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. Numerical illustrations are shown to demonstrate the efficiency of the scheme by obtaining an optimal order of convergence.

翻译：本文针对一维及二维空间中涉及指数$\alpha\in (1,2)$的分数阶拉普拉斯算子的分数阶Korteweg-de Vries方程，提出了一种局部间断Galerkin（LDG）方法。通过将分数阶拉普拉斯算子分解为一阶导数和分数阶积分，我们证明了结合适当界面通量与边界通量的半离散LDG格式具有$L^2$稳定性。通过分析线性对流项并利用相关估计，我们推导了广义非线性通量的误差估计，证明了收敛阶为$\mathcal{O}(h^{k+1/2})$。此外，稳定性与误差分析结果已推广至多维空间情形。数值算例通过获得最优收敛阶验证了该格式的有效性。