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A fast fractional block-centered finite difference method for two-sided space-fractional diffusion equations on general nonuniform grids

Meijie Kong, Hongfei Fu

Abstract

In this paper, a two-sided variable-coefficient space-fractional diffusion equation with fractional Neumann boundary condition is considered. To conquer the weak singularity caused by nonlocal space-fractional differential operators, a fractional block-centered finite difference (BCFD) method on general nonuniform grids is proposed. However, this discretization still results in an unstructured dense coefficient matrix with huge memory requirement and computational complexity. To address this issue, a fast version fractional BCFD algorithm by employing the well-known sum-of-exponentials (SOE) approximation technique is also proposed. Based upon the Krylov subspace iterative methods, fast matrix-vector multiplications of the resulting coefficient matrices with any vector are developed, in which they can be implemented in only $\mathcal{O}(MN_{exp})$ operations per iteration without losing any accuracy compared to the direct solvers, where $N_{exp}\ll M$ is the number of exponentials in the SOE approximation. Moreover, the coefficient matrices do not necessarily need to be generated explicitly, while they can be stored in $\mathcal{O}(MN_{exp})$ memory by only storing some coefficient vectors. Numerical experiments are provided to demonstrate the efficiency and accuracy of the method.

A fast fractional block-centered finite difference method for two-sided space-fractional diffusion equations on general nonuniform grids

Abstract

In this paper, a two-sided variable-coefficient space-fractional diffusion equation with fractional Neumann boundary condition is considered. To conquer the weak singularity caused by nonlocal space-fractional differential operators, a fractional block-centered finite difference (BCFD) method on general nonuniform grids is proposed. However, this discretization still results in an unstructured dense coefficient matrix with huge memory requirement and computational complexity. To address this issue, a fast version fractional BCFD algorithm by employing the well-known sum-of-exponentials (SOE) approximation technique is also proposed. Based upon the Krylov subspace iterative methods, fast matrix-vector multiplications of the resulting coefficient matrices with any vector are developed, in which they can be implemented in only operations per iteration without losing any accuracy compared to the direct solvers, where is the number of exponentials in the SOE approximation. Moreover, the coefficient matrices do not necessarily need to be generated explicitly, while they can be stored in memory by only storing some coefficient vectors. Numerical experiments are provided to demonstrate the efficiency and accuracy of the method.
Paper Structure (3 sections, 2 theorems, 63 equations)

This paper contains 3 sections, 2 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. A fractional CN-BCFD method on general nonuniform grids
  3. A fast fractional CN-BCFD method on general nonuniform grids

Key Result

lemma thmcounterlemma

For given $\alpha\in(1,2)$, an absolute tolerance error $\epsilon$, a cut-off restriction $\Delta x>0$ and a given position $X>0$, there exists a positive integer $N_{exp}$, positive quadrature points $\left\lbrace \lambda_s\right\rbrace_{s=1}^{N_{exp}}$ and corresponding positive weights $\left\lbr where the number of exponentials satisfies

Theorems & Definitions (6)

  • remark thmcounterremark
  • remark thmcounterremark
  • remark thmcounterremark
  • lemma thmcounterlemma: JZZZ17
  • remark thmcounterremark
  • lemma thmcounterlemma