Correction of High-Order L_k Approximation for Subdiffusion

The subdiffusion equations with a Caputo fractional derivative of order α∈ (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 ( k≤ 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 τ _8 approximation of Bose-Einstein integral, the desired (k+1-α ) 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.