A superfast-preconditioned iterative method for steady-state space-fractional diffusion equations

  • Authors:

  • Hong Wang;Ning Du

  • Affiliations:

  • School of Mathematics, Shandong University, Jinan 250100, China and Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA;School of Mathematics, Shandong University, Jinan 250100, China

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2013

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Abstract

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the classical second-order diffusion equations. Because of the nonlocal property of fractional differential operators, the corresponding numerical methods have full coefficient matrices which require storage of O(N^2) and computational cost of O(N^3) for a problem of size N. We develop a superfast-preconditioned conjugate gradient squared method for the efficient solution of steady-state space-fractional diffusion equations. The method reduces the computational work from O(N^2) to O(NlogN) per iteration and reduces the memory requirement from O(N^2) to O(N). Furthermore, the method significantly reduces the number of iterations to be mesh size independent. Preliminary numerical experiments for a one-dimensional steady-state diffusion equation with 2^1^3 nodes show that the fast method reduces the overall CPU time from 3h and 27min for the Gaussian elimination to 0.39s for the fast method while retaining the accuracy of Gaussian elimination. In contrast, the regular conjugate gradient squared method diverges after 2days of simulations and more than 20,000 iterations.