We present a fully discrete Crank-Nicolson Fourier-spectral-Galerkin (FSG) scheme for approximating solutions of the fractional Korteweg-de Vries (KdV) equation, which involves a fractional Laplacian with exponent $\alpha \in [1,2]$ and a small dispersion coefficient of order $\varepsilon^2$. The solution in the limit as $\varepsilon \to 0$ is known as the zero dispersion limit. We demonstrate that the semi-discrete FSG scheme conserves the first three integral invariants, thereby structure preserving, and that the fully discrete FSG scheme is $L^2$-conservative, ensuring stability. Using a compactness argument, we constructively prove the convergence of the approximate solution to the unique solution of the fractional KdV equation in $C([0,T]; H_p^{1+\alpha}(\mathbb{R}))$ for the periodic initial data in $H_p^{1+\alpha}(\mathbb{R})$. The devised scheme achieves spectral accuracy for the initial data in $H_p^r,$ $r \geq 1+\alpha$ and exponential accuracy for the analytic initial data. Additionally, we establish that the approximation of the zero dispersion limit obtained from the fully discrete FSG scheme converges to the solution of the Hopf equation in $L^2$ as $\varepsilon \to 0$, up to the gradient catastrophe time $t_c$. Beyond $t_c$, numerical investigations reveal that the approximation converges to the asymptotic solution, which is weakly described by the Whitham's averaged equation within the oscillatory zone for $\alpha = 2$. Numerical results are provided to demonstrate the convergence of the scheme and to validate the theoretical findings.

翻译：本文提出了一种全离散的Crank-Nicolson Fourier谱伽辽金（FSG）格式，用于逼近分数阶Korteweg-de Vries（KdV）方程的数值解。该方程包含指数$\alpha \in [1,2]$的分数阶拉普拉斯算子以及阶数为$\varepsilon^2$的小色散系数。当$\varepsilon \to 0$时的极限解被称为零色散极限。我们证明了半离散FSG格式能够保持前三个积分不变量，从而具有结构保持性；而全离散FSG格式具有$L^2$守恒性，确保了数值稳定性。利用紧性论证，我们构造性地证明了对于周期初值属于$H_p^{1+\alpha}(\mathbb{R})$的情形，近似解在$C([0,T]; H_p^{1+\alpha}(\mathbb{R}))$空间中收敛到分数阶KdV方程的唯一解。所设计的格式对属于$H_p^r$（$r \geq 1+\alpha$）的初值具有谱精度，对解析初值则具有指数精度。此外，我们证明了当$\varepsilon \to 0$时，由全离散FSG格式获得的零色散极限近似在梯度灾难时刻$t_c$之前于$L^2$范数下收敛到Hopf方程的解。超过$t_c$后，数值研究表明对于$\alpha = 2$的情形，近似解在振荡区域内弱收敛于由Whitham平均方程描述的渐近解。文中提供了数值结果以验证格式的收敛性并支持理论结论。