A FAST HIGH ORDER METHOD FOR THE TIME-FRACTIONAL DIFFUSION EQUATION

In this paper, we present a fast (3 - alpha)-order numerical method for the Caputo fractional derivative based on the L2 scheme and the sum-of-exponentials (SOE) approximation to the convolution kernel involved in the fractional derivative. This work can be regarded as a continuation of previous works reported by one of the authors in [C.W. Lv and C.J. Xu, SIAM T. Sci. Comput., 38 (2016), pp. A2699-A2724], which constructed and analyzed a (3-alpha)-order L2 time stepping scheme for the time-fractional diffusion equation. It is now extended to take into account the fast SOE evaluation method, which allows us to reduce the storage and overall computational cost from O(N-T) and O(N-T(2)) for the L2 scheme to O(N-epsilon) and O(NTN epsilon), respectively, with N-T being the number of time steps and N-epsilon being the number of fast evaluation terms. The proposed method is then used for the time-fractional diffusion equation in bounded domains. The stability as well as the accuracy of the resulting scheme are rigorously analyzed. Several numerical examples are provided to validate the theoretical results and to demonstrate the efficiency of the proposed method. Finally, we extend the discussion to a graded mesh to make the scheme more suitable for problems having weakly singular solutions at the initial time.