Correction of High-Order Lk Approximation for Subdiffusion

Abstract

The subdiffusion equations with a Caputo fractional derivative of order $\alpha \in (0, 1)$ arise in a wide variety of practical problems, which describe 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 $k ( \leq 6)$ convolution quadrature, called $L_{k}$ approximation, for the subdiffusion. 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 kth-order BDFk 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.