维分数阶Korteweg-de Vries方程，提出了一种局部间断Galerkin（LDG）方法。通过将分数阶Laplacian分解为一阶导数与分数阶积分，我们证明了结合适当界面和边界通量的半离散LDG格式的$L^2$稳定性。在考虑线性对流项的基础上分析误差估计，进而推导出一般非线性通量下的误差估计，并得到收敛阶$\mathcal{O}(h^{k+1/2})$。此外，该稳定性与误差分析被推广至多维空间情形。进一步地，我们采用四阶四步Runge-Kutta方法设计了全离散LDG格式，并通过建立三步强稳定性估计，证明了该格式在适当时间步长约束下的强稳定性。数值实验通过获得最优收敛阶展示了该格式的有效性。