Correction of high-order $L_k$ approximation for subdiffusion

TL;DR

This paper develops correction schemes for high-order $L_k$ convolution quadrature to improve the accuracy of subdiffusion equations with Caputo derivatives, achieving higher convergence rates even with nonsmooth data.

Contribution

The authors derive explicit correction algorithms for $L_k$ approximation of subdiffusion, enabling higher-order convergence on variable grids with practical implementation.

Findings

Achieved $(k+1- ext{alpha})$th-order convergence with correction schemes.

Validated theoretical results through numerical experiments with spectral methods.

Extended the applicability of $L_k$ approximation to nonsmooth data scenarios.

Abstract

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.