We propose a local discontinuous Galerkin (LDG) method for the fractional Korteweg-de Vries (KdV) equation, involving the fractional Laplacian with exponent $\alpha \in (1,2)$ in one and multiple space dimensions. By decomposing the fractional Laplacian into first-order derivatives and a fractional integral, we prove the $L^2$-stability of the semi-discrete LDG scheme incorporating suitable interface and boundary fluxes. We derive the optimal error estimate for linear flux and demonstrate an error estimate with an order of convergence $\mathcal{O}(h^{k+\frac{1}{2}})$ for general nonlinear flux utilizing the Gauss-Radau projections. Moreover, we extend the stability and error analysis to the multiple space dimensional case. Additionally, we discretize time using the Crank-Nicolson method to devise a fully discrete stable LDG scheme, and obtain a similar order error estimate as in the semi-discrete scheme. Numerical illustrations are provided to demonstrate the efficiency of the scheme, confirming an optimal order of convergence.

翻译：本文针对含指数$\alpha \in (1,2)$的分数阶拉普拉斯算子的单维及多维空间分数阶Korteweg-de Vries (KdV) 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和边界通量的半离散LDG格式的$L^2$稳定性。对于线性通量情形，我们推导了最优误差估计；对于一般非线性通量情形，利用Gauss-Radau投影，我们证明了收敛阶为$\mathcal{O}(h^{k+\frac{1}{2}})$的误差估计。此外，我们将稳定性和误差分析推广到多维空间情形。进一步地，我们采用Crank-Nicolson方法对时间进行离散，设计出全离散的稳定LDG格式，并得到了与半离散格式相似的误差估计阶。数值算例验证了该格式的有效性，并确认了其最优收敛阶。