The subdiffusion equations with a Caputo fractional derivative of order $\alpha \in (0,1)$ arise in a wide variety of practical problems, which is describing the transport processes, in the force-free limit, slower than Brownian diffusion. In this work, we derive the correction schemes of the Lagrange interpolation with degree $k$ ($k\leq 6$) convolution quadrature, called $L_k$ approximation, for the subdiffusion, which are easy to implement on variable grids. The key step of designing correction algorithm is to calculate the explicit form of the coefficients of $L_k$ approximation by the polylogarithm function or Bose-Einstein integral. To construct a $\tau_8$ approximation of Bose-Einstein integral, the desired $(k+1-\alpha)$th-order convergence rate can be proved for the correction $L_k$ scheme with nonsmooth data, which is higher than $k$th-order BDF$k$ method in [Jin, Li, and Zhou, SIAM J. Sci. Comput., 39 (2017), A3129--A3152; Shi and Chen, J. Sci. Comput., (2020) 85:28]. The numerical experiments with spectral method are given to illustrate theoretical results.
翻译:以 $ ALpha 的 Caputo 分源衍生物 $ ALpha $ ( 0, 1) 的 子扩散方程, 产生于 各种各样的实际问题, 这些问题正在描述无力限制或Bose- Einstein 组成部分的运输过程, 速度慢于 Brownian 扩散。 在这项工作中, 我们用 $k (kleq 6美元) 的 递增方块来得出拉格兰特内插法的校正方案, 称为 $L_k美元近似值, 用于在变量网格上实施 。 设计校正算算法的关键步骤是计算以 $_k美元 的明显系数形式, 以 $_k$ 的, 以 pool- 无力限制 函数 或 Bose- Einstein 集成的 。 建造 $\ tau_ $ 8 美元 Bose- Einstein 集成, $ (k+1- 1 - leq- alpha) suffa, suffal supple grouplegilal data- data, $ liv.