A direct O(Nlog2N) finite difference method for fractional diffusion equations

  • Authors:

  • Hong Wang;Kaixin Wang;Treena Sircar

  • Affiliations:

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

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2010

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Abstract

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that can not be modeled accurately by the second-order diffusion equations. Because of the nonlocal property of fractional differential operators, the numerical methods have full coefficient matrices which require storage of O(N^2) and computational cost of O(N^3) where N is the number of grid points. In this paper we develop a fast finite difference method for fractional diffusion equations, which only requires storage of O(N) and computational cost of O(Nlog^2N) while retaining the same accuracy and approximation property as the regular finite difference method. Numerical experiments are presented to show the utility of the method.