This paper develops a fully discrete Fourier spectral Galerkin (FSG) method for the fractional Zakharov-Kuznetsov (fZK) equation posed on a two-dimensional periodic domain. The equation generalizes the classical ZK model to incorporate nonlocal dispersion through a fractional Laplacian of order $\alpha \in [1,2]$. We first propose a semi-discrete FSG scheme in space that preserves the discrete analogs of mass, momentum, and energy. The existence and uniqueness of semi-discrete solutions are established. Using compactness arguments, we prove the uniform convergence of the semi-discrete approximations to the unique solution of the fZK equation for the periodic initial data in $H^{1+\alpha}_{\mathrm{per}}(\Omega)$. The method achieves spectral convergence of order $\mathcal{O}(N^{-r})$ for initial data in $H^r_{\mathrm{per}}$ with $r \geq \alpha+1$, and exponential convergence for analytic solutions utilizing a modified projection. An efficient integrating-factor Runge-Kutta time discretization is designed to handle the stiff fractional term, and an error analysis is presented. Numerical experiments validate the theoretical results and demonstrate the method's effectiveness across various fractional orders.

翻译：本文针对定义在二维周期区域上的分数阶Zakharov-Kuznetsov（fZK）方程，提出了一种全离散傅里叶谱伽辽金（FSG）方法。该方程通过引入阶数 $\alpha \in [1,2]$ 的分数阶拉普拉斯算子，将经典ZK模型推广至包含非局部色散效应。我们首先在空间上提出一种半离散FSG格式，该格式保持了离散形式的质量、动量和能量守恒律。我们建立了半离散解的存在性与唯一性。利用紧致性论证，我们证明了对于周期初始数据属于 $H^{1+\alpha}_{\mathrm{per}}(\Omega)$ 的情形，半离散近似解一致收敛于fZK方程的唯一解。对于属于 $H^r_{\mathrm{per}}$（$r \geq \alpha+1$）的初始数据，该方法实现了 $\mathcal{O}(N^{-r})$ 阶的谱收敛；对于解析解，通过采用修正投影可达到指数收敛。为处理刚性的分数阶项，我们设计了一种高效的积分因子龙格-库塔时间离散格式，并给出了误差分析。数值实验验证了理论结果，并展示了该方法在不同分数阶数下的有效性。