Correction of high-order L_k approximation for subdiffusion
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The subdiffusion equations with a Caputo fractional derivative of order α∈ (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≤ 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 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 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.
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