A second-order accurate time-stepping scheme for solving a time-fractional Fokker–Planck equation of order $\alpha \in (0, 1)$, with a general driving force, is investigated. A stability bound for the semidiscrete solution is obtained for $\alpha \in (1/2,1)$ via a novel and concise approach. Our stability estimate is $\alpha $-robust in the sense that it remains valid in the limiting case where $\alpha $ approaches $1$ (when the model reduces to the classical Fokker–Planck equation), a limit that presents practical importance. Concerning the error analysis, we obtain an optimal second-order accurate estimate for $\alpha \in (1/2,1)$. A time-graded mesh is used to compensate for the singular behavior of the continuous solution near the origin. The time-stepping scheme scheme is associated with a standard spatial Galerkin finite element discretization to numerically support our theoretical contributions. We employ the resulting fully discrete computable numerical scheme to perform some numerical tests. These tests suggest that the imposed time-graded meshes assumption could be further relaxed, and we observe second-order accuracy even for the case $\alpha \in (0,1/2]$, that is, outside the range covered by the theory.

中文翻译：

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时间分数 Fokker-Planck 方程的二阶精确数值格式

研究了求解具有一般驱动力的$\alpha \in (0, 1)$ 阶时间分数Fokker-Planck 方程的二阶精确时间步长方案。对于$\alpha \in (1/2,1)$，通过一种新颖而简洁的方法获得了半离散解的稳定性界。我们的稳定性估计是 $\alpha $-robust 的，因为它在 $\alpha $ 接近 $1$ 的极限情况下仍然有效（当模型简化为经典的 Fokker-Planck 方程时），这个极限具有实际重要性。关于误差分析，我们获得了$\alpha \in (1/2,1)$ 的最优二阶准确估计。时间分级网格用于补偿原点附近连续解的奇异行为。时间步长方案方案与标准空间 Galerkin 有限元离散化相关联，以在数值上支持我们的理论贡献。我们采用得到的完全离散的可计算数值方案来执行一些数值测试。这些测试表明，施加的时间分级网格假设可以进一步放宽，即使对于 $\alpha \in (0,1/2]$ 的情况，我们也观察到二阶精度，也就是说，超出了理论。